1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
2873
2874
2875
2876
2877
2878
2879
2880
2881
2882
2883
2884
2885
2886
2887
2888
2889
2890
2891
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908
2909
2910
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
3037
3038
3039
3040
3041
3042
3043
3044
3045
3046
3047
3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
3077
3078
3079
3080
3081
3082
3083
3084
3085
3086
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
3108
3109
3110
3111
3112
3113
3114
3115
3116
3117
3118
3119
3120
3121
3122
3123
3124
3125
3126
3127
3128
3129
3130
3131
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161
3162
3163
3164
3165
3166
3167
3168
3169
3170
3171
3172
3173
3174
3175
3176
3177
3178
3179
3180
3181
3182
3183
3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
3204
3205
3206
3207
3208
3209
3210
3211
3212
3213
3214
3215
3216
3217
3218
3219
3220
3221
3222
3223
3224
3225
3226
3227
3228
3229
3230
3231
3232
3233
3234
3235
3236
3237
3238
3239
3240
3241
3242
3243
3244
3245
3246
3247
3248
3249
3250
3251
3252
3253
3254
3255
3256
3257
3258
3259
3260
3261
3262
3263
3264
3265
3266
3267
3268
3269
3270
3271
3272
3273
3274
3275
3276
3277
3278
3279
3280
3281
3282
3283
3284
3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
3300
3301
3302
3303
3304
3305
3306
3307
3308
3309
3310
3311
3312
3313
3314
3315
3316
3317
3318
3319
3320
3321
3322
3323
3324
3325
3326
3327
3328
3329
3330
3331
3332
3333
3334
3335
3336
3337
3338
3339
3340
3341
3342
3343
3344
3345
3346
3347
3348
3349
3350
3351
3352
3353
3354
3355
3356
3357
3358
3359
3360
3361
3362
3363
3364
3365
3366
3367
3368
3369
3370
3371
3372
3373
3374
3375
3376
3377
3378
3379
3380
3381
3382
3383
3384
3385
3386
3387
3388
3389
3390
3391
3392
3393
3394
3395
3396
3397
3398
3399
3400
3401
3402
3403
3404
3405
3406
3407
3408
3409
3410
3411
3412
3413
3414
3415
3416
3417
3418
3419
3420
3421
3422
3423
3424
3425
3426
3427
3428
3429
3430
3431
3432
3433
3434
3435
3436
3437
3438
3439
3440
3441
3442
3443
3444
3445
3446
3447
3448
3449
3450
3451
3452
3453
3454
3455
3456
3457
3458
3459
3460
3461
3462
3463
3464
3465
3466
3467
3468
3469
3470
3471
3472
3473
3474
3475
3476
3477
3478
3479
3480
3481
3482
3483
3484
3485
3486
3487
3488
3489
3490
3491
3492
3493
3494
3495
3496
3497
3498
3499
3500
3501
3502
3503
3504
3505
3506
3507
3508
3509
3510
3511
3512
3513
3514
3515
3516
3517
3518
3519
3520
3521
3522
3523
3524
3525
3526
3527
3528
3529
3530
3531
3532
3533
3534
3535
3536
3537
3538
3539
3540
3541
3542
3543
3544
3545
3546
3547
3548
3549
3550
3551
3552
3553
3554
3555
3556
3557
3558
3559
3560
3561
3562
3563
3564
3565
3566
3567
3568
3569
3570
3571
3572
3573
3574
3575
3576
3577
3578
3579
3580
3581
3582
3583
3584
3585
3586
3587
3588
3589
3590
3591
3592
3593
3594
3595
3596
3597
3598
3599
3600
3601
3602
3603
3604
3605
3606
3607
3608
3609
3610
3611
3612
3613
3614
3615
3616
3617
3618
3619
3620
3621
3622
3623
3624
3625
3626
3627
3628
3629
3630
3631
3632
3633
3634
3635
3636
3637
3638
3639
3640
3641
3642
3643
3644
3645
3646
3647
3648
3649
3650
3651
3652
3653
3654
3655
3656
3657
3658
3659
3660
3661
3662
3663
3664
3665
3666
3667
3668
3669
3670
3671
3672
3673
3674
3675
3676
3677
3678
3679
3680
3681
3682
3683
3684
3685
3686
3687
3688
3689
3690
3691
3692
3693
3694
3695
3696
3697
3698
3699
3700
3701
3702
3703
3704
3705
3706
3707
3708
3709
3710
3711
3712
3713
3714
3715
3716
3717
3718
3719
3720
3721
3722
3723
3724
3725
3726
3727
3728
3729
3730
3731
3732
3733
3734
3735
3736
3737
3738
3739
3740
3741
3742
3743
3744
3745
3746
3747
3748
3749
3750
3751
3752
3753
3754
3755
3756
3757
3758
3759
3760
3761
3762
3763
3764
3765
3766
3767
3768
3769
3770
3771
3772
3773
3774
3775
3776
3777
3778
3779
3780
3781
3782
3783
3784
3785
3786
3787
3788
3789
3790
3791
3792
3793
3794
3795
3796
3797
3798
3799
3800
3801
3802
3803
3804
3805
3806
3807
3808
3809
3810
3811
3812
3813
3814
3815
3816
3817
3818
3819
3820
3821
3822
3823
3824
3825
3826
3827
3828
3829
3830
3831
3832
3833
3834
3835
3836
3837
3838
3839
3840
3841
3842
3843
3844
3845
3846
3847
3848
3849
3850
3851
3852
3853
3854
3855
3856
3857
3858
3859
3860
3861
3862
3863
3864
3865
3866
3867
3868
3869
3870
3871
3872
3873
3874
3875
3876
3877
3878
3879
3880
3881
3882
3883
3884
3885
3886
3887
3888
3889
3890
3891
3892
3893
3894
3895
3896
3897
3898
3899
3900
3901
3902
3903
3904
3905
3906
3907
3908
3909
3910
3911
3912
3913
3914
3915
3916
3917
3918
3919
3920
3921
3922
3923
3924
3925
3926
3927
3928
3929
3930
3931
3932
3933
3934
3935
3936
3937
3938
3939
3940
3941
3942
3943
3944
3945
3946
3947
3948
3949
3950
3951
3952
3953
3954
3955
3956
3957
3958
3959
3960
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<!-- This file copyright Persistence of Vision Raytracer Pty. Ltd. 2009-2011 -->
<html lang="en">
<head>
<meta http-equiv=Content-Type content="text/html; charset=windows-1252">
<title>Reference Section 5</title>
<link rel="StyleSheet" href="povray.css" type="text/css">
<link rel="shortcut icon" href="favicon.ico">
<!-- NOTE: In order to help users find information about POV-Ray using web -->
<!-- search engines, we ask that you *not* let them index documentation -->
<!-- mirrors because effectively, when searching, users will get hundreds of -->
<!-- results containing the same information! For this reason, these meta tags -->
<!-- below disable archiving of this page by search engines. -->
<meta name="robots" content="noarchive">
<meta http-equiv="Pragma" content="no-cache">
<meta http-equiv="expires" content="0">
</head>
<body>
<div class="Page">
<!-- NavPanel Begin -->
<div class="NavPanel">
<table class="NavTable">
<tr>
<td class="FixedPanelHeading"><a title="3.5" href="#r3_5">The Object Zoo</a></td>
</tr>
<tr>
<td><div class="divh2"><strong><a title="3.5.1" href="#r3_5_1">Object</a></strong></div></td>
</tr>
<tr>
<td><div class="divh3"><a title="3.5.1.1" href="#r3_5_1_1">Finite Solid Primitives</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.1" href="#r3_5_1_1_1">Blob</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.2" href="#r3_5_1_1_2">Box</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.3" href="#r3_5_1_1_3">Cone</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.4" href="#r3_5_1_1_4">Cylinder</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.5" href="#r3_5_1_1_5">Height Field</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.6" href="#r3_5_1_1_6">Isosurface</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.7" href="#r3_5_1_1_7">Julia Fractal</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.8" href="#r3_5_1_1_8">Lathe</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.9" href="#r3_5_1_1_9">Lemon</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.10" href="#r3_5_1_1_10">Ovus</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.11" href="#r3_5_1_1_11">Parametric</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.12" href="#r3_5_1_1_12">Prism</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.13" href="#r3_5_1_1_13">Sphere</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.14" href="#r3_5_1_1_14">Sphere Sweep</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.15" href="#r3_5_1_1_15">Superquadric Ellipsoid</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.16" href="#r3_5_1_1_16">Surface of Revolution</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.17" href="#r3_5_1_1_17">Text</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.1.18" href="#r3_5_1_1_18">Torus</a></div></td>
</tr>
<tr>
<td><div class="divh3"><a title="3.5.1.2" href="#r3_5_1_2">Finite Patch Primitives</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.2.1" href="#r3_5_1_2_1">Bicubic Patch</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.2.2" href="#r3_5_1_2_2">Disc</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.2.3" href="#r3_5_1_2_3">Mesh</a></div></td>
</tr>
<tr>
<td><div class="divh5"><a title="3.5.1.2.3.1" href="#r3_5_1_2_3_1">Solid Mesh</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.2.4" href="#r3_5_1_2_4">Mesh2</a></div></td>
</tr>
<tr>
<td><div class="divh5"><a title="3.5.1.2.4.1" href="#r3_5_1_2_4_1">Smooth and Flat triangles in the same mesh</a></div></td>
</tr>
<tr>
<td><div class="divh5"><a title="3.5.1.2.4.2" href="#r3_5_1_2_4_2">Mesh Triangle Textures</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.2.5" href="#r3_5_1_2_5">Polygon</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.2.6" href="#r3_5_1_2_6">Triangle</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.2.7" href="#r3_5_1_2_7">Smooth Triangle</a></div></td>
</tr>
<tr>
<td><div class="divh3"><a title="3.5.1.3" href="#r3_5_1_3">Infinite Solid Primitives</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.3.1" href="#r3_5_1_3_1">Plane</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.3.2" href="#r3_5_1_3_2">Poly</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.3.3" href="#r3_5_1_3_3">Cubic</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.3.4" href="#r3_5_1_3_4">Quartic</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.3.5" href="#r3_5_1_3_5">Polynomial</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.3.6" href="#r3_5_1_3_6">Quadric</a></div></td>
</tr>
<tr>
<td><div class="divh3"><a title="3.5.1.4" href="#r3_5_1_4">Constructive Solid Geometry</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.4.1" href="#r3_5_1_4_1">Inside and Outside</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.4.2" href="#r3_5_1_4_2">Union</a></div></td>
</tr>
<tr>
<td><div class="divh5"><a title="3.5.1.4.2.1" href="#r3_5_1_4_2_1">Split_Union</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.4.3" href="#r3_5_1_4_3">Intersection</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.4.4" href="#r3_5_1_4_4">Difference</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.4.5" href="#r3_5_1_4_5">Merge</a></div></td>
</tr>
<tr>
<td><div class="divh3"><a title="3.5.1.5" href="#r3_5_1_5">Object Modifiers</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.5.1" href="#r3_5_1_5_1">Clipped By Object Modifier</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.5.2" href="#r3_5_1_5_2">Bounded By Object Modifier</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.5.3" href="#r3_5_1_5_3">Material</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.5.4" href="#r3_5_1_5_4">Hollow Object Modifier</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.5.5" href="#r3_5_1_5_5">Inverse Object Modifier</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.5.6" href="#r3_5_1_5_6">No Shadow Object Modifier</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.5.7" href="#r3_5_1_5_7">No Image Object Modifier</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.5.8" href="#r3_5_1_5_8">No Reflection Object Modifier</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.5.9" href="#r3_5_1_5_9">Double Illuminate Object Modifier</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.5.10" href="#r3_5_1_5_10">No Radiosity Object Modifier</a></div></td>
</tr>
<tr>
<td><div class="divh4"><a title="3.5.1.5.11" href="#r3_5_1_5_11">Sturm Object Modifier</a></div></td>
</tr>
<tr>
<td><div class="divh1"> </div></td>
</tr>
<tr>
<td><div class="divh1"> </div></td>
</tr>
</table>
</div>
<!-- NavPanel End -->
<div class="Content">
<table class="HeaderFooter" width="100%">
<tr>
<td colspan=5 align="left" class="HeaderFooter">
POV-Ray for Unix <strong class="HeaderFooter">version 3.8</strong>
</td>
</tr>
<tr >
<td colspan=5>
<hr align="right" width="70%">
</td>
</tr>
<tr>
<td width="30%"></td>
<td class="NavBar"><a href="index.html" title="The Front Door">Home</a></td>
<td class="NavBar"><a href="u1_0.html" title="Unix Table of Contents">POV-Ray for Unix</a></td>
<td class="NavBar"><a href="t2_0.html" title="Tutorial Table of Contents">POV-Ray Tutorial</a></td>
<td class="NavBar"><a href="r3_0.html" title="Reference Table of Contents">POV-Ray Reference</a></td>
</tr>
</table>
<a name="r3_5"></a>
<div class="content-level-h2" contains="The Object Zoo" id="r3_5">
<h2>3.5 The Object Zoo</h2>
<p>Quick Links:</p>
<ul>
<li><a href="r3_5.html#r3_5_1_1">Finite Solid Primitives</a></li>
<li><a href="r3_5.html#r3_5_1_2">Finite Patch Primitives</a></li>
<li><a href="r3_5.html#r3_5_1_3">Infinite Solid Primitives</a></li>
<li><a href="r3_5.html#r3_5_1_4">Constructive Solid Geometry</a></li>
<li><a href="r3_5.html#r3_5_1_5">Object Modifiers</a></li>
</ul></div>
<a name="r3_5_1"></a>
<div class="content-level-h3" contains="Object" id="r3_5_1">
<h3>3.5.1 Object</h3>
<p>Objects are the building blocks of your scene. There are a lot of different types of objects supported by POV-Ray. In the following sections, we describe <a href="r3_5.html#r3_5_1_1">Finite Solid Primitives</a>, <a href="r3_5.html#r3_5_1_2">Finite Patch Primitives</a> and <a href="r3_5.html#r3_5_1_3">Infinite Solid Primitives</a>. These primitive shapes may be combined into complex shapes using <a href="r3_5.html#r3_5_1_4">Constructive Solid Geometry</a> (also known as CSG).</p>
<p>
The basic syntax of an object is a keyword describing its type, some floats,
vectors or other parameters which further define its location and/or shape
and some optional object modifiers such as texture, interior_texture, pigment, normal, finish,
interior, bounding, clipping or transformations. Specifically the syntax
is:</p>
<pre>
OBJECT:
FINITE_SOLID_OBJECT | FINITE_PATCH_OBJECT |
INFINITE_SOLID_OBJECT | CSG_OBJECT | LIGHT_SOURCE |
object { OBJECT_IDENTIFIER [OBJECT_MODIFIERS...] }
FINITE_SOLID_OBJECT:
BLOB | BOX | CONE | CYLINDER | HEIGHT_FIELD | ISOSURFACE | JULIA_FRACTAL |
LATHE | OVUS | PARAMETRIC | PRISM | SPHERE | SPHERE_SWEEP | SUPERELLIPSOID |
SOR | TEXT | TORUS
FINITE_PATCH_OBJECT:
BICUBIC_PATCH | DISC | MESH | MESH2 | POLYGON | TRIANGLE |
SMOOTH_TRIANGLE
INFINITE_SOLID_OBJECT:
PLANE | POLY | CUBIC | QUARTIC | QUADRIC
CSG_OBJECT:
UNION | INTERSECTION | DIFFERENCE | MERGE
</pre>
<p>Object identifiers may be declared to make scene files more readable and
to parameterize scenes so that changing a single declaration changes many
values. An identifier is declared as follows.</p>
<pre>
OBJECT_DECLARATION:
#declare IDENTIFIER = OBJECT |
#local IDENTIFIER = OBJECT
</pre>
<p>Where <em>IDENTIFIER</em> is the name of the identifier up to 40
characters long and <em>OBJECT</em> is any valid object. To invoke
an object identifier, you wrap it in an <code>object{...}</code> statement.
You use the <code>object</code> statement regardless of what type of object
it originally was. Although early versions of POV-Ray required this <code>
object</code> wrapper all of the time, now it is only used with <em>
OBJECT_IDENTIFIERS</em>.</p>
<p>
Object modifiers are covered in detail later. However here is a brief
overview.</p>
<p>
The texture describes the surface properties of the object. Complete details
are in <a href="r3_6.html#r3_6_1">textures</a>. Textures are combinations of pigments, normals,
and finishes. In the section <a href="r3_6.html#r3_6_1_1">pigment</a> you will learn how to
specify the color or pattern of colors inherent in the material. In <a href="r3_6.html#r3_6_1_2">normal</a>, we describe a method of simulating various patterns of bumps, dents, ripples
or waves by modifying the surface normal vector. The section on <a href="r3_6.html#r3_6_1_3">finish</a> describes the reflective properties of the surface. The <a href="r3_7.html#r3_7_2_1">Interior</a> is a feature introduced in POV-Ray 3.1. It contains information
about the interior of the object which was formerly contained in the finish
and halo parts of a texture. Interior items are no longer part of the
texture. Instead, they attach directly to the objects. The halo feature has
been discontinued and replaced with a new feature called <a href="r3_7.html#r3_7_2">Media</a> which replaces both halo and atmosphere.</p>
<p>
Bounding shapes are finite, invisible shapes which wrap around complex, slow
rendering shapes in order to speed up rendering time. Clipping shapes are
used to cut away parts of shapes to expose a hollow interior. Transformations
tell the ray-tracer how to move, size or rotate the shape and/or the texture
in the scene.</p></div>
<a name="r3_5_1_1"></a>
<div class="content-level-h4" contains="Finite Solid Primitives" id="r3_5_1_1">
<h4>3.5.1.1 Finite Solid Primitives</h4>
<p>There are seventeen different solid finite primitive shapes: <a href="r3_5.html#r3_5_1_1_1">blob</a>, <a href="r3_5.html#r3_5_1_1_2">box</a>,
<a href="r3_5.html#r3_5_1_1_3">cone</a>, <a href="r3_5.html#r3_5_1_1_4">cylinder</a>, <a href="r3_5.html#r3_5_1_1_5">height field</a>, <a href="r3_5.html#r3_5_1_1_6">isosurface</a>, <a href="r3_5.html#r3_5_1_1_7">Julia fractal</a>, <a href="r3_5.html#r3_5_1_1_8">lathe</a>, <a href="r3_5.html#r3_5_1_1_10">ovus</a>, <a href="r3_5.html#r3_5_1_1_11">parametric</a>, <a href="r3_5.html#r3_5_1_1_12">prism</a>, <a href="r3_5.html#r3_5_1_1_13">sphere</a>, <a href="r3_5.html#r3_5_1_1_14">sphere_sweep</a>, <a href="r3_5.html#r3_5_1_1_15">superellipsoid</a>, <a href="r3_5.html#r3_5_1_1_16">surface of revolution</a>, <a href="r3_5.html#r3_5_1_1_17">text</a> and <a href="r3_5.html#r3_5_1_1_18">torus</a>. These have a
well-defined <em>inside</em> and can be used in CSG: see <a href="r3_5.html#r3_5_1_4">Constructive Solid Geometry</a>. They are finite and respond to automatic bounding. You may specify an interior for these objects.</p></div>
<a name="r3_5_1_1_1"></a>
<div class="content-level-h5" contains="Blob" id="r3_5_1_1_1">
<h5>3.5.1.1.1 Blob</h5>
<p>Blobs are an interesting and flexible object type. Mathematically they are iso-surfaces of scalar fields, i.e. their surface is defined by the strength of the field in each point. If this strength is equal to a threshold value you are on the surface otherwise you are not.</p>
<p>Picture each blob component as an object floating in space. This object is <em> filled</em> with a field that has its maximum at the center of the object and drops off to zero at the object's surface. The field strength of all those components are added together to form the field of the blob. Now POV-Ray looks for points where this field has a given value, the threshold
value. All these points form the surface of the blob object. Points with a greater field value than the threshold value are considered to be inside while points with a smaller field value are outside.</p>
<p>There's another, simpler way of looking at blobs. They can be seen as a union of flexible components that attract or repel each other to form a blobby organic looking shape. The components' surfaces actually stretch out smoothly and connect as if they were made of honey or something similar.</p>
<p>The syntax for <code>blob</code> is defined as follows:</p>
<pre>
BLOB:
blob { BLOB_ITEM... [BLOB_MODIFIERS...]}
BLOB_ITEM:
sphere{<Center>, Radius,
[ strength ] Strength[COMPONENT_MODIFIER...] } |
cylinder{<End1>, <End2>, Radius,
[ strength ] Strength [COMPONENT_MODIFIER...] } |
component Strength, Radius, <Center> |
threshold Amount
COMPONENT_MODIFIER:
TEXTURE | PIGMENT | NORMAL | FINISH | TRANSFORMATION
BLOB_MODIFIER:
hierarchy [Boolean] | sturm [Boolean] | OBJECT_MODIFIER
</pre>
<p>Blob default values:</p>
<pre>
hierarchy : on
sturm : off
threshold : 1.0
</pre>
<p>The <code>threshold</code> keyword is followed by a float value which determines the total field strength value that POV-Ray is looking for. The default value if none is specified is <code>threshold 1.0</code>. By following the ray out into space and looking at how each blob component affects the ray, POV-Ray will find the points in space where the field strength is equal to the threshold value. The following list shows some things you should know about the threshold value.</p>
<ol>
<li>The threshold value must be positive.</li>
<li>A component disappears if the threshold value is greater than its strength.</li>
<li>As the threshold value gets larger, the surface you see gets closer to the centers of the components.</li>
<li>As the threshold value gets smaller, the surface you see gets closer to the surface of the components.</li>
</ol>
<p>Cylindrical components are specified by a <code>cylinder</code> statement. The center of the end-caps of the cylinder is defined by the vectors <em><code><End1></code></em> and <em><code> <End2></code></em>. Next is the float value of the <em>Radius</em> followed by the float <em>Strength</em>. These vectors and floats are required and should be separated
by commas. The keyword <code> strength</code> may optionally precede the strength value. The cylinder has hemispherical caps at each end.</p>
<p>Spherical components are specified by a <code>sphere</code> statement. The location is defined by the vector <em><code><Center></code></em>. Next is the float value of the <em> Radius</em> followed by the float <em>
Strength</em>. These vector and float values are required and should be separated by commas. The keyword <code> strength</code> may optionally precede the strength value.</p>
<p>You usually will apply a single texture to the entire blob object, and you typically use transformations to change its size, location, and orientation. However both the <code>cylinder</code> and <code>sphere</code> statements may have individual texture, pigment, normal, finish, and transformations applied to them. You may not apply separate <code>interior</code> statements to the components but you may specify one for the entire blob.</p>
<p class="Note"><strong>Note:</strong> By unevenly scaling a spherical component you can create ellipsoidal components. The
tutorial section on <a href="t2_3.html#t2_3_3_1">Blob Object</a> illustrates individually textured blob components and many other blob examples.</p>
<p>The <code>component</code> keyword is an obsolete method for specifying a spherical component and is only used for compatibility with earlier POV-Ray versions. It may not have textures or transformations individually applied to it.</p>
<p>The <code>strength</code> parameter of either type of blob component is a float value specifying the field strength at the center of the object. The strength may be positive or negative. A positive value will make that component attract other components while a negative value will make it repel other components. Components in different, separate blob shapes do not affect each other.</p>
<p>You should keep the following things in mind.</p>
<ol>
<li>The strength value may be positive or negative. Zero is a bad value, as the net result is that no field was added -- you might just as well have not used this component.</li>
<li>If strength is positive, then POV-Ray will add the component's field to the space around the center of the component. If this adds enough field strength to be greater than the threshold value you will see a surface.</li>
<li>If the strength value is negative, then POV-Ray will subtract the component's field from the space around the center of the component. This will only do something if there happen to be positive components nearby. The surface around any nearby positive components will be dented away from the center of the negative component.</li>
</ol>
<p>After all components and the optional <code>threshold</code> value have been specified you may specify zero or more blob modifiers. A blob modifier is any regular object modifier or the <code>hierarchy</code> or <code>
sturm</code> keywords.</p>
<p>The components of each blob object are internally bounded by a spherical bounding hierarchy to speed up blob intersection tests and other operations. Using the optional keyword <code>hierarchy</code> followed by an optional boolean float value will turn it off or on. By default it is on.</p>
<p>If additional accuracy is required you can add the <code><a href="r3_5.html#r3_5_1_5_11">sturm</a></code> object modifier.</p>
<p>An example of a three component blob is:</p>
<pre>
BLOB:
blob {
threshold 0.6
sphere { <.75, 0, 0>, 1, 1 }
sphere { <-.375, .64952, 0>, 1, 1 }
sphere { <-.375, -.64952, 0>, 1, 1 }
scale 2
}
</pre>
<p>If you have a single blob component then the surface you see will just look like the object used, i.e. a sphere or a cylinder, with the surface being somewhere inside the surface specified for the component. The exact surface location can be determined from the blob equation listed below (you will probably never need to know this, blobs are more for visual appeal than
for exact modeling).</p>
<p>For the more mathematically minded, here's the formula used internally by POV-Ray to create blobs. You do not need to understand this to use blobs. The density of the blob field of a single component is:</p>
<table class="centered" width="405x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="385px" src="images/6/69/RefImgBlobdens.png">
</td>
</tr>
<tr>
<td>
<p class="caption">Blob Density</p>
</td>
</tr>
</table>
<p>Where <em>distance</em> is the distance of a given point from the spherical blob's center or cylinder blob's axis. This formula has a convenient property that it is exactly equal to the strength parameter at the center of the component and drops off to exactly 0 at a distance from the center of the component that is equal to the radius value. The density formula for more than one blob component is just the sum of the individual component densities.</p></div>
<a name="r3_5_1_1_2"></a>
<div class="content-level-h5" contains="Box" id="r3_5_1_1_2">
<h5>3.5.1.1.2 Box</h5>
<p>A simple box can be defined by listing two corners of the box using the
following syntax for a <code>box</code> statement:</p>
<pre>
BOX:
box {
<Corner_1>, <Corner_2>
[OBJECT_MODIFIERS...]
}
</pre>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/9/98/RefImgBoxgeom.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">The geometry of a box.</p>
</td>
</tr>
</table>
<p>Where <em><code><Corner_1></code></em> and <em><code>
<Corner_2></code></em> are vectors defining the x, y, z coordinates of
the opposite corners of the box.</p>
<p class="Note"><strong>Note:</strong> All boxes are defined with their faces parallel to the coordinate axes. They may later be rotated to any orientation using the <code>rotate</code> keyword.</p>
<p>
Boxes are calculated efficiently and make good bounding shapes (if manually
bounding seems to be necessary).</p></div>
<a name="r3_5_1_1_3"></a>
<div class="content-level-h5" contains="Cone" id="r3_5_1_1_3">
<h5>3.5.1.1.3 Cone</h5>
<p>The <code>cone</code> statement creates a finite length cone or a <em>
frustum</em> (a cone with the point cut off). The syntax is:</p>
<pre>
CONE:
cone {
<Base_Point>, Base_Radius, <Cap_Point>, Cap_Radius
[ open ][OBJECT_MODIFIERS...]
}
</pre>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/9/93/RefImgConegeom.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">The geometry of a cone.</p>
</td>
</tr>
</table>
<p>Where <em><code><Base_Point></code></em> and <em><code><
Cap_Point></code></em> are vectors defining the x, y, z coordinates of the
center of the cone's base and cap and <em><code> Base_Radius</code></em>
and <em><code>Cap_Radius</code></em> are float values for the corresponding
radii.</p>
<p>Normally the ends of a cone are closed by flat discs that are parallel to
each other and perpendicular to the length of the cone. Adding the optional
keyword <code>open</code> after <em><code>Cap_Radius</code></em> will remove
the end caps and results in a tapered hollow tube like a megaphone or
funnel.</p></div>
<a name="r3_5_1_1_4"></a>
<div class="content-level-h5" contains="Cylinder" id="r3_5_1_1_4">
<h5>3.5.1.1.4 Cylinder</h5>
<p>The <code>cylinder</code> statement creates a finite length cylinder with
parallel end caps The syntax is:</p>
<pre>
CYLINDER:
cylinder {
<Base_Point>, <Cap_Point>, Radius
[ open ][OBJECT_MODIFIERS...]
}
</pre>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/1/18/RefImgCylgeom.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">The geometry of a cylinder.</p>
</td>
</tr>
</table>
<p>Where <em><code><Base_Point></code></em> and <em><code>
<Cap_Point></code></em> are vectors defining the x, y, z coordinates of
the cylinder's base and cap and <em><code>Radius</code></em> is a float
value for the radius.</p>
<p>
Normally the ends of a cylinder are closed by flat discs that are parallel
to each other and perpendicular to the length of the cylinder. Adding the
optional keyword <code>open</code> after the radius will remove the end caps
and results in a hollow tube.</p></div>
<a name="r3_5_1_1_5"></a>
<div class="content-level-h5" contains="Height Field" id="r3_5_1_1_5">
<h5>3.5.1.1.5 Height Field</h5>
<p>Height fields are fast, efficient objects that are generally used to
create mountains or other raised surfaces out of hundreds of triangles in a
mesh. The <code>height_field</code> statement syntax is:</p>
<pre>
HEIGHT_FIELD:
height_field {
[HF_TYPE] "filename" [gamma GAMMA] [premultiplied BOOL] | [HF_FUNCTION]
[HF_MODIFIER...]
[OBJECT_MODIFIER...]
}
HF_TYPE:
exr | gif | hdr | jpeg | pgm | png | pot | ppm | sys | tga | tiff
GAMMA:
Float_Value | srgb | bt709 | bt2020
HF_FUNCTION:
function FieldResolution_X, FieldResolution_Y { UserDefined_Function }
HF_MODIFIER:
smooth & water_level Level
OBJECT_MODIFIER:
hierarchy [Boolean]
</pre>
<p>Height_field default values:</p>
<pre>
hierarchy : on
smooth : off
water_level : 0.0
</pre>
<p>A height field is essentially a one unit wide by one unit long square with
a mountainous surface on top. The height of the mountain at each point is
taken from the color number or palette index of the pixels in a graphic image
file. The maximum height is one, which corresponds to the maximum possible
color or palette index value in the image file.</p>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/0/0c/RefImgUnhfield.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">The size and orientation of an unscaled height field.</p>
</td>
</tr>
</table>
<p>The mesh of triangles corresponds directly to the pixels in the image
file. Each square formed by four neighboring pixels is divided into two
triangles. An image with a resolution of <em><code>N*M</code></em> pixels has
<em><code>(N-1)*(M-1)</code></em> squares that are divided into <em> <code>
2*(N-1)*(M-1)</code></em> triangles.</p>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/3/3c/RefImgPixhfld.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">Relationship of pixels and triangles in a height field.</p>
</td>
</tr>
</table>
<p>The resolution of the height field is influenced by two factors: the
resolution of the image and the resolution of the color/index values. The
size of the image determines the resolution in the x- and z-direction. A
larger image uses more triangles and looks smoother. The resolution of the
color/index value determines the resolution along the y-axis. A height field
made from an 8-bit image can have 256 different height levels while one made
from a 16-bit image can have up to 65536 different height levels. Thus the
second height field will look much smoother in the y-direction if the height
field is created appropriately.</p>
<p>
The size/resolution of the image does not affect the size of the height
field. The unscaled height field size will always be 1 by 1 by 1. Higher
resolution image files will create smaller triangles, not larger height
fields.</p>
<p>The image file type used to create a height field is specified by one of the keywords <a href="r3_5.html#r3_5_1_1_5">listed</a> above. Specifying the file type is optional. If it is not defined the same file type will be assumed as the one that is set as the output file type. This is useful when the source for the <code>height_field</code> is also generated with POV-Ray.</p>
<p>The GIF, PNG, PGM, TIFF and possibly SYS format files are the only
ones that can be created using a standard paint program. Though there are
paint programs for creating TGA image files they will not be of much use for
creating the special 16 bit TGA files used by POV-Ray (see below and
<a href="r3_4.html#r3_4_1_4">HF_Gray_16</a> for more details).</p>
<p>
In an image file that uses a color palette, like GIF, the color number is the
palette index at a given pixel. Use a paint program to look at the palette of
a GIF image. The first color is palette index zero, the second is index one,
the third is index two and so on. The last palette entry is index 255.
Portions of the image that use low palette entries will result in lower parts
of the height field. Portions of the image that use higher palette entries
will result in higher parts of the height field.</p>
<p>
Height fields created from GIF files can only have 256 different height
levels because the maximum number of colors in a GIF file is 256.</p>
<p>
The color of the palette entry does not affect the height of the pixel.
Color entry 0 could be red, blue, black or orange but the height of any pixel
that uses color entry 0 will always be 0. Color entry 255 could be indigo,
hot pink, white or sky blue but the height of any pixel that uses color entry
255 will always be 1.</p>
<p>
You can create height field GIF images with a paint program or a fractal program like <em>Fractint</em>.</p>
<p>
A POT file is essentially a GIF file with a 16 bit palette. The maximum
number of colors in a POT file is 65536. This means a POT height field can
have up to 65536 possible height values. This makes it possible to have much
smoother height fields.</p>
<p class="Note"><strong>Note:</strong> The maximum height of the field is still 1
even though more intermediate values are possible.</p>
<p> At the time of this writing the only program that created POT files was a freeware MS-Dos/Windows program called <em>Fractint</em>. POT files generated with this fractal program create fantastic landscapes.</p>
<p>
The TGA and PPM file formats may be used as a storage device for 16 bit
numbers rather than an image file. These formats use the red and green bytes
of each pixel to store the high and low bytes of a height value. These files
are as smooth as POT files but they must be generated with special
custom-made programs. Several programs can create TGA heightfields in the
format POV uses, such as <em>Gforge</em> and <em>Terrain Maker</em>.</p>
<p>
PNG format heightfields are usually stored in the form of a grayscale image
with black corresponding to lower and white to higher parts of the height
field. Because PNG files can store up to 16 bits in grayscale images they
will be as smooth as TGA and PPM images. Since they are grayscale images you
will be able to view them with a regular image viewer. <em>Gforge</em>
can create 16-bit heightfields in PNG format. Color PNG images will be used
in the same way as TGA and PPM images.</p>
<p>
SYS format is a platform specific file format. See your platform specific
documentation for details.</p>
<p>
In addition to all the usual object modifiers, there are three additional
height field modifiers available.</p>
<p>The optional <code>water_level</code> parameter may be added after the file
name. It consists of the keyword <code>water_level</code> followed by a
float value telling the program to ignore parts of the height field below
that value. The default value is zero and legal values are between zero and
one. For example <code>water_level 0.5</code> tells POV-Ray to only render
the top half of the height field. The other half is <em>below the water</em>
and could not be seen anyway. Using <code>water_level</code> renders
faster than cutting off the lower part using CSG or clipping. This term comes
from the popular use of height fields to render landscapes. A height field
would be used to create islands and another shape would be used to simulate
water around the islands. A large portion of the height field would be
obscured by the water so the <code>water_level</code> parameter was
introduced to allow the ray-tracer to ignore the unseen parts of the height
field. <code>water_level</code> is also used to cut away unwanted lower
values in a height field. For example if you have an image of a fractal on a
solid colored background, where the background color is palette entry 0, you
can remove the background in the height field by specifying, <code>
water_level 0.001</code>.</p>
<p>Normally height fields have a rough, jagged look because they are made of
lots of flat triangles. Adding the keyword <code>smooth</code> causes
POV-Ray to modify the surface normal vectors of the triangles in such a way
that the lighting and shading of the triangles will give a smooth look. This
may allow you to use a lower resolution file for your height field than would
otherwise be needed. However, smooth triangles will take longer to render.
The default value is off.</p>
<p>In order to speed up the intersection tests a one-level bounding hierarchy
is available. By default it is always used but it can be switched off using
<code>hierarchy off</code> to improve the rendering speed for small height
fields (i.e. low resolution images). You may optionally use a boolean value
such as <code>hierarchy on</code> or <code>hierarchy off</code>.</p>
<p>While POV-Ray will normally interpret the height field input file as a container of linear data irregardless of file type, this can be overridden for any individual height field input file by specifying <code>gamma</code> GAMMA immediately after the file name. For example:</p>
<pre>
height field {
jpeg "foobar.jpg" gamma 1.8
}
</pre>
<p>This will cause POV-Ray to perform gamma adjustment or decoding on the input file data before building the height field. Alternatively to a numerical value, <code>srgb</code> may be specified to denote that the file format is pre-corrected or encoded using the ''sRGB transfer function'' instead of a power-law gamma function. <font class="New">New</font> in version 3.8, other valid special values are <code>bt709</code> and <code>bt2020</code>, denoting that the file is encoded or pre-corrected using the ITU-R BT.709 or BT.2020 transfer function, respectively. See section <a href="t2_3.html#t2_3_4">Gamma Handling</a> for more details.</p>
<p>The height field object also allows for substituting a <a href="r3_6.html#r3_6_2_1_12">user defined function</a> instead of specifying an image. That function can either be in it's literal form, or it can be a call to a function that you have predeclared. The user supplied parameters <code>FieldResolution_X</code> and <code>FieldResolution_Y</code> are integer values that affect the resolution of the color/index values, <em>not</em> size of the unscaled height field.</p></div>
<a name="r3_5_1_1_6"></a>
<div class="content-level-h5" contains="Isosurface" id="r3_5_1_1_6">
<h5>3.5.1.1.6 Isosurface</h5>
<p>See the <code>isosurface</code> <a href="t2_3.html#t2_3_3_3">tutorial</a> for more about working with <em>isosurfaces</em>.</p>
<p><font class="New">New</font> in version 3.8 a <code>potential</code> pattern has been added to define a pattern based on the <em>potential</em> field of a <code>blob</code> or <code>isosurface</code> object. See also: <a href="r3_6.html#r3_6_2_4_3">Potential Pattern</a>.</p>
<p>The syntax basics are as follows:</p>
<pre>
isosurface {
function { FUNCTION_ITEMS }
[contained_by { SPHERE | BOX }]
[threshold FLOAT_VALUE]
[accuracy FLOAT_VALUE]
[max_gradient FLOAT_VALUE]
[evaluate P0, P1, P2]
[open]
[max_trace INTEGER] | [all_intersections]
[polarity on | +VALUE | off | -VALUE]
[OBJECT_MODIFIERS...]
}
</pre>
<p>Isosurface default values:</p>
<pre>
contained_by : box{-1,1}
threshold : 0.0
accuracy : 0.001
max_gradient : 1.1
polarity : off
</pre>
<p>Since <em>isosurfaces</em> are defined by a user supplied <code>function { ... }</code> it must be specified as the first item of the <code>isosurface</code> statement. Here you place all the mathematical functions that will describe the surface.</p>
<p>The <code>contained_by</code> <em>object</em> limits the area where POV-Ray samples for the surface of the function. This container can either be a <code>sphere</code> or a <code>box</code> both of which use the standard POV-Ray syntax. If nothing is specified a default <code>box</code>, as noted above, will be used. See additional usage examples below:</p>
<pre>
contained_by { sphere { CENTER, RADIUS } }
contained_by { box { CORNER1, CORNER2 } }
</pre>
<p>Using <code>threshold</code> specifies how much strength, or substance to give the <code>isosurface</code>. The surface appears where the <code>function</code> value equals the <code>threshold</code> value. See above for the listed default value.</p>
<p>The <code>isosurface</code> resolver uses a recursive subdivision method. This subdivision goes on until the length of the interval where POV-Ray finds a surface point is less than the specified <code>accuracy</code>. See above for the listed default value. Be aware that smaller values produce more accurate surfaces, but takes longer to render.</p>
<p>POV-Ray can find the first intersecting point between a ray and the <code>isosurface</code> of any continuous function if the maximum gradient of the function is known. To that end you can specify a <code>max_gradient</code> for the function. See above for the listed default value. When the <code>max_gradient</code> used to find the intersecting point is too high, the render slows down considerably. Conversely, when it is too low, artifacts or holes may appear on the isosurface, and in some cases when it is way too low, the surface does not show at all. While rendering the <code>isosurface</code> POV-Ray stores the found gradient values and issues a warning, if these values are either higher or much lower than the specified <code>max_gradient</code> value:</p>
<pre>
Warning: The maximum gradient found was 5.257, but max_gradient of
the isosurface was set to 5.000. The isosurface may contain holes!
Adjust max_gradient to get a proper rendering of the isosurface.
</pre>
<pre>
Warning: The maximum gradient found was 5.257, but max_gradient of
the isosurface was set to 7.000. Adjust max_gradient to
get a faster rendering of the isosurface.
</pre>
<p>For best performance you should specify a value close to the real maximum gradient.</p>
<p><em>Isosurfaces</em> can also dynamically adapt the used <code>max_gradient</code>. To activate this technique you have to specify the <code>evaluate</code> keyword followed by these three parameters:</p>
<ul>
<li><strong>P0:</strong> the minimum <code>max_gradient</code> in the estimation process.</li>
<li><strong>P1:</strong> an over-estimating factor. That is, the <code>max_gradient</code> is multiplied by the <em>P1</em> parameter.</li>
<li><strong>P2:</strong> an attenuation parameter of 1 or less</li>
</ul>
<p>In this case POV-Ray starts with the <code>max_gradient</code> value <em>P0</em> and dynamically changes it during the render using <em>P1</em> and <em>P2</em>. In the evaluation process, the <em>P1</em> and <em>P2</em> parameters are used in
quadratic functions. This means that over-estimation increases more rapidly with higher values and attenuates more rapidly with lower values. It should also be noted that when using dynamic <code>max_gradient</code>, there can be artifacts or holes.</p>
<p>If you are unsure what values to use, just start a render <em>without</em> using <code>evaluate</code> to get a value for the <code>max_gradient</code>, then use that value with <code>evaluate</code> like this:</p>
<ul>
<li><strong>P0:</strong> <code>max_gradient * Min_Factor</code></li>
<li><strong>P1:</strong> <code>sqrt(max_gradient/(max_gradient * Min_Factor))</code></li>
<li><strong>P2:</strong> should be 1 or less use 0.7 as a good starting point.</li>
</ul>
<p>Where <em>Min_Factor</em> is a float between 0 and 1 to reduce the <code>max_gradient</code> to a <em>minimum</em> <code>max_gradient</code>. The ideal value for <em>P0</em> would be the average of the found maximum gradients, but we do not
have access to that information. A good starting point for <em>Min_Factor</em> is 0.6</p>
<p>If there are artifacts or holes in the <code>isosurface</code>, you can just increase <em>Min_Factor</em> and / or <em>P2</em>. For Example: when the first run gives a <em>found</em> <code>max_gradient</code> of 356, start with:</p>
<pre>
#declare Min_factor= 0.6;
isosurface {
...
evaluate 356*Min_factor, sqrt(356/(356*Min_factor)), 0.7
//evaluate 213.6, 1.29, 0.7
...
}
</pre>
<p>This method is only an approximation of what happens internally, but it gives faster rendering speeds with the majority of <em>isosurfaces</em>.</p>
<p>When the <code>isosurface</code> is not fully contained within the <code>contained_by</code> object, there will be a cross section. When this happens, you will see the surface of the container. Using the <code>open</code> keyword, these cross section surfaces are removed, and the inside of the <code>isosurface</code> becomes visible.</p>
<p class="Note"><strong>Note:</strong> Using <code>open</code> slows down the render speed, and it is not recommended for use with CSG operations.</p>
<p><em>Isosurfaces</em> can be used in CSG shapes since they are solid finite objects - if not finite by themselves, they are through the cross section with the container. By default POV-Ray searches only for the first surface which the ray intersects. However, when using an <code>isosurface</code> in CSG operations, the other surfaces must also be found. Consequently, the keyword <code>max_trace</code> followed by an integer value, must be added to the <code>isosurface</code> statement. To check for all surfaces, use the keyword <code>all_intersections</code> instead. With <code>max_trace</code> it only checks until that number is reached.</p>
<p class="Note"><strong>Note:</strong> The current implementation has a <em>limit</em> of 10 <em>intersections</em> in all cases.</p>
<p>By default, the inside of an <code>isosurface</code> is defined as the set of all points inside the <code>contained_by</code> shape where the function values are below the threshold. <font class="New">New</font> in version 3.8 this can be changed via the <code>polarity</code> keyword. Specifying a <em>positive</em> setting or <code>on</code> will instead cause function values <em>above</em> the threshold to be considered inside. Specifying a <em>negative</em> setting or <code>off</code> will give the default behavior.</div>
<a name="r3_5_1_1_7"></a>
<div class="content-level-h5" contains="Julia Fractal" id="r3_5_1_1_7">
<h5>3.5.1.1.7 Julia Fractal</h5>
<p>A <em>julia fractal</em> object is a 3-D <em>slice</em> of a 4-D object
created by generalizing the process used to create the classic Julia sets.
You can make a wide variety of strange objects using the <code>
julia_fractal</code> statement including some that look like bizarre blobs of
twisted taffy. The <code>julia_fractal</code> syntax is:</p>
<pre>
JULIA_FRACTAL:
julia_fractal {
<4D_Julia_Parameter>
[JF_ITEM...] [OBJECT_MODIFIER...]
}
JF_ITEM:
ALGEBRA_TYPE | FUNCTION_TYPE | max_iteration Count |
precision Amt | slice <4D_Normal>, Distance
ALGEBRA_TYPE:
quaternion | hypercomplex
FUNCTION_TYPE:
QUATERNATION:
sqr | cube
HYPERCOMPLEX:
sqr | cube | exp | reciprocal | sin | asin | sinh |
asinh | cos | acos | cosh | acosh | tan | atan |tanh |
atanh | ln | pwr( X_Val, Y_Val )
</pre>
<p>Julia Fractal default values:</p>
<pre>
ALGEBRA_TYPE : quaternion
FUNCTION_TYPE : sqr
max_iteration : 20
precision : 20
slice, DISTANCE : <0,0,0,1>, 0.0
</pre>
<p>The required 4-D vector <em><code><4D_Julia_Parameter></code></em>
is the classic Julia parameter <em><code>p</code></em> in the iterated
formula <em><code>f(h) + p</code></em>. The julia fractal object is
calculated by using an algorithm that determines whether an arbitrary point
<em><code>h(0)</code></em> in 4-D space is inside or outside the object. The
algorithm requires generating the sequence of vectors <em><code>h(0), h(1),
...</code></em> by iterating the formula <em><code>h(n+1) = f(h(n)) + p (n =
0, 1, ..., max_iteration-1)</code></em> where <em><code> p</code></em> is the
fixed 4-D vector parameter of the julia fractal and <em><code>f()</code></em>
is one of the functions <code>sqr</code>, <code> cube</code>, ... specified
by the presence of the corresponding keyword. The point <em><code>
h(0)</code></em> that begins the sequence is considered inside the julia
fractal object if none of the vectors in the sequence escapes a hypersphere
of radius 4 about the origin before the iteration number reaches the integer
<code>max_iteration</code> value. As you increase <code>max_iteration</code>,
some points escape that did not previously escape, forming the julia fractal.
Depending on the <em><code> <4D_Julia_Parameter></code></em>, the julia
fractal object is not necessarily connected; it may be scattered fractal
dust. Using a low <code> max_iteration</code> can fuse together the dust to
make a solid object. A high <code>max_iteration</code> is more accurate but
slows rendering. Even though it is not accurate, the solid shapes you get
with a low <code>max_iteration</code> value can be quite interesting. If
none is specified, the default is <code>max_iteration 20</code>.</p>
<p>Since the mathematical object described by this algorithm is four-dimensional
and POV-Ray renders three dimensional objects, there must be a way to reduce
the number of dimensions of the object from four dimensions to three. This is
accomplished by intersecting the 4-D fractal with a 3-D plane defined by the <code>slice</code> modifier and then projecting the
intersection to 3-D space. The keyword is followed by 4-D vector and a float
separated by a comma. The slice plane is the 3-D space that is perpendicular
to <em><code> <4D_Normal></code></em> and is <em><code>
Distance</code></em> units from the origin. Zero length <em><code>
<4D_Normal></code></em> vectors or a <em><code>
<4D_Normal></code></em> vector with a zero fourth component are
illegal. If none is specified, the default is <code> slice
<0,0,0,1>,0</code>.</p>
<p>
You can get a good feel for the four dimensional nature of a julia fractal by
using POV-Ray's animation feature to vary a slice's <em><code>
Distance</code></em> parameter. You can make the julia fractal appear from
nothing, grow, then shrink to nothing as <em><code>
Distance</code></em> changes, much as the cross section of a 3-D object
changes as it passes through a plane.</p>
<p>The <code> precision</code> parameter is a tolerance used in the
determination of whether points are inside or outside the fractal object.
Larger values give more accurate results but slower rendering. Use as low a
value as you can without visibly degrading the fractal object's
appearance but note values less than 1.0 are clipped at 1.0. The default if
none is specified is <code>precision 20</code>.</p>
<p>The presence of the keywords <code> quaternion</code> or <code>
hypercomplex</code> determine which 4-D algebra is used to calculate the
fractal. The default is <code>quaternion</code>. Both are 4-D generalizations
of the complex numbers but neither satisfies all the field properties (all
the properties of real and complex numbers that many of us slept through in
high school). Quaternions have non-commutative multiplication and
hypercomplex numbers can fail to have a multiplicative inverse for some
non-zero elements (it has been proved that you cannot successfully generalize
complex numbers to four dimensions with all the field properties intact, so
something has to break). Both of these algebras were discovered in the 19th
century. Of the two, the quaternions are much better known, but one can argue
that hypercomplex numbers are more useful for our purposes, since complex
valued functions such as sin, cos, etc. can be generalized to work for
hypercomplex numbers in a uniform way.</p>
<p>
For the mathematically curious, the algebraic properties of these two
algebras can be derived from the multiplication properties of the unit basis
vectors 1 = <1,0,0,0>, i=< 0,1,0,0>, j=<0,0,1,0> and k=<
0,0,0,1>. In both algebras 1 x = x 1 = x for any x (1 is the
multiplicative identity). The basis vectors 1 and i behave exactly like the
familiar complex numbers 1 and i in both algebras.</p>
<table SUMMARY="Quaternion basis vector multiplication rules" width="75%">
<tr>
<td width="33%"><code>ij = k</code></td>
<td width="33%"><code>jk = i</code></td>
<td width="33%"><code>ki = j</code></td>
</tr>
<tr>
<td><code>ji = -k</code></td>
<td><code>kj = -i</code></td>
<td><code>ik = -j</code></td>
</tr>
<tr>
<td><code>ii = jj = kk = -1</code></td>
<td><code>ijk = -1</code></td>
<td> </td>
</tr>
</table>
<table SUMMARY="Hypercomplex basis vector multiplication rules" width="75%">
<tr>
<td width="33%"><code>ij = k</code></td>
<td width="33%"><code>jk = -i</code></td>
<td width="33%"><code>ki = -j</code></td>
</tr>
<tr>
<td><code>ji = k</code></td>
<td><code>kj = -i</code></td>
<td><code>ik = -j</code></td>
</tr>
<tr>
<td><code>ii = jj = kk = -1</code></td>
<td><code>ijk = 1</code></td>
<td> </td>
</tr>
</table>
<p>A distance estimation calculation is used with the quaternion calculations
to speed them up. The proof that this distance estimation formula works does
not generalize from two to four dimensions but the formula seems to work well
anyway, the absence of proof notwithstanding!</p>
<p>The presence of one of the function keywords <code>sqr</code>, <code>
cube</code>, etc. determines which function is used for <em><code>
f(h)</code></em> in the iteration formula <em> <code>h(n+1) = f(h(n)) +
p</code></em>. The default is <code>sqr.</code> Most of the function keywords
work only if the <code>hypercomplex</code> keyword is present. Only <code>
sqr</code> and <code>cube</code> work with <code> quaternion</code>. The
functions are all familiar complex functions generalized to four dimensions.
Function Keyword Maps 4-D value h to:</p>
<table SUMMARY="Function Keyword Maps 4-D value of h" width="75%">
<tr>
<td width="30%"><code>sqr</code></td>
<td width="70%">h*h</td>
</tr>
<tr>
<td><code>cube</code></td>
<td>h*h*h</td>
</tr>
<tr>
<td><code>exp</code></td>
<td>e raised to the power h</td>
</tr>
<tr>
<td><code>reciprocal</code></td>
<td>1/h</td>
</tr>
<tr>
<td><code>sin</code></td>
<td>sine of h</td>
</tr>
<tr>
<td><code>asin</code></td>
<td>arcsine of h</td>
</tr>
<tr>
<td><code>sinh</code></td>
<td>hyperbolic sine of h</td>
</tr>
<tr>
<td><code>asinh</code></td>
<td>inverse hyperbolic sine of h</td>
</tr>
<tr>
<td><code>cos</code></td>
<td>cosine of h</td>
</tr>
<tr>
<td><code>acos</code></td>
<td>arccosine of h</td>
</tr>
<tr>
<td><code>cosh</code></td>
<td>hyperbolic cos of h</td>
</tr>
<tr>
<td><code>acosh</code></td>
<td>inverse hyperbolic cosine of h</td>
</tr>
<tr>
<td><code>tan</code></td>
<td>tangent of h</td>
</tr>
<tr>
<td><code>atan</code></td>
<td>arctangent of h</td>
</tr>
<tr>
<td><code>tanh</code></td>
<td>hyperbolic tangent of h</td>
</tr>
<tr>
<td><code>atanh</code></td>
<td>inverse hyperbolic tangent of h</td>
</tr>
<tr>
<td><code>ln</code></td>
<td>natural logarithm of h</td>
</tr>
<tr>
<td><code>pwr(x,y)</code></td>
<td>h raised to the complex power x+iy</td>
</tr>
</table>
<p>A simple example of a julia fractal object is:</p>
<pre>
julia_fractal {
<-0.083,0.0,-0.83,-0.025>
quaternion
sqr
max_iteration 8
precision 15
}
</pre>
<p>The first renderings of julia fractals using quaternions were done by Alan
Norton and later by John Hart in the '80's. This POV-Ray
implementation follows <em>Fractint</em> in pushing beyond what is known
in the literature by using hypercomplex numbers and by generalizing the
iterating formula to use a variety of transcendental functions instead of
just the classic Mandelbrot <em>z2 + c</em> formula. With an extra two
dimensions and eighteen functions to work with, intrepid explorers should be
able to locate some new fractal beasts in hyperspace, so have at it!</p></div>
<a name="r3_5_1_1_8"></a>
<div class="content-level-h5" contains="Lathe" id="r3_5_1_1_8">
<h5>3.5.1.1.8 Lathe</h5>
<p>The <code>lathe</code> is an object generated from rotating a two-dimensional curve about an axis. This curve is defined by a set of points which are connected by linear, quadratic, cubic or bezier spline curves. The syntax is:</p>
<pre>
LATHE:
lathe {
[SPLINE_TYPE] Number_Of_Points, <Point_1>
<Point_2>... <Point_n>
[LATHE_MODIFIER...]
}
SPLINE_TYPE:
linear_spline | quadratic_spline | cubic_spline | bezier_spline
LATHE_MODIFIER:
sturm | OBJECT_MODIFIER
</pre>
<p>Lathe default values:</p>
<pre>
SPLINE_TYPE : linear_spline
sturm : off
</pre>
<p>The first item is a keyword specifying the type of spline. The default if none is specified is <code>linear_spline</code>. The required integer value <em><code>Number_Of_Points</code></em> specifies how many two-dimensional points are used to define the curve. The points follow and are specified by 2-D vectors. The curve is not automatically closed, i.e. the first and last
points are not automatically connected. You will have to do this yourself if you want a closed curve. The curve thus defined is rotated about the y-axis to form the lathe object, centered at the origin.</p>
<p>The following example creates a simple lathe object that looks like a thick cylinder, i.e. a cylinder with a thick wall:</p>
<pre>
lathe {
linear_spline
5,
<2, 0>, <3, 0>, <3, 5>, <2, 5>, <2, 0>
pigment {Red}
}
</pre>
<p>The cylinder has an inner radius of 2 and an outer radius of 3, giving a wall width of 1. It's height is 5 and it's located at the origin pointing up, i.e. the rotation axis is the y-axis.</p>
<p class="Note"><strong>Note:</strong> The first and last point are equal to get a closed curve.</p>
<p>The splines that are used by the lathe and prism objects are a little bit difficult to understand. The basic concept of splines is to draw a curve through a given set of points in a determined way. The default <code>linear_spline</code> is the simplest spline because it's nothing more than connecting consecutive points with a line. This means the curve that is drawn between two points only depends on those two points. No additional information is taken into account. The other splines are different in that they do take other points into account when connecting two points. This creates a smooth curve and, in the case of the cubic spline, produces smoother transitions at each point.</p>
<p>The <code>quadratic_spline</code> keyword creates splines that are made of quadratic curves. Each of them connects two consecutive points. Since those two points (call them second and third point) are not sufficient to describe a quadratic curve, the predecessor of the second point is taken into account when the curve is drawn. Mathematically, the relationship (their relative locations on the 2-D plane) between the first and second point determines the slope of the curve at the second point. The slope of the curve at the third point is out of control. Thus quadratic splines look much smoother than linear splines but
the transitions at each point are generally not smooth because the slopes on both sides of the point are different.</p>
<p>The <code>cubic_spline</code> keyword creates splines which overcome the transition problem of quadratic splines because they also take a fourth point into account when drawing the curve between the second and third point. The slope at the fourth point is under control now and allows a smooth transition at each point. Thus cubic splines produce the most flexible and
smooth curves.</p>
<p>The <code>bezier_spline</code> is an alternate kind of cubic spline. Points 1 and 4 specify the end points of a segment and points 2 and 3 are control points which specify the slope at the endpoints. Points 2 and 3 do not actually lie on the spline. They adjust the slope of the spline. If you draw an imaginary line between point 1 and 2, it represents the slope at point 1.
It is a line tangent to the curve at point 1. The greater the distance between 1 and 2, the flatter the curve. With a short tangent the spline can bend more. The same holds true for control point 3 and endpoint 4. If you want the spline to be smooth between segments, points 3 and 4 on one segment and points 1 and 2 on the next segment must form a straight line and point 4 of one segment must be the same as point 1 on the next segment.</p>
<p>You should note that the number of spline segments, i. e. curves between two points, depends on the spline type used. For linear splines you get n-1 segments connecting the points P[i], i=1,...,n. A quadratic spline gives you n-2 segments because the last point is only used for determining the slope, as explained above (thus you will need at least three points to define a
quadratic spline). The same holds for cubic splines where you get n-3 segments with the first and last point used only for slope calculations (thus needing at least four points). The bezier spline requires 4 points per segment, creating n/4 segments.</p>
<p>If you want to get a closed quadratic and cubic spline with smooth transitions at the end points you have to make sure that in the cubic case P[n-1] = P[2] (to get a closed curve), P[n] = P[3] and P[n-2] = P[1] (to smooth the transition). In the quadratic case P[n-1] = P[1] (to close the curve) and P[n] = P[2].</p>
<p>The surface normal determination for lathes depends upon the order in which the splines points are specified. For <code>interior_texture</code> to work as expected, the outline of the underlying 2D shape must be specified in counter-clockwise order, except for holes which must be specified in clockwise order, and they must not self-intersect.</p>
<p>The following code will render with the color Red on the outside and the color Blue on the inside.</p>
<pre>
#declare Lathe_InitialOrder = lathe {
bezier_spline
16,
<0.45,0>,<0.45,0.0828427>,<0.382843,0.15>,<0.3,0.15>
<0.3,0.15>,<0.217157,0.15>,<0.15,0.0828427>,<0.15,0>
<0.15,0>,<0.15,-0.0828427>,<0.217157,-0.15>,<0.3,-0.15>
<0.3,-0.15>,<0.382843,-0.15>,<0.45,-0.0828427>,<0.45,0>
sturm
texture { pigment { Red } }
interior_texture { pigment { Blue } }
}
</pre>
<p>While the following example will render inside-out, with the color Blue on the outside and the color Red on the inside.</p>
<pre>
#declare Lathe_ReverseOrder = lathe {
bezier_spline
16,
<0.45,0>,<0.45,-0.0828427>,<0.382843,-0.15>,<0.3,-0.15>
<0.3,-0.15>,<0.217157,-0.15>,<0.15,-0.0828427>,<0.15,0>
<0.15,0>,<0.15,0.0828427>,<0.217157,0.15>,<0.3,0.15>
<0.3,0.15>,<0.382843,0.15>,<0.45,0.0828427>,<0.45,0>
sturm
texture { pigment { Red } }
interior_texture { pigment { Blue } }
}
</pre>
<p>The actual normal determination is more complicated for complex splines. If the surface normal is important to the visual result, it is best to check how the lathe is being rendered by testing with substantially different inside and outside textures.</p>
<p class="Note"><strong>Note:</strong> With the bezier spline, unlike all the other spline types used with the lathe, it is possible to create independent closed curves within a single lathe definition.</p>
<p>If additional accuracy is required you can add the <code><a href="r3_5.html#r3_5_1_5_11">sturm</a></code> object modifier.</p></div>
<a name="r3_5_1_1_9"></a>
<div class="content-level-h5" contains="Lemon" id="r3_5_1_1_9">
<h5>3.5.1.1.9 Lemon</h5>
<table class="left" width="700px" cellpadding="0" cellspacing="20">
<tr>
<td>
<p><font class="New">New</font> to version 3.8 the <code>lemon</code> object has been added. It is similar to the <code>cone</code> in that it's basically described the same way but with these differences:</p>
<ul>
<li>end points are connected along their axis via the revolution surface that's generated by the circular arc of the <em>Inner_Radius</em></li>
<li>for the minimal value of <em>Inner_Radius</em> the surface is a spherical segment where the center of the arc lies along the end points axis</li>
<li>with larger values for <em>Inner_Radius</em> the surface is the inner part of a self intersecting torus or <a href="https://en.wikipedia.org/wiki/Frustum">frustum</a></li>
<li>if the given <em>Inner_Radius</em> is too small the minimal value is used instead and a warning is issued</li>
<li><em>both</em> or <em>either</em> of the end points radii can be zero</li>
</ul>
</td>
<td>
<img class="centered" width="220px" src="images/5/51/RefImgLemon.png">
</td>
</tr>
<tr>
<td>
</td>
<td>
<p class="caption">example lemon objects</p>
</td>
</tr>
</table>
<p class="Note"><strong>Note:</strong> The minimal <em>Inner_Radius</em> value is defined as: sqrt ( Radius<sup>2</sup> + ( DistanceBetweenEndPoints/2 )<sup>2</sup> )</p>
<p>The syntax is as follows:</p>
<pre>
lemon {
<Base_Point>, Base_Radius, <Cap_Point>, Cap_Radius, Inner_Radius
[LEMON_MODIFIERS]
}
LEMON_MODIFIERS:
sturm | open | OBJECT_MODIFIER
</pre>
<p>The following example produced the above image:</p>
<pre>
#version 3.8;
global_settings { assumed_gamma 1.0 }
camera {
location -560*z
direction z
up y
rotate x*20
right image_width*x/image_height
angle 5
}
#include "colors.inc"
#macro LightSource (LightColor)
light_source { <0,0,-200>, LightColor area_light x*15, z*15 20,20 adaptive 1 jitter orient circular }
#end
background { White }
plane { y,-20 pigment { White } }
object { LightSource (1.0) rotate x*40 }
object { LightSource (0.8) rotate x*-40 rotate y*330 }
object { LightSource (1.2) rotate x*80 rotate y*210 }
lemon { <0,-20,0>, 0, <0,20,0>, 0, 30
texture { pigment { color srgb <1,0,0> }}
translate x*-12
sturm
}
lemon { <0,0,0>, 5, <0,-20,0>, 0, 15
texture { pigment { color srgb <0,1,0> }}
translate y*10
translate <3,-10,-10>
sturm
}
lemon { <0,0,0>, 5, <0,-20,0>, 5, 15
open
texture { pigment { color srgb <0,0,1> }}
interior_texture { pigment { srgb <1,1,0> }}
translate y*10
translate <12,2,12>
sturm
}
</pre>
<p>If additional accuracy is required you can add the <code><a href="r3_5.html#r3_5_1_5_11">sturm</a></code> object modifier.</p>
<p>See also the <code><a href="r3_5.html#r3_5_1_1_3">cone</a></code> object.</p></div>
<a name="r3_5_1_1_10"></a>
<div class="content-level-h5" contains="Ovus" id="r3_5_1_1_10">
<h5>3.5.1.1.10 Ovus</h5>
<p>An <code>ovus</code> is a shape that looks like an egg. A <font class="Change">Change</font> in version 3.8 has extended the syntax of the <code>ovus</code> object by adding <code>radius</code>, <code>distance</code> and <code>precision</code> controls.</p>
<p>The syntax is as follows:</p>
<pre>
ovus {
Bottom_radius, Top_radius [radius Inner_radius] [distance Vertical_distance] [precision Root_tolerance]
[OBJECT_MODIFIERS...]
}
</pre>
<p>Where <em>Bottom_radius</em> is a float value giving the radius of the bottom sphere and <em>Top_radius</em> is a float specifying the radius of the top sphere. The top and bottom spheres are connected together with a suitably truncated lemon, or self intersection of a torus, that is automatically computed so as to provide the needed continuity to the shape. The <code>distance</code> is a float value that represents the length between the center of the two spheres, defaulting to <em>Bottom_radius</em>. The <code>radius</code> float value represents the inner circle of the connecting torus and it's default is twice the greater of either <em>Top_radius</em> or <em>Bottom_radius</em>. The <code>precision</code> float value is the tolerance used for the root solving of the connecting torus, and it's default is <code>1e-4</code>. If additional accuracy is required you can now add the <code><a href="r3_5.html#r3_5_1_5_11">sturm</a></code> object modifier.</p>
<table class="centered" width="700px" cellpadding="0" cellspacing="10">
<tr>
<td><img class="left" width="125px" src="images/1/12/RefImgOvus2D.png"></td>
<td>
<ul>
<li>The center of the top sphere lies on the top of the bottom sphere</li>
<li>The bottom sphere of the <code>ovus</code> is centered at the origin</li>
<li>The top sphere of the <code>ovus</code> lies on the y-axis</li>
<li>The minor radius of the lemon is twice the largest radius</li>
<li>The <code>distance</code> <em>must</em> be greater than or equal to <em>Bottom_radius</em></li>
<li>The <code>radius</code> <em>must</em> be greater than or equal to half the sum of <em>Bottom_radius</em>, <em>Top_radius</em> and <em>Vertical_distance</em></li>
</ul>
</td>
</tr>
<tr>
<td><p class="caption">An ovus 2D section</p></td>
<td></td>
</tr>
</table>
<table class="centered" width="640px" cellpadding="0" cellspacing="10">
<tr>
<td><img class="centered" width="620px" src="images/7/72/RefImgOvus3D.png"></td>
</tr>
<tr>
<td><p class="caption">The ovus and it's constituent 3D shapes</p></td>
</tr>
</table>
<p class="Note"><strong>Note:</strong> According to the ratio of the radius, the pointy part is the smallest radius, but is <em>not</em> always on top!</p>
<table class="centered" width="690px" cellpadding="0" cellspacing="10">
<tr>
<td><img class="centered" width="670px" src="images/4/46/RefImgDemoOvus.jpg"></td>
</tr>
<tr>
<td><p class="caption">Evolution of ratio from 0 to 1.95 in 0.15 steps.</p></td>
</tr>
</table>
<p>See also: <a href="r3_6.html#r3_6_1_7_1">UV Mapping</a>.</p>
<p class="Note"><strong>Note:</strong> See the following <em>MathWorld</em> references for more information about the math behind how the <code>ovus</code> object is constructed.</p>
<ul>
<li><a href="http://mathworld.wolfram.com/Torus.html">Torus</a></li>
<li><a href="http://mathworld.wolfram.com/Lemon.html">Lemon</a></li>
<li><a href="http://mathworld.wolfram.com/SpindleTorus.html">SpindleTorus</a></li>
</ul></div>
<a name="r3_5_1_1_11"></a>
<div class="content-level-h5" contains="Parametric" id="r3_5_1_1_11">
<h5>3.5.1.1.11 Parametric</h5>
<p>While the <code>isosurface</code> object uses implicit surface functions like <strong><em>F(x,y,z) = 0</em></strong> the <code>parametric</code> object uses is a set of equations for a surface expressed in the form of the parameters that locate points on the surface. For example: <code>x(u,v)</code>, <code>y(u,v)</code>, <code>z(u,v)</code>. Each of the pairs of values for <code>u</code> and <code>v</code> gives a single point <code><x,y,z></code> in 3d space.</p>
<p>The <code>parametric</code> object is not a solid it is <em>hollow</em>, like a thin shell. The syntax is as follows:</p>
<pre>
parametric {
function { FUNCTION_ITEMS },
function { FUNCTION_ITEMS },
function { FUNCTION_ITEMS }
<u1,v1>, <u2,v2>
[contained_by { SPHERE | BOX }]
[max_gradient FLOAT_VALUE]
[accuracy FLOAT_VALUE]
[precompute DEPTH, VarList]
}
</pre>
<p>The default values are:</p>
<pre>
accuracy : 0.001
contained_by : box {<-1,-1,-1>, <1,1,1>}
</pre>
<ol>
<li>The first function calculates the <code>x</code> value of the surface, the second <code>y</code> and the third the <code>z</code> value. Any function that results in a float is allowed.</li>
<li>The <code><u1,v1></code> and the <code><u2,v2></code> boundaries of the <code>(u,v)</code> space, in which the surface <em>has</em> to be calculated.</li>
<li>The <code>contained_by</code> <em>object</em> limits the area where POV-Ray samples for the surface of the function. The container can either be a <code>sphere</code> or a <code>box</code>.</li>
<li>The <code>max_gradient</code> is the maximum magnitude of all six partial derivatives over the specified ranges of u and v.</li>
<ol type="a">
<li>Take <code>dx/du</code>, <code>dx/dv</code>, <code>dy/du</code>, <code>dy/dv</code>, <code>dz/du</code>, and <code>dz/dv</code> and calculate them over the entire range</li>
<li>The <code>max_gradient</code> should be at least the maximum (absolute value) of all of those values.</li>
<li>Choosing a <em>too small</em> of a value will create holes or artifacts in the object.</li>
</ol>
<li>For <code>accuracy</code> smaller values produces more accurate surfaces, but take longer to render.</li>
<li>Using <code>precompute</code> can speedup the rendering of parametric surfaces by simply dividing the parametric surfaces into smaller ones</li>
<ol type="a">
<li>The maximum value for <em>DEPTH</em> is 20. High values of depth can produce arrays that use a lot of memory, take longer to parse and render.</li>
<li>It <em>precomputes</em> the ranges for the <em>VarList</em> variables (x,y,z)</li>
<li>If you declare a <code>parametric</code> surface using <code>precompute</code> and then use it twice, all arrays are in memory only once.</li>
</ol>
</ol>
<p>Example, a unit sphere:</p>
<pre>
parametric {
function { sin(u)*cos(v) }
function { sin(u)*sin(v) }
function { cos(u) }
<0,0>, <2*pi,pi>
contained_by { sphere{0, 1.1} }
max_gradient ??
accuracy 0.0001
precompute 10 x,y,z
pigment {rgb 1}
}
</pre></div>
<a name="r3_5_1_1_12"></a>
<div class="content-level-h5" contains="Prism" id="r3_5_1_1_12">
<h5>3.5.1.1.12 Prism</h5>
<p>The <code>prism</code> is an object generated by specifying one or more two-dimensional, closed curves in the x-z plane and sweeping them along y axis. These curves are defined by a set of points which are connected by linear, quadratic, cubic or bezier splines.</p>
<p>The syntax for the prism is:</p>
<pre>
PRISM:
prism {
[PRISM_ITEMS...] Height_1, Height_2, Number_Of_Points,
<Point_1>, <Point_2>, ... <Point_n>
[ open ] [PRISM_MODIFIERS...]
}
PRISM_ITEM:
linear_spline | quadratic_spline | cubic_spline |
bezier_spline | linear_sweep | conic_sweep
PRISM_MODIFIER:
sturm | OBJECT_MODIFIER
</pre>
<p>Prism default values:</p>
<pre>
SPLINE_TYPE : linear_spline
SWEEP_TYPE : linear_sweep
sturm : off
</pre>
<p>The first items specify the spline type and sweep type. The defaults if none is specified is <code>linear_spline</code> and <code> linear_sweep</code>. This is followed by two float values <em><code>Height_1</code></em> and <em> <code>Height_2</code></em> which are the y
coordinates of the top and bottom of the prism. This is followed by a float value specifying the number of 2-D points you will use to define the prism.This includes all control points needed for quadratic, cubic and bezier splines. This is followed by the specified number of 2-D vectors which define the shape in the x-z plane.</p>
<p>The interpretation of the points depends on the spline type. The prism object allows you to use any number of sub-prisms inside one prism statement, they are of the same spline and sweep type. Wherever an even number of sub-prisms overlaps a hole appears.</p>
<p class="Note"><strong>Note:</strong> You need not have multiple sub-prisms and they need not overlap as in the following examples.</p>
<p>In the <code>linear_spline</code> the first point specified is the start of the first sub-prism. The following points are connected by straight lines. If you specify a value identical to the first point, this closes the sub-prism and next point starts a new one. When you specify the value of that sub-prism's start, then it is closed. Each of the sub-prisms has to be closed by repeating the first point of a sub-prism at the end of the sub-prism's point sequence.</p>
<p>In the following example, there are two rectangular sub-prisms nested inside each other to create a frame.</p>
<pre>
prism {
linear_spline
0, 1, 10,
<0,0>, <6,0>, <6,8>, <0,8>, <0,0>, //outer rim
<1,1>, <5,1>, <5,7>, <1,7>, <1,1> //inner rim
}
</pre>
<p>The last sub-prism of a linear spline prism is automatically closed, just like the last sub-polygon in the polygon statement, if the first and last point of the sub-polygon's point sequence are not the same. This makes it very easy to convert between polygons and prisms. Quadratic, cubic and bezier splines are <em>never</em> automatically closed.</p>
<p>In the <code> quadratic_spline</code>, each sub-prism needs an additional control point at the beginning of each sub-prisms point sequence to determine the slope at the start of the curve. The first point specified is the control point which is not actually part of the spline. The second point is the start of the spline. The sub-prism ends when this second point is duplicated. The next point is the control point of the next sub-prism. The point after that is the first point of the second sub-prism.</p>
<p>Here is an example:</p>
<pre>
prism {
quadratic_spline
0, 1, 12,
<1,-1>, <0,0>, <6,0>, //outer rim; <1,-1> is control point and
<6,8>, <0,8>, <0,0>, //<0,0> is first & last point
<2,0>, <1,1>, <5,1>, //inner rim; <2,0> is control point and
<5,7>, <1,7>, <1,1> //<1,1> is first & last point
}
</pre>
<p>In the <code>cubic_spline</code>, each sub-prism needs two additional control points, one at the beginning of each sub-prisms point sequence to determine the slope at the start of the curve and one at the end. The first point specified is the control point which is not actually part of the spline. The second point is the start of the spline. The sub-prism ends when this second point is duplicated. The next point is the control point of the end of the first sub-prism. Next is the beginning control point of the next sub-prism. The point after that is the first point of the second sub-prism.</p>
<p>Here is an example:</p>
<pre>
prism {
cubic_spline
0, 1, 14,
<1,-1>, <0,0>, <6,0>, //outer rim; First control is <1,-1> and
<6,8>, <0,8>, <0,0>, //<0,0> is first & last point.
<-1,1>, //Last control of first spline is <-1,1>
<2,0>, <1,1>, <5,1>, //inner rim; First control is <2,0> and
<5,7>, <1,7>, <1,1>, //<1,1> is first & last point
<0,2> //Last control of first spline is <0,2>
}
</pre>
<p>The <code>bezier_spline</code> is an alternate kind of cubic spline. Points 1 and 4 specify the end points of a segment and points 2 and 3 are control points which specify the slope at the endpoints. Points 2 and 3 do not actually lie on the spline. They adjust the slope of the spline. If you draw an imaginary line between point 1 and 2, it represents the slope at point 1. It is a line tangent to the curve at point 1. The greater the distance between 1 and 2, the flatter the curve. With a short tangent the spline can bend more. The same holds true for control point 3 and endpoint 4. If you want the spline to be smooth between segments, point 3 and 4 on one segment and point 1 and 2 on the next segment must form a straight line and point 4 of one segment must be the same as point one on the next segment.</p>
<p>By default linear sweeping is used to create the prism, that is, the prism's walls are perpendicular to the x-z plane. The size of the curve does not change during the sweep. You can also use <code>conic_sweep</code> that leads to a prism with cone-like walls by scaling the curve down during the sweep.</p>
<p>Like cylinders the prism is normally closed. You can remove the caps on the prism by using the <code>open</code> keyword. If you do, you should not use it in CSG operations, because the result may not be as expected.</p>
<p>The surface normal determination for prism sides depends upon the order in which the splines points are specified. Prism ends have normals which face outward at one end and inward at the other end. For <code>interior_texture</code> to work as expected, the outline of the underlying 2D shape must be specified in counter-clockwise order, except for holes which must be specified in clockwise order, and they must not self-intersect.</p>
<p>The following code will render sides with the color Red on the outside and the color Blue on the inside.</p>
<pre>
#declare Prism_InitialOrder = prism {
linear_spline
linear_sweep
0,1,5,
<0.5,-0.5>,<0.5,0.5>,<0.3,0.5>,<0.3,-0.5>,<0.5,-0.5>
texture { pigment { Red } }
interior_texture { pigment { Blue } }
}
</pre>
<p>While the following example will render sides inside-out with the color Blue on the outside and the color Red on the inside. Surface normals for the prism ends are unchanged.</p>
<pre>
#declare Prism_ReverseOrder = prism {
linear_spline
linear_sweep
0,1,5,
<0.5,-0.5>,<0.3,-0.5>,<0.3,0.5>,<0.5,0.5>,<0.5,-0.5>
texture { pigment { Red } }
interior_texture { pigment { Blue } }
}
</pre>
<p>The actual normal determination is more complicated for complex splines. If the surface normal is important to the visual result, it is best to check how the prism is being rendered by testing with substantially different inside and outside textures.</p>
<p>If additional accuracy is required you can add the <code><a href="r3_5.html#r3_5_1_5_11">sturm</a></code> object modifier.</p>
<p>For an explanation of the spline concept read the description for the <a href="r3_5.html#r3_5_1_1_8">Lathe</a> object.</p>
<p>See also the tutorials on <a href="t2_3.html#t2_3_1_1">Lathe</a> and <a href="t2_3.html#t2_3_1_3">Prism</a> objects.</p></div>
<a name="r3_5_1_1_13"></a>
<div class="content-level-h5" contains="Sphere" id="r3_5_1_1_13">
<h5>3.5.1.1.13 Sphere</h5>
<p>The syntax of the <code>sphere</code> object is:</p>
<pre>
SPHERE:
sphere {
<Center>, Radius
[OBJECT_MODIFIERS...]
}
</pre>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/b/b2/RefImgSphgeom.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">The geometry of a sphere.</p>
</td>
</tr>
</table>
<p>Where <em><code><Center></code></em> is a vector specifying the x,
y, z coordinates of the center of the sphere and <em><code>
Radius</code></em> is a float value specifying the radius. Spheres may be
scaled unevenly giving an ellipsoid shape.</p>
<p>
Because spheres are highly optimized they make good bounding shapes (if
manual bounding seems to be necessary).</p></div>
<a name="r3_5_1_1_14"></a>
<div class="content-level-h5" contains="Sphere Sweep" id="r3_5_1_1_14">
<h5>3.5.1.1.14 Sphere Sweep</h5>
<p>The syntax of the <code>sphere_sweep</code> object is:</p>
<pre>
SPHERE_SWEEP:
sphere_sweep {
linear_spline | b_spline | cubic_spline
NUM_OF_SPHERES,
CENTER, RADIUS,
CENTER, RADIUS,
...
CENTER, RADIUS
[tolerance DEPTH_TOLERANCE]
[OBJECT_MODIFIERS]
}
</pre>
<p>Sphere_sweep default values:</p>
<pre>
tolerance : 1.0e-6 (0.000001)
</pre>
<p>A Sphere Sweep is the envelope of a moving sphere with varying radius, or, in other words, the
space a sphere occupies during its movement along a spline.
<br>Sphere Sweeps are modeled by specifying a list of single spheres which are then interpolated.
<br>Three kinds of interpolation are supported:</p><ul>
<li><code>linear_spline</code> : Interpolating the input data with a linear function, which means
that the single spheres are connected by straight tubes.</li>
<li><code>b_spline</code> : Approximating the input data using a cubic b-spline function, which
results in a curved object.</li>
<li><code>cubic_spline</code> : Approximating the input data using a cubic spline,
which results in a curved object.</li></ul>
<p>The sphere list (center and radius of each sphere) can take as many spheres as you like to describe
the object, but you will need at least two spheres for a <code>linear_spline</code>, and four spheres
for <code>b_spline</code> or <code>cubic_spline</code>.</p>
<p>Optional: The depth tolerance that should be used for the intersection calculations. This is done by
adding the <code>tolerance</code> keyword and the desired value: the default distance is
1.0e-6 (0.000001) and should do for most sphere sweep objects.
<br>You should change this when you see dark spots on the surface of the object. These are probably
caused by an effect called <em>self-shading</em>. This means that the object casts shadows onto itself at some
points because of calculation errors. A ray tracing program usually defines the minimal distance a ray
must travel before it actually hits another (or the same) object to avoid this effect. If this distance is
chosen too small, self-shading may occur.
<br>If so, specify <code>tolerance 1.0e-4</code> or higher.</p>
<p class="Note"><strong>Note:</strong> If these dark spots remain after raising the tolerance, you might get rid of these spots by
using adaptive super-sampling (method 2) for anti-aliasing. Images look better with anti-aliasing anyway.</p>
<p class="Note"><strong>Note:</strong> The merge CSG operation is not recommended with Sphere Sweeps: there could be a small gap
between the merged objects!</p></div>
<a name="r3_5_1_1_15"></a>
<div class="content-level-h5" contains="Superquadric Ellipsoid" id="r3_5_1_1_15">
<h5>3.5.1.1.15 Superquadric Ellipsoid</h5>
<p>The <code>superellipsoid</code> object creates a shape known as a <em>
superquadric ellipsoid</em> object. It is an extension of the quadric
ellipsoid. It can be used to create boxes and cylinders with round edges and
other interesting shapes. Mathematically it is given by the equation:</p>
<table class="centered" width="465x" cellpadding="0" cellspacing="10">
<tr>
<td>
<!--<img src="ref_tex/sqemath.tex" alt="">---><img class="center" width="445px" src="images/5/5b/RefImgSqemath.png">
</td>
</tr>
<tr>
<td>
<p class="caption">Superquadric Ellipsoid Formula</p>
</td>
</tr>
</table>
<p>The values of <em><code>e</code></em> and <em><code>n</code></em>, called
the <em>east-west</em> and <em>north-south</em> exponent, determine the shape
of the superquadric ellipsoid. Both have to be greater than zero. The sphere
is given by <em>e = 1</em> and <em>n = 1</em>.</p>
<p>
The syntax of the superquadric ellipsoid is:</p>
<pre>
SUPERELLIPSOID:
superellipsoid {
<Value_E, Value_N>
[OBJECT_MODIFIERS...]
}
</pre>
<p>The 2-D vector specifies the <em><code>e</code></em> and <em><code>
n</code></em> values in the equation above. The object sits at the origin and
occupies a space about the size of a <code>
box{<-1,-1,-1>,<1,1,1>}</code>.</p>
<p>
Two useful objects are the rounded box and the rounded cylinder. These are
declared in the following way.</p>
<pre>
#declare Rounded_Box = superellipsoid { <Round, Round> }
#declare Rounded_Cylinder = superellipsoid { <1, Round> }
</pre>
<p>The roundedness value <code>Round</code> determines the roundedness of the
edges and has to be greater than zero and smaller than one. The smaller you
choose the values, the smaller and sharper the edges will get.</p>
<p>
Very small values of <em><code>e</code></em> and <em><code>n</code></em>
might cause problems with the root solver (the Sturmian root solver cannot be
used).</p></div>
<a name="r3_5_1_1_16"></a>
<div class="content-level-h5" contains="Surface of Revolution" id="r3_5_1_1_16">
<h5>3.5.1.1.16 Surface of Revolution</h5>
<p>The <code>sor</code> object is a <em>surface of revolution</em> generated by rotating the graph of a function about the y-axis. This function describes the dependence of the radius from the position on the rotation axis.</p>
<p>The syntax is:</p>
<pre>
SOR:
sor {
Number_Of_Points, <Point_1>, <Point_2>, ... <Point_n>
[ open ] [SOR_MODIFIERS...]
}
SOR_MODIFIER:
sturm | OBJECT_MODIFIER
</pre>
<p>SOR default values:</p>
<pre>
sturm : off
</pre>
<p>The float value <em><code>Number_Of_Points</code></em> specifies the number of 2-D vectors which follow. The points <em><code>
<Point_1></code></em> through <em><code><Point_n></code></em> are two-dimensional vectors consisting of the radius and the corresponding height, i.e. the position on the rotation axis. These points are smoothly connected (the curve is passing through the specified points) and rotated about the y-axis to form the SOR object. The first and last points are only used to determine the slopes of the function at the start and end point. They do not actually lie on the curve. The function used for the SOR object is similar to the splines used for the lathe object. The difference is that the SOR object is less flexible because it underlies the restrictions of any mathematical function, i.e. to any given point y on the rotation axis belongs at most one function value, i.e. one radius value. You cannot rotate closed curves with the SOR object. Also, make sure that the curve does not cross zero (y-axis) as this can result in 'less than perfect' bounding cylinders. POV-Ray will very likely fail to render large chunks of the part of the spline contained in such an interval.</p>
<p>The optional keyword <code>open</code> allows you to remove the caps on the SOR object. If you do this you should not use it with CSG because the results may be wrong.</p>
<p> The SOR object is useful for creating bottles, vases, and things like that. A simple vase could look like this:</p>
<pre>
#declare Vase = sor {
7,
<0.000000, 0.000000>
<0.118143, 0.000000>
<0.620253, 0.540084>
<0.210970, 0.827004>
<0.194093, 0.962025>
<0.286920, 1.000000>
<0.468354, 1.033755>
open
}
</pre>
<p>One might ask why there is any need for a SOR object if there is already a lathe object which is much more flexible. The reason is quite simple. The intersection test with a SOR object involves solving a cubic polynomial while the test with a lathe object requires to solve a 6th order polynomial (you need a cubic spline for the same smoothness). Since most SOR and lathe objects will have several segments this will make a great difference in speed. The roots of the 3rd order polynomial will also be more accurate and easier to find.</p>
<p>If additional accuracy is required you can add the <code><a href="r3_5.html#r3_5_1_5_11">sturm</a></code> object modifier.</p>
<p>The following explanations are for the mathematically interested reader who wants to know how the surface of revolution is calculated. Though it is not necessary to read on it might help in understanding the SOR object.</p>
<p>The function that is rotated about the y-axis to get the final SOR object is given by:</p>
<table class="centered" width="380x" cellpadding="0" cellspacing="10">
<tr>
<td>
<!--<img src="ref_tex/sormath.tex" alt="">---><img class="center" width="360px" src="images/a/af/RefImgSormath.png">
</td>
</tr>
<tr>
<td>
<p class="caption">Surface of Revolution Formula</p>
</td>
</tr>
</table>
<p>with radius <em><code>r</code></em> and height <em><code>h</code></em>. Since this is a cubic function in h it has enough flexibility to allow smooth curves.</p>
<p>The curve itself is defined by a set of n points P(i), i=0...n-1, which are interpolated using one function for every segment of the curve. A segment j, j=1...n-3, goes from point P(j) to point P(j+1) and uses points P(j-1) and P(j+2) to determine the slopes at the endpoints. If there are n points we will have n-3 segments. This means that we need at least four points to get a proper curve. The coefficients A(j), B(j), C(j) and D(j) are calculated for every segment using the equation</p>
<table class="centered" width="470x" cellpadding="0" cellspacing="10">
<tr>
<td>
<!--<img src="ref_tex/curvmath.tex" alt="">---><img class="center" width="450px" src="images/a/af/RefImgCurvmath.png">
</td>
</tr>
<tr>
<td>
<p class="caption">Curve Math</p>
</td>
</tr>
</table>
<p>Where r(j) is the radius and h(j) is the height of point P(j).</p>
<p>The figure below shows the configuration of the points P(i), the location of segment j, and the curve that is defined by this segment.</p>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/e/e7/RefImgSegmpts.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">Points on a surface of revolution.</p>
</td>
</tr>
</table></div>
<a name="r3_5_1_1_17"></a>
<div class="content-level-h5" contains="Text" id="r3_5_1_1_17">
<h5>3.5.1.1.17 Text</h5>
<p>A <code>text</code> object creates 3-D text as an extruded block letter. Currently only TrueType fonts (ttf) and TrueType Collections (ttc) are supported but the syntax allows for other font types to be added in the future. If TrueType Collections are used, the first font found in the collection will be used. The syntax is:</p>
<pre>
TEXT_OBECT:
text {
ttf "fontname.ttf/ttc" "String_of_Text"
Thickness, <Offset>
[OBJECT_MODIFIERS...]
}
</pre>
<p>Where <code>fontname.ttf</code> or <code>fontname.ttc</code> is the name of the TrueType font file. It is a quoted string literal or string expression. The string expression which follows is the actual text of the string object. It too may be a quoted string literal or string expression. See section <a href="r3_3.html#r3_3_1_9">Strings</a> for more on string expressions.</p>
<p>In version 3.7 several fonts are now <em>built-in</em>. It should be noted that this is only a preliminary solution so the benchmark program will run without installing POV-Ray. Future versions may lack this mechanism, so in scene files (other than the built-in benchmark) you should continue to reference the external font files as before. Consequently, the following <em>alternate</em> syntax is available:</p>
<pre>
TEXT_OBECT:
text {
internal Font_Number "String_of_Text"
Thickness, <Offset>
[OBJECT_MODIFIERS] }
}
</pre>
<p>Where <code><em>Font_Number</em></code> is one of the <em>integer</em> values from the list below:</p>
<ol start=0>
<li>povlogo.ttf</li>
<li>timrom.ttf</li>
<li>cyrvetic.ttf</li>
<li>crystal.ttf</li>
</ol>
<p class="Note"><strong>Note:</strong> An out of range <code><em>Font_Number</em></code> value will default to 0.</p>
<p>The text will start with the origin at the lower left, front of the first character and will extend in the +x-direction. The baseline of the text follows the x-axis and descender drop into the -y-direction. The front of the character sits in the x-y-plane and the text is extruded in the +z-direction. The front-to-back thickness is specified by the required value <em><code>
Thickness</code></em>.</p>
<p>Characters are generally sized so that 1 unit of vertical spacing is correct. The characters are about 0.5 to 0.75 units tall.</p>
<p>The horizontal spacing is handled by POV-Ray internally including any kerning information stored in the font. The required vector <em><code><Offset></code></em> defines any extra translation between each character. Normally you should specify a zero for this value. Specifying <code>0.1*x</code> would put additional 0.1 units of space between each character. Here is an example:</p>
<pre>
text {
ttf "timrom.ttf" "POV-Ray" 1, 0
pigment { Red }
}
</pre>
<p>Only printable characters are allowed in text objects. Characters such as return, line feed, tabs, backspace etc. are not supported.</p>
<p>For easy access to your fonts, set the <a href="r3_2.html#r3_2_5_3">Library_Path</a> to the directory that contains your font collection.</p></div>
<a name="r3_5_1_1_18"></a>
<div class="content-level-h5" contains="Torus" id="r3_5_1_1_18">
<h5>3.5.1.1.18 Torus</h5>
<p>A <code>torus</code> is a 4th order quartic polynomial shape that looks like a donut or inner tube. Because this shape is so useful and quartics are difficult to define, POV-Ray lets you take a short-cut and define a torus by:</p>
<pre>
TORUS:
torus {
Major, Minor [SPINDLE_MODE]
[TORUS_MODIFIER...]
}
TORUS_MODIFIER:
sturm | OBJECT_MODIFIER
SPINDLE_MODE:
difference | intersection | merge | union
</pre>
<p>Torus default values:</p>
<pre>
union
sturm : off
</pre>
<p>where <em><code>Major</code></em> is a float value giving the major radius and <em><code>Minor</code></em> is a float specifying the minor radius. The major radius extends from the center of the hole to the mid-line of the rim while the minor radius is the radius of the cross-section of the rim. The torus is centered at the origin and lies in the x-z-plane with the y-axis
sticking through the hole.</p>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/2/29/RefImgMimxrtor.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">Major and minor radius of a torus.</p>
</td>
</tr>
</table>
<p>The torus is internally bounded by two cylinders and two rings forming a thick cylinder. With this bounding cylinder the performance of the torus intersection test is vastly increased. The test for a valid torus intersection, i.e. solving a 4th order polynomial, is only performed if the bounding cylinder is hit. Thus a lot of slow root solving calculations are avoided.</p>
<p><font class="New">New</font> in version 3.8 is a torus with a minor radius greater than the major radius (aka <em>spindle torus</em>) will self-intersect in a spindle-shaped region. The behavior with respect to the spindle can be controlled by specifying either of the <code>difference</code>,<code>intersection</code>, <code>merge</code> or <code>union</code> keywords, which act similar to the corresponding CSG operations:</p>
<ul>
<li>Using the <code>difference</code> keyword, the self-intersecting portion is cut away from the torus, so that the spindle volume is considered <em>outside</em> the primitive; the spindle surface is visible (provided you cut open the torus, or make it semi-transparent).</li>
<li>Using the <code>intersection</code> keyword, the resulting shape consists of <em>only</em> the self-intersecting portion, so that only the spindle volume is considered inside the primitive, and only the spindle surface is visible.</li>
<li>Using the <code>merge</code> keyword, the surface within the self-intersecting portion is hidden, so that the spindle surface is <em>not</em> visible; the spindle volume is considered inside the primitive.</li>
<li>Using the <code>union</code> keyword, the entire torus surface remains visible and the spindle volume is considered inside the primitive (this is the default).</li>
</ul>
<p>In a non-spindle torus, the <code>intersection</code> keyword will cause a "possible parse error" warning and make the torus invisible, while the other spindle mode keywords will have no effect whatsoever.</p>
<p class="Note"><strong>Note:</strong> The <code>difference</code> spindle mode does <em>not</em> affect the behavior with respect to the
<code><a href="r3_6.html#r3_6_1_9">interior_texture</a></code> keyword. An <code>interior_texture</code> will always be applied to the side of the spindle surface facing the spindle volume.</p>
<table class="centered" width="420px" cellpadding="0" cellspacing="10">
<tr>
<td><img class="centered" width="200px" src="images/5/5c/SpindleTorusDifference.png"></td>
<td><img class="centered" width="200px" src="images/2/24/SpindleTorusIntersection.png"></td>
<td><img class="centered" width="200px" src="images/b/b1/SpindleTorusMerge.png"></td>
<td><img class="centered" width="200px" src="images/0/0c/SpindleTorusUnion.png"></td>
</tr>
<tr>
<td colspan="4"><p class="caption">cutaway view of spindle torus using the <code>difference</code>, <code>intersection</code>, <code>merge</code> and <code>union</code> mode, respectively</p></td>
</tr>
</table>
<p>If additional accuracy is required you can add the <code><a href="r3_5.html#r3_5_1_5_11">sturm</a></code> object modifier.</p></div>
<a name="r3_5_1_2"></a>
<div class="content-level-h4" contains="Finite Patch Primitives" id="r3_5_1_2">
<h4>3.5.1.2 Finite Patch Primitives</h4>
<p>There are six totally thin, finite objects which have no well-defined inside. They are <a href="r3_5.html#r3_5_1_2_1">bicubic patch</a>, <a href="r3_5.html#r3_5_1_2_2">disc</a>, <a href="r3_5.html#r3_5_1_2_7">smooth triangle</a>, <a href="r3_5.html#r3_5_1_2_6">triangle</a>, <a href="r3_5.html#r3_5_1_2_5">polygon</a>, <a href="r3_5.html#r3_5_1_2_3">mesh</a>, and <a href="r3_5.html#r3_5_1_2_4">mesh2</a>. They may be combined in CSG union, but cannot be used inside a <code>clipped_by</code> statement.</p>
<p class="Note"><strong>Note:</strong> Patch objects <em>may</em> give unexpected results when used in differences and intersections.</p>
<p>These conditions apply:</p>
<ol>
<li>Solids may be differenced from bicubic patches with the expected results.</li>
<li>Differencing a bicubic patch from a solid <em>may</em> give unexpected results.</li>
<ul>
<li>Especially if the inverse keyword is used!</li>
</ul>
<li>Intersecting a solid and a bicubic patch will give the expected results.</li>
<ul>
<li>The parts of the patch that intersect the solid object will be visible.</li>
</ul>
<li>Merging a solid and a bicubic patch will remove the parts of the bicubic patch that intersect the solid.</li>
</ol>
<p>Because these types are finite POV-Ray can use automatic bounding on them to speed up rendering time. As with all shapes they can be translated, rotated and scaled.</p></div>
<a name="r3_5_1_2_1"></a>
<div class="content-level-h5" contains="Bicubic Patch" id="r3_5_1_2_1">
<h5>3.5.1.2.1 Bicubic Patch</h5>
<p>A <code>bicubic_patch</code> is a 3D curved surface created from a mesh of
triangles. POV-Ray supports a type of bicubic patch called a <em>Bezier
patch</em>. A bicubic patch is defined as follows:</p>
<pre>
BICUBIC_PATCH:
bicubic_patch {
PATCH_ITEMS...
<Point_1>,<Point_2>,<Point_3>,<Point_4>,
<Point_5>,<Point_6>,<Point_7>,<Point_8>,
<Point_9>,<Point_10>,<Point_11>,<Point_12>,
<Point_13>,<Point_14>,<Point_15>,<Point_16>
[OBJECT_MODIFIERS...]
}
PATCH_ITEMS:
type Patch_Type | u_steps Num_U_Steps | v_steps Num_V_Steps |
flatness Flatness
</pre>
<p>Bicubic patch default values:</p>
<pre>
flatness : 0.0
u_steps : 0
v_steps : 0
</pre>
<p>The keyword <code>type</code> is followed by a float <em><code>
Patch_Type</code></em> which currently must be either 0 or 1. For type 0 only
the control points are retained within POV-Ray. This means that a minimal
amount of memory is needed but POV-Ray will need to perform many extra
calculations when trying to render the patch. Type 1 preprocesses the patch
into many subpatches. This results in a significant speedup in rendering at
the cost of memory.</p>
<p>The four parameters <code>type</code>, <code>flatness</code>, <code>
u_steps</code> and <code>v_steps</code> may appear in any order. Only
<code>type</code> is required. They are followed by 16 vectors (4 rows
of 4) that define the x, y, z coordinates of the 16 control points which
define the patch. The patch touches the four corner points <em><code>
<Point_1></code></em>, <em><code><Point_4></code></em>, <em>
<code><Point_13></code></em> and <em> <code>
<Point_16></code></em> while the other 12 points pull and stretch the
patch into shape. The Bezier surface is enclosed by the convex hull formed by
the 16 control points, this is known as the <em>convex hull property</em>.</p>
<p>The keywords <code>u_steps</code> and <code>v_steps</code> are each followed
by integer values which tell how many rows and columns of triangles are the
minimum to use to create the surface, both default to 0. The maximum number of individual pieces
of the patch that are tested by POV-Ray can be calculated from the following:
<em>pieces = 2^u_steps * 2^v_steps</em>.</p>
<p>This means that you really should keep <code>u_steps</code> and <code>
v_steps</code> under 4. Most patches look just fine with <code>u_steps
3</code> and <code>v_steps 3</code>, which translates to 64 sub-patches (128
smooth triangles).</p>
<p>As POV-Ray processes the Bezier patch it makes a test of the current piece
of the patch to see if it is flat enough to just pretend it is a rectangle.
The statement that controls this test is specified with the <code>
flatness</code> keyword followed by a float. Typical flatness values range
from 0 to 1 (the lower the slower). The default if none is specified is
0.0.</p>
<p>If the value for flatness is 0 POV-Ray will always subdivide the patch to
the extend specified by <code>u_steps</code> and <code>v_steps</code>. If
flatness is greater than 0 then every time the patch is split, POV-Ray will
check to see if there is any need to split further.</p>
<p>There are both advantages and disadvantages to using a non-zero flatness.
The advantages include:</p>
<p>- If the patch is not very curved, then this will be detected and
POV-Ray will not waste a lot of time looking at the wrong pieces.</p>
<p>- If the patch is only highly curved in a couple of places, POV-Ray will keep
subdividing there and concentrate its efforts on the hard part.</p>
<p>The biggest disadvantage is that if POV-Ray stops subdividing at a
particular level on one part of the patch and at a different level on an
adjacent part of the patch there is the potential for cracking. This is
typically visible as spots within the patch where you can see through. How
bad this appears depends very highly on the angle at which you are viewing
the patch.</p>
<p>
Like triangles, the bicubic patch is not meant to be generated by hand. These
shapes should be created by a special utility. You may be able to acquire
utilities to generate these shapes from the same source from which you
obtained POV-Ray. Here is an example:</p>
<pre>
bicubic_patch {
type 0
flatness 0.01
u_steps 4
v_steps 4
<0, 0, 2>, <1, 0, 0>, <2, 0, 0>, <3, 0,-2>,
<0, 1 0>, <1, 1, 0>, <2, 1, 0>, <3, 1, 0>,
<0, 2, 0>, <1, 2, 0>, <2, 2, 0>, <3, 2, 0>,
<0, 3, 2>, <1, 3, 0>, <2, 3, 0>, <3, 3, -2>
}
</pre>
<p>The triangles in a POV-Ray <code>bicubic_patch</code> are automatically
smoothed using normal interpolation but it is up to the user (or the
user's utility program) to create control points which smoothly stitch
together groups of patches.</p></div>
<a name="r3_5_1_2_2"></a>
<div class="content-level-h5" contains="Disc" id="r3_5_1_2_2">
<h5>3.5.1.2.2 Disc</h5>
<p>Another flat, finite object available with POV-Ray is the <code>
disc</code>. The disc is infinitely thin, it has no thickness. If you want a
disc with true thickness you should use a very short cylinder. A disc shape
may be defined by:</p>
<pre>
DISC:
disc {
<Center>, <Normal>, Radius [, Hole_Radius]
[OBJECT_MODIFIERS...]
}
</pre>
<p>Disc default values:</p>
<pre>
HOLE RADIUS : 0.0
</pre>
<p>The vector <em><code><Center></code></em> defines the x, y, z
coordinates of the center of the disc. The <em><code>
<Normal></code></em> vector describes its orientation by describing its
surface normal vector. This is followed by a float specifying the <em> <code>
Radius</code></em>. This may be optionally followed by another float
specifying the radius of a hole to be cut from the center of the disc.</p>
<p class="Note"><strong>Note:</strong> The inside of a disc is the inside of the plane that contains
the disc. Also note that it is not constrained by the radius of the disc.</p></div>
<a name="r3_5_1_2_3"></a>
<div class="content-level-h5" contains="Mesh" id="r3_5_1_2_3">
<h5>3.5.1.2.3 Mesh</h5>
<p>The <code>mesh</code> object can be used to efficiently store large numbers of triangles.</p>
<p>The syntax is:</p>
<pre>
MESH:
mesh {
MESH_TRIANGLE...
[MESH_MODIFIER...]
}
MESH_TRIANGLE:
triangle {
<Corner_1>, <Corner_2>, <Corner_3>
[uv_vectors <uv_Corner_1>, <uv_Corner_2>, <uv_Corner_3>]
[MESH_TEXTURE]
} |
smooth_triangle {
<Corner_1>, <Normal_1>,
<Corner_2>, <Normal_2>,
<Corner_3>, <Normal_3>
[uv_vectors <uv_Corner_1>, <uv_Corner_2>, <uv_Corner_3>]
[MESH_TEXTURE]
}
MESH_MODIFIER:
inside_vector <direction> | hierarchy [ Boolean ] |
OBJECT_MODIFIER
MESH_TEXTURE:
texture { TEXTURE_IDENTIFIER }
texture_list {
TEXTURE_IDENTIFIER TEXTURE_IDENTIFIER TEXTURE_IDENTIFIER
}
</pre>
<p>Mesh default values:</p>
<pre>
hierarchy : on
</pre>
<p>Any number of <code>triangle</code> and/or <code>smooth_triangle</code> statements can be used and each of those triangles can be individually textured by assigning a texture identifier to it. The texture has to be declared before the mesh is parsed. It is not possible to use texture definitions inside the triangle or smooth triangle statements. This is a restriction that is necessary for an efficient storage of the assigned textures. See <a href="r3_6.html#r3_6_1_8">Triangle and Smooth Triangle</a> for more information on triangles.</p>
<p>The <code>mesh</code> object can support <code>uv_mapping</code>. For this, per triangle the keyword <code>uv_vectors</code> has to be given, together with three 2D uv-vectors. Each vector specifies a location in the xy-plane from which the texture has to be mapped to the matching points of the triangle. Also see the section <a href="r3_6.html#r3_6_1_7">uv_mapping</a>.</p>
<p>The mesh's components are internally bounded by a bounding box hierarchy to speed up intersection testing. The bounding hierarchy can be turned off with the <code>hierarchy off</code> keyword. This should only be done if memory is short or the mesh consists of only a few triangles. The default is <code>hierarchy on</code>.</p>
<p>Copies of a mesh object refer to the same triangle data and thus consume very little memory. You can easily trace a hundred copies of a 10000 triangle mesh without running out of memory (assuming the first mesh fits into memory). The mesh object has two advantages over a union of triangles: it needs less memory and it is transformed faster. The memory requirements are reduced by efficiently storing the triangles vertices and normals. The parsing time for transformed meshes is reduced because only the mesh object has to be transformed and not every single triangle as it is necessary for unions.</p>
</div>
<a name="r3_5_1_2_3_1"></a>
<div class="content-level-h6" contains="Solid Mesh" id="r3_5_1_2_3_1">
<h6>3.5.1.2.3.1 Solid Mesh</h6>
<p>The triangle mesh objects <code>mesh</code> (and <code>mesh2</code>) can be used in CSG objects such as difference and intersect. Adding the <code>inside_vector</code> they do have a defined <em>inside</em>. This will only work for well-behaved meshes, which are completely closed volumes. If meshes have any holes in them, this might work, but the results are not guaranteed.</p>
<p>To determine if a point is inside a triangle mesh, POV-Ray shoots a ray from the point in some arbitrary direction. If this vector intersects an odd number of triangles, the point is inside the mesh. If it intersects an even number of triangles, the point is outside of the mesh. You can specify the direction of this vector. For example, to use <code>+z</code> as the direction, you would add the following line to the triangle mesh description (following all other mesh data, but before the object modifiers).</p>
<pre>
inside_vector <0, 0, 1>
</pre>
<p>This change does not have any effect on unions of triangles, these will still be always hollow.</p></div>
<a name="r3_5_1_2_4"></a>
<div class="content-level-h5" contains="Mesh2" id="r3_5_1_2_4">
<h5>3.5.1.2.4 Mesh2</h5>
<p>The new mesh syntax is designed for use in conversion from other file formats.</p>
<pre>
MESH2 :
mesh2{
VECTORS...
LISTS... |
INDICES... |
MESH_MODIFIERS
}
VECTORS :
vertex_vectors {
number_of_vertices,
<vertex1>, <vertex2>, ...
}|
normal_vectors {
number_of_normals,
<normal1>, <normal2>, ...
}|
uv_vectors {
number_of_uv_vectors,
<uv_vect1>, <uv_vect2>, ...
}
LISTS :
texture_list {
number_of_textures,
texture { Texture1 },
texture { Texture2 }, ...
}|
INDICES :
face_indices {
number_of_faces,
<index_a, index_b, index_c> [,texture_index [,
texture_index, texture_index]],
<index_d, index_e, index_f> [,texture_index [,
texture_index, texture_index]],
...
}|
normal_indices {
number_of_faces,
<index_a, index_b, index_c>,
<index_d, index_e, index_f>,
...
}|
uv_indices {
number_of_faces,
<index_a, index_b, index_c>,
<index_d, index_e, index_f>,
...
}
MESH_MODIFIER :
inside_vector <direction> | OBJECT_MODIFIERS
</pre>
<p>Best practices dictate that the <code>mesh2</code> object definition <em>SHOULD</em> be specified in the following order:</p>
<ul>
<li>VECTORS</li>
<li>LISTS</li>
<li>INDICES</li>
</ul>
<p>However, the actual implementation also allows the <code>texture_list</code> to be placed <em>BEFORE</em> <code>vertex_vectors</code>.</p>
<p>The <code>normal_vectors</code>, <code>uv_vectors</code>, and <code>texture_list</code> sections are optional.
If the number of normals equals the number of vertices then the normal_indices section is optional and the indexes from the <code>face_indices</code> section are used instead. Likewise for the <code>uv_indices</code> section.</p>
<p class="Note"><strong>Note:</strong> The <code>texture_list</code> section is optional <em>only</em> if <code>face_indices</code> doesn't contain any texture index values.</p>
<p>For example:</p>
<pre>
face_indices {
number_of_faces,
<index_a, index_b, index_c>,
<index_d, index_e, index_f>,
...
}
</pre>
<p class="Note"><strong>Note:</strong> The numbers of <code>uv_indices</code> must equal number of faces.</p>
<p>The indexes are <em>zero based</em>, so the first item in each list has an index of zero.</p>
</div>
<a name="r3_5_1_2_4_1"></a>
<div class="content-level-h6" contains="Smooth and Flat triangles in the same mesh" id="r3_5_1_2_4_1">
<h6>3.5.1.2.4.1 Smooth and Flat triangles in the same mesh</h6>
<p>You can specify both flat and smooth triangles in the same mesh. To do this, specify
the smooth triangles first in the <code>face_indices</code>
section, followed by the flat triangles. Then, specify normal indices (in the
<code>normal_indices</code> section) for only the
smooth triangles. Any remaining triangles that do not have normal indices associated with
them will be assumed to be flat triangles.</p>
</div>
<a name="r3_5_1_2_4_2"></a>
<div class="content-level-h6" contains="Mesh Triangle Textures" id="r3_5_1_2_4_2">
<h6>3.5.1.2.4.2 Mesh Triangle Textures</h6>
<p>To specify a texture for an individual mesh triangle, specify a single integer texture
index following the face-index vector for that triangle.</p>
<p>To specify three textures for vertex-texture interpolation, specify three integer
texture indices (separated by commas) following the face-index vector for that triangle.</p>
<p>Vertex-texture interpolation and textures for an individual triangle can be mixed in the same mesh.</p></div>
<a name="r3_5_1_2_5"></a>
<div class="content-level-h5" contains="Polygon" id="r3_5_1_2_5">
<h5>3.5.1.2.5 Polygon</h5>
<p>The <code>polygon</code> object is useful for creating rectangles, squares
and other planar shapes with more than three edges. Their syntax is:</p>
<pre>
POLYGON:
polygon {
Number_Of_Points, <Point_1> <Point_2>... <Point_n>
[OBJECT_MODIFIER...]
}
</pre>
<p>The float <em><code>Number_Of_Points</code></em> tells how many points are
used to define the polygon. The points <em><code><Point_1></code></em>
through <em><code><Point_n></code></em> describe the polygon or
polygons. A polygon can contain any number of sub-polygons, either
overlapping or not. In places where an even number of polygons overlaps a
hole appears. When you repeat the first point of a sub-polygon, it closes it
and starts a new sub-polygon's point sequence. This means that all points
of a sub-polygon are different.</p>
<p>
If the last sub-polygon is not closed a warning is issued and the program
automatically closes the polygon. This is useful because polygons imported
from other programs may not be closed, i.e. their first and last point are
not the same.</p>
<p>
All points of a polygon are three-dimensional vectors that have to lay on
the same plane. If this is not the case an error occurs. It is common to use
two-dimensional vectors to describe the polygon. POV-Ray assumes that the z
value is zero in this case.</p>
<p>
A square polygon that matches the default planar image map is simply:</p>
<pre>
polygon {
4,
<0, 0>, <0, 1>, <1, 1>, <1, 0>
texture {
finish { ambient 1 diffuse 0 }
pigment { image_map { gif "test.gif" } }
}
//scale and rotate as needed here
}
</pre>
<p>The sub-polygon feature can be used to generate complex shapes like the
letter "P", where a hole is cut into another polygon:</p>
<pre>
#declare P = polygon {
12,
<0, 0>, <0, 6>, <4, 6>, <4, 3>, <1, 3>, <1,0>, <0, 0>,
<1, 4>, <1, 5>, <3, 5>, <3, 4>, <1, 4>
}
</pre>
<p>The first sub-polygon (on the first line) describes the outer shape of the
letter "P". The second sub-polygon (on the second line) describes
the rectangular hole that is cut in the top of the letter "P". Both
rectangles are closed, i.e. their first and last points are the same.</p>
<p>
The feature of cutting holes into a polygon is based on the polygon
inside/outside test used. A point is considered to be inside a polygon if a
straight line drawn from this point in an arbitrary direction crosses an odd
number of edges, this is known as <em>Jordan's curve theorem</em>.</p>
<p>
Another very complex example showing one large triangle with three small
holes and three separate, small triangles is given below:</p>
<pre>
polygon {
28,
<0, 0> <1, 0> <0, 1> <0, 0> // large outer triangle
<.3, .7> <.4, .7> <.3, .8> <.3, .7> // small outer triangle #1
<.5, .5> <.6, .5> <.5, .6> <.5, .5> // small outer triangle #2
<.7, .3> <.8, .3> <.7, .4> <.7, .3> // small outer triangle #3
<.5, .2> <.6, .2> <.5, .3> <.5, .2> // inner triangle #1
<.2, .5> <.3, .5> <.2, .6> <.2, .5> // inner triangle #2
<.1, .1> <.2, .1> <.1, .2> <.1, .1> // inner triangle #3
}
</pre></div>
<a name="r3_5_1_2_6"></a>
<div class="content-level-h5" contains="Triangle" id="r3_5_1_2_6">
<h5>3.5.1.2.6 Triangle</h5>
<p>The <code>triangle</code> primitive is available in order to make more
complex objects than the built-in shapes will permit. Triangles are usually
not created by hand but are converted from other files or generated by
utilities. A triangle is defined by</p>
<pre>
TRIANGLE:
triangle {
<Corner_1>, <Corner_2>, <Corner_3>
[OBJECT_MODIFIER...]
}
</pre>
<p>where <em><code><Corner_n></code></em> is a vector defining the x,
y, z coordinates of each corner of the triangle.</p>
<p>
Because triangles are perfectly flat surfaces it would require extremely
large numbers of very small triangles to approximate a smooth, curved
surface. However much of our perception of smooth surfaces is dependent upon
the way light and shading is done. By artificially modifying the surface
normals we can simulate a smooth surface and hide the sharp-edged seams
between individual triangles.</p></div>
<a name="r3_5_1_2_7"></a>
<div class="content-level-h5" contains="Smooth Triangle" id="r3_5_1_2_7">
<h5>3.5.1.2.7 Smooth Triangle</h5>
<p>The <code>smooth_triangle</code> primitive is used for just such purposes.
The smooth triangles use a formula called Phong normal interpolation to
calculate the surface normal for any point on the triangle based on normal
vectors which you define for the three corners. This makes the triangle
appear to be a smooth curved surface. A smooth triangle is defined by</p>
<pre>
SMOOTH_TRIANGLE:
smooth_triangle {
<Corner_1>, <Normal_1>, <Corner_2>,
<Normal_2>, <Corner_3>, <Normal_3>
[OBJECT_MODIFIER...]
}
</pre>
<p>where the corners are defined as in regular triangles and <em><code>
<Normal_n></code></em> is a vector describing the direction of the
surface normal at each corner.</p>
<p>
These normal vectors are prohibitively difficult to compute by hand.
Therefore smooth triangles are almost always generated by utility programs.
To achieve smooth results, any triangles which share a common vertex should
have the same normal vector at that vertex. Generally the smoothed normal
should be the average of all the actual normals of the triangles which share
that point.</p>
<p>
The <code>mesh</code> object is a way to combine many <code>triangle</code>
and <code>smooth_triangle</code> objects together in a very efficient way.
See <a href="r3_5.html#r3_5_1_2_3">Mesh</a> for details.</p></div>
<a name="r3_5_1_3"></a>
<div class="content-level-h4" contains="Infinite Solid Primitives" id="r3_5_1_3">
<h4>3.5.1.3 Infinite Solid Primitives</h4>
<p>There are six polynomial primitive shapes that are possibly infinite and
do not respond to automatic bounding. They are <a href="r3_5.html#r3_5_1_3_1">plane</a>, <a href="r3_5.html#r3_5_1_3_3">cubic</a>, <a href="r3_5.html#r3_5_1_3_2">poly</a>, <a href="r3_5.html#r3_5_1_3_4">quartic</a>, <a href="r3_5.html#r3_5_1_3_5">polynomial</a>,
and <a href="r3_5.html#r3_5_1_3_6">quadric</a>. They do have a well defined inside and may be used in CSG and
inside a <code>clipped_by</code> statement. As with all shapes they can be
translated, rotated and scaled.</p></div>
<a name="r3_5_1_3_1"></a>
<div class="content-level-h5" contains="Plane" id="r3_5_1_3_1">
<h5>3.5.1.3.1 Plane</h5>
<p>The <code>plane</code> primitive is a simple way to define an infinite
flat surface. The plane is not a thin boundary or can be compared to a sheet
of paper. A plane is a solid object of infinite size that divides POV-space
in two parts, inside and outside the plane. The plane is specified as follows:</p>
<pre>
PLANE:
plane {
<Normal>, Distance
[OBJECT_MODIFIERS...]
}
</pre>
<p>The <em><code><Normal></code></em> vector defines the surface normal
of the plane. A surface normal is a vector which points up from the surface
at a 90 degree angle. This is followed by a float value that gives the
distance along the normal that the plane is from the origin (that is only
true if the normal vector has unit length; see below). For example:</p>
<pre>
plane { <0, 1, 0>, 4 }
</pre>
<p>This is a plane where straight up is defined in the positive y-direction.
The plane is 4 units in that direction away from the origin. Because most
planes are defined with surface normals in the direction of an axis you will
often see planes defined using the <code>x</code>, <code>y</code> or <code>
z</code> built-in vector identifiers. The example above could be specified
as:</p>
<pre>
plane { y, 4 }
</pre>
<p>The plane extends infinitely in the x- and z-directions. It effectively
divides the world into two pieces. By definition the normal vector points to
the outside of the plane while any points away from the vector are defined as
inside. This inside/outside distinction is important when using planes in CSG
and <code>clipped_by</code>. It is also important when using fog or
atmospheric media. If you place a camera on the "inside" half of
the world, then the fog or media will not appear. Such issues arise in any
solid object but it is more common with planes. Users typically know when
they have accidentally placed a camera inside a sphere or box but
"inside a plane" is an unusual concept. In general you can reverse the
inside/outside properties of an object by adding the object modifier <code>
inverse</code>. See <a href="r3_5.html#r3_5_1_5_5">Inverse</a> and <a href="r3_7.html#r3_7_2_1_2">Empty and Solid Objects</a> for details.</p>
<p>
A plane is called a <em>polynomial</em> shape because it is defined by a
first order polynomial equation. Given a plane:</p>
<pre>
plane { <A, B, C>, D }
</pre>
<p>it can be represented by the equation <em><code>A*x + B*y + C*z - D*sqrt(A^2 + B^2 + C^2) = 0</code></em>.</p>
<p>
Therefore our example <code>plane{y,4}</code> is actually the polynomial
equation y=4. You can think of this as a set of all x, y, z points where all
have y values equal to 4, regardless of the x or z values.</p>
<p>
This equation is a first order polynomial because each term contains only
single powers of x, y or z. A second order equation has terms like x^2, y^2,
z^2, xy, xz and yz. Another name for a 2nd order equation is a quadric
equation. Third order polys are called cubics. A 4th order equation is a
quartic. Such shapes are described in the sections below.</p></div>
<a name="r3_5_1_3_2"></a>
<div class="content-level-h5" contains="Poly" id="r3_5_1_3_2">
<h5>3.5.1.3.2 Poly</h5>
<p>Higher order polynomial surfaces may be defined by the use of a <code>poly</code> shape.</p>
<p>The syntax is</p>
<pre>
POLY:
poly {
Order, <A1, A2, A3,... An>
[POLY_MODIFIERS...]
}
POLY_MODIFIERS:
sturm | OBJECT_MODIFIER
</pre>
<p>Poly default values:</p>
<pre>
sturm : off
</pre>
<p>Where <em><code>Order</code></em> is an integer number from 2 to 35 inclusively that specifies the order of the equation. <em><code>A1, A2, ...An</code></em> are float values for the coefficients of the equation. There are <em><code>n</code></em> such terms where <em><code>n = ((Order+1)*(Order+2)*(Order+3))/6.</code></em></p>
<p>If additional accuracy is required you can add the <code><a href="r3_5.html#r3_5_1_5_11">sturm</a></code> object modifier.</p></div>
<a name="r3_5_1_3_3"></a>
<div class="content-level-h5" contains="Cubic" id="r3_5_1_3_3">
<h5>3.5.1.3.3 Cubic</h5>
<p>The <code>cubic</code> object
is an alternate way to specify 3rd order polys. Its syntax is:</p>
<pre>
CUBIC:
cubic {
<A1, A2, A3,... A20>
[POLY_MODIFIERS...]
}
</pre>
<p>If additional accuracy is required you can add the <code><a href="r3_5.html#r3_5_1_5_11">sturm</a></code> object modifier.</p></div>
<a name="r3_5_1_3_4"></a>
<div class="content-level-h5" contains="Quartic" id="r3_5_1_3_4">
<h5>3.5.1.3.4 Quartic</h5>
<p>Also 4th order equations may be specified with the <code>quartic</code> object.</p>
<p>Its syntax is:</p>
<pre>
QUARTIC:
quartic {
<A1, A2, A3,... A35>
[POLY_MODIFIERS...]
}
</pre>
<p>If additional accuracy is required you can add the <code><a href="r3_5.html#r3_5_1_5_11">sturm</a></code> object modifier.</p></div>
<a name="r3_5_1_3_5"></a>
<div class="content-level-h5" contains="Polynomial" id="r3_5_1_3_5">
<h5>3.5.1.3.5 Polynomial</h5>
<p>Poly, cubic and quartics are just like quadrics in that you do not have
to understand one to use one. The file <code>shapesq.inc</code> has
plenty of pre-defined quartics for you to play with.</p>
<p>For convenience an alternate syntax is available as <code>polynomial</code>. It doesn't care about the order of the coefficients, as long as you do not define them more than once, otherwise only the value of the last definition is kept. Additionally the default with all coefficients is 0, which can be especially useful typing shortcut.</p>
<p>See the <a href="t2_3.html#t2_3_3_4_3">tutorial</a> section for more examples of the simplified syntax.</p>
<pre>
POLYNOMIAL:
polynomial {
Order, [COEFFICIENTS...]
[POLY_MODIFIERS...]
}
COEFFICIENTS:
xyz(<x_power>,<y_power>,<z_power>):<value>[,]
POLY_MODIFIERS:
sturm | OBJECT_MODIFIER
</pre>
<p>Same as the torus above, but with the polynomial syntax:</p>
<pre>
// Torus having major radius sqrt(40), minor radius sqrt(12)
polynomial { 4,
xyz(4,0,0):1,
xyz(2,2,0):2,
xyz(2,0,2):2,
xyz(2,0,0):-104,
xyz(0,4,0):1,
xyz(0,2,2):2,
xyz(0,2,0):56,
xyz(0,0,4):1,
xyz(0,0,2):-104,
xyz(0,0,0):784
sturm
}
</pre>
<p>The following table shows which polynomial terms correspond to which x,y,z
factors for the orders 2 to 7. Remember <code>cubic</code> is actually a 3rd order polynomial and
<code>quartic</code> is 4th order.</p>
<table class="matte" SUMMARY="Cubic and quartic polynomial terms" width="700px">
<tr>
<!-- That this is only 99% is intentional! [trf] -->
<th width="5%"> </th>
<th width="5%">2<sup>nd</sup></th>
<th width="6%">3<sup>rd</sup></th>
<th width="7%">4<sup>th</sup></th>
<th width="8%">5<sup>th</sup></th>
<th width="9%">6<sup>th</sup></th>
<th width="10%">7<sup>th</sup></th>
<th width="6%"> </th>
<th width="8%">5<sup>th</sup></th>
<th width="9%">6<sup>th</sup></th>
<th width="10%">7<sup>th</sup></th>
<th width="6%"> </th>
<th width="5%">6<sup>th</sup></th>
<th width="5%">7<sup>th</sup></th>
</tr>
<tr>
<td>A<sub>1</sub></td>
<td>x<sup>2</sup></td>
<td>x<sup>3</sup></td>
<td>x<sup>4</sup></td>
<td>x<sup>5</sup></td>
<td>x<sup>6</sup></td>
<td>x<sup>7</sup></td>
<td>A<sub>41</sub></td>
<td>y<sup>3</sup></td>
<td>xy<sup>3</sup></td>
<td>x<sup>2</sup>y<sup>3</sup></td>
<td>A<sub>81</sub></td>
<td>z<sup>3</sup></td>
<td>xz<sup>3</sup></td>
</tr>
<tr>
<td>A<sub>2</sub></td>
<td>xy</td>
<td>x<sup>2</sup>y</td>
<td>x<sup>3</sup>y</td>
<td>x<sup>4</sup>y</td>
<td>x<sup>5</sup>y</td>
<td>x<sup>6</sup>y</td>
<td>A<sub>42</sub></td>
<td>y<sup>2</sup>z<sup>3</sup></td>
<td>xy<sup>2</sup>z<sup>3</sup></td>
<td>x<sup>2</sup>y<sup>2</sup>z<sup>3</sup></td>
<td>A<sub>82</sub></td>
<td>z<sup>2</sup></td>
<td>xz<sup>2</sup></td>
</tr>
<tr>
<td>A<sub>3</sub></td>
<td>xz</td>
<td>x<sup>2</sup>z</td>
<td>x<sup>3</sup>z</td>
<td>x<sup>4</sup>z</td>
<td>x<sup>5</sup>z</td>
<td>x<sup>6</sup>z</td>
<td>A<sub>43</sub></td>
<td>y<sup>2</sup>z<sup>2</sup></td>
<td>xy<sup>2</sup>z<sup>2</sup></td>
<td>x<sup>2</sup>y<sup>2</sup>z<sup>2</sup></td>
<td>A<sub>83</sub></td>
<td>z</td>
<td>xz</td>
</tr>
<tr>
<td>A<sub>4</sub></td>
<td>x</td>
<td>x<sup>2</sup></td>
<td>x<sup>3</sup></td>
<td>x<sup>4</sup></td>
<td>x<sup>5</sup></td>
<td>x<sup>6</sup></td>
<td>A<sub>44</sub></td>
<td>y<sup>2</sup>z</td>
<td>xy<sup>2</sup>z</td>
<td>x<sup>2</sup>y<sup>2</sup>z</td>
<td>A<sub>84</sub></td>
<td>1</td>
<td>x</td>
</tr>
<tr>
<td>A<sub>5</sub></td>
<td>y<sup>2</sup></td>
<td>xy<sup>2</sup></td>
<td>x<sup>2</sup>y<sup>2</sup></td>
<td>x<sup>3</sup>y<sup>2</sup></td>
<td>x<sup>4</sup>y<sup>2</sup></td>
<td>x<sup>5</sup>y<sup>2</sup></td>
<td>A<sub>45</sub></td>
<td>y<sup>2</sup></td>
<td>xy<sup>2</sup></td>
<td>x<sup>2</sup>y<sup>2</sup></td>
<td>A<sub>85</sub></td>
<td> </td>
<td>y<sup>7</sup></td>
</tr>
<tr>
<td>A<sub>6</sub></td>
<td>yz</td>
<td>xyz</td>
<td>x<sup>2</sup>yz</td>
<td>x<sup>3</sup>yz</td>
<td>x<sup>4</sup>yz</td>
<td>x<sup>5</sup>yz</td>
<td>A<sub>46</sub></td>
<td>yz<sup>4</sup></td>
<td>xyz<sup>4</sup></td>
<td>x<sup>2</sup>yz<sup>4</sup></td>
<td>A<sub>86</sub></td>
<td> </td>
<td>y<sup>6</sup>z</td>
</tr>
<tr>
<td>A<sub>7</sub></td>
<td>y</td>
<td>xy</td>
<td>x<sup>2</sup>y</td>
<td>x<sup>3</sup>y</td>
<td>x<sup>4</sup>y</td>
<td>x<sup>5</sup>y</td>
<td>A<sub>47</sub></td>
<td>yz<sup>3</sup></td>
<td>xyz<sup>3</sup></td>
<td>x<sup>2</sup>yz<sup>3</sup></td>
<td>A<sub>87</sub></td>
<td> </td>
<td>y<sup>6</sup></td>
</tr>
<tr>
<td>A<sub>8</sub></td>
<td>z<sup>2</sup></td>
<td>xz<sup>2</sup></td>
<td>x<sup>2</sup>z<sup>2</sup></td>
<td>x<sup>3</sup>z<sup>2</sup></td>
<td>x<sup>4</sup>z<sup>2</sup></td>
<td>x<sup>5</sup>z<sup>2</sup></td>
<td>A<sub>48</sub></td>
<td>yz<sup>2</sup></td>
<td>xyz<sup>2</sup></td>
<td>x<sup>2</sup>yz<sup>2</sup></td>
<td>A<sub>88</sub></td>
<td> </td>
<td>y<sup>5</sup>z<sup>2</sup></td>
</tr>
<tr>
<td>A<sub>9</sub></td>
<td>z</td>
<td>xz</td>
<td>x<sup>2</sup>z</td>
<td>x<sup>3</sup>z</td>
<td>x<sup>4</sup>z</td>
<td>x<sup>5</sup>z</td>
<td>A<sub>49</sub></td>
<td>yz</td>
<td>xyz</td>
<td>x<sup>2</sup>yz</td>
<td>A<sub>89</sub></td>
<td> </td>
<td>y<sup>5</sup>z</td>
</tr>
<tr>
<td>A<sub>10</sub></td>
<td>1</td>
<td>x</td>
<td>x<sup>2</sup></td>
<td>x<sup>3</sup></td>
<td>x<sup>4</sup></td>
<td>x<sup>5</sup></td>
<td>A<sub>50</sub></td>
<td>y</td>
<td>xy</td>
<td>x<sup>2</sup>y</td>
<td>A<sub>90</sub></td>
<td> </td>
<td>y<sup>5</sup></td>
</tr>
<tr>
<td>A<sub>11</sub></td>
<td> </td>
<td>y<sup>3</sup></td>
<td>xy<sup>3</sup></td>
<td>x<sup>2</sup>y<sup>3</sup></td>
<td>x<sup>3</sup>y<sup>3</sup></td>
<td>x<sup>4</sup>y<sup>3</sup></td>
<td>A<sub>51</sub></td>
<td>z<sup>5</sup></td>
<td>xz<sup>5</sup></td>
<td>x<sup>2</sup>z<sup>5</sup></td>
<td>A<sub>91</sub></td>
<td> </td>
<td>y<sup>4</sup>z<sup>3</sup></td>
</tr>
<tr>
<td>A<sub>12</sub></td>
<td> </td>
<td>y<sup>2</sup>z</td>
<td>xy<sup>2</sup>z</td>
<td>x<sup>2</sup>y<sup>2</sup>z</td>
<td>x<sup>3</sup>y<sup>2</sup>z</td>
<td>x<sup>4</sup>y<sup>2</sup>z</td>
<td>A<sub>52</sub></td>
<td>z<sup>4</sup></td>
<td>xz<sup>4</sup></td>
<td>x<sup>2</sup>z<sup>4</sup></td>
<td>A<sub>92</sub></td>
<td> </td>
<td>y<sup>4</sup>z<sup>2</sup></td>
</tr>
<tr>
<td>A<sub>13</sub></td>
<td> </td>
<td>y<sup>2</sup></td>
<td>xy<sup>2</sup></td>
<td>x<sup>2</sup>y<sup>2</sup></td>
<td>x<sup>3</sup>y<sup>2</sup></td>
<td>x<sup>4</sup>y<sup>2</sup></td>
<td>A<sub>53</sub></td>
<td>z<sup>3</sup></td>
<td>xz<sup>3</sup></td>
<td>x<sup>2</sup>z<sup>3</sup></td>
<td>A<sub>93</sub></td>
<td> </td>
<td>y<sup>4</sup>z</td>
</tr>
<tr>
<td>A<sub>14</sub></td>
<td> </td>
<td>yz<sup>2</sup></td>
<td>xyz<sup>2</sup></td>
<td>x<sup>2</sup>yz<sup>2</sup></td>
<td>x<sup>3</sup>yz<sup>2</sup></td>
<td>x<sup>4</sup>yz<sup>2</sup></td>
<td>A<sub>54</sub></td>
<td>z<sup>2</sup></td>
<td>xz<sup>2</sup></td>
<td>x<sup>2</sup>z<sup>2</sup></td>
<td>A<sub>94</sub></td>
<td> </td>
<td>y<sup>4</sup></td>
</tr>
<tr>
<td>A<sub>15</sub></td>
<td> </td>
<td>yz</td>
<td>xyz</td>
<td>x<sup>2</sup>yz</td>
<td>x<sup>3</sup>yz</td>
<td>x<sup>4</sup>yz</td>
<td>A<sub>55</sub></td>
<td>z</td>
<td>xz</td>
<td>x<sup>2</sup>z</td>
<td>A<sub>95</sub></td>
<td> </td>
<td>y<sup>3</sup>z<sup>4</sup></td>
</tr>
<tr>
<td>A<sub>16</sub></td>
<td> </td>
<td>y</td>
<td>xy</td>
<td>x<sup>2</sup>y</td>
<td>x<sup>3</sup>y</td>
<td>x<sup>4</sup>y</td>
<td>A<sub>56</sub></td>
<td>1</td>
<td>x</td>
<td>x<sup>2</sup></td>
<td>A<sub>96</sub></td>
<td> </td>
<td>y<sup>3</sup>z<sup>3</sup></td>
</tr>
<tr>
<td>A<sub>17</sub></td>
<td> </td>
<td>z<sup>3</sup></td>
<td>xz<sup>3</sup></td>
<td>x<sup>2</sup>z<sup>3</sup></td>
<td>x<sup>3</sup>z<sup>3</sup></td>
<td>x<sup>4</sup>z<sup>3</sup></td>
<td>A<sub>57</sub></td>
<td> </td>
<td>y<sup>6</sup></td>
<td>xy<sup>6</sup></td>
<td>A<sub>97</sub></td>
<td> </td>
<td>y<sup>3</sup>z<sup>2</sup></td>
</tr>
<tr>
<td>A<sub>18</sub></td>
<td> </td>
<td>z<sup>2</sup></td>
<td>xz<sup>2</sup></td>
<td>x<sup>2</sup>z<sup>2</sup></td>
<td>x<sup>3</sup>z<sup>2</sup></td>
<td>x<sup>4</sup>z<sup>2</sup></td>
<td>A<sub>58</sub></td>
<td> </td>
<td>y<sup>5</sup>z</td>
<td>xy<sup>5</sup>z</td>
<td>A<sub>98</sub></td>
<td> </td>
<td>y<sup>3</sup>z</td>
</tr>
<tr>
<td>A<sub>19</sub></td>
<td> </td>
<td>z</td>
<td>xz</td>
<td>x<sup>2</sup>z</td>
<td>x<sup>3</sup>z</td>
<td>x<sup>4</sup>z</td>
<td>A<sub>59</sub></td>
<td> </td>
<td>y<sup>5</sup></td>
<td>xy<sup>5</sup></td>
<td>A<sub>99</sub></td>
<td> </td>
<td>y<sup>3</sup></td>
</tr>
<tr>
<td>A<sub>20</sub></td>
<td> </td>
<td>1</td>
<td>x</td>
<td>x<sup>2</sup></td>
<td>x<sup>3</sup></td>
<td>x<sup>4</sup></td>
<td>A<sub>60</sub></td>
<td> </td>
<td>y<sup>4</sup>z<sup>2</sup></td>
<td>xy<sup>4</sup>z<sup>2</sup></td>
<td>A<sub>100</sub></td>
<td> </td>
<td>y<sup>2</sup>z<sup>5</sup></td>
</tr>
<tr>
<td>A<sub>21</sub></td>
<td> </td>
<td> </td>
<td>y<sup>4</sup></td>
<td>xy<sup>4</sup></td>
<td>x<sup>2</sup>y<sup>4</sup></td>
<td>x<sup>3</sup>y<sup>4</sup></td>
<td>A<sub>61</sub></td>
<td> </td>
<td>y<sup>4</sup>z</td>
<td>xy<sup>4</sup>z</td>
<td>A<sub>101</sub></td>
<td> </td>
<td>y<sup>2</sup>z<sup>4</sup></td>
</tr>
<tr>
<td>A<sub>22</sub></td>
<td> </td>
<td> </td>
<td>y<sup>3</sup>z</td>
<td>xy<sup>3</sup>z</td>
<td>x<sup>2</sup>y<sup>3</sup>z</td>
<td>x<sup>3</sup>y<sup>3</sup>z</td>
<td>A<sub>62</sub></td>
<td> </td>
<td>y<sup>4</sup></td>
<td>xy<sup>4</sup></td>
<td>A<sub>102</sub></td>
<td> </td>
<td>y<sup>2</sup>z<sup>3</sup></td>
</tr>
<tr>
<td>A<sub>23</sub></td>
<td> </td>
<td> </td>
<td>y<sup>3</sup></td>
<td>xy<sup>3</sup></td>
<td>x<sup>2</sup>y<sup>3</sup></td>
<td>x<sup>3</sup>y<sup>3</sup></td>
<td>A<sub>63</sub></td>
<td> </td>
<td>y<sup>3</sup>z<sup>3</sup></td>
<td>xy<sup>3</sup>z<sup>3</sup></td>
<td>A<sub>103</sub></td>
<td> </td>
<td>y<sup>2</sup>z<sup>2</sup></td>
</tr>
<tr>
<td>A<sub>24</sub></td>
<td> </td>
<td> </td>
<td>y<sup>2</sup>z<sup>2</sup></td>
<td>xy<sup>2</sup>z<sup>2</sup></td>
<td>x<sup>2</sup>y<sup>2</sup>z<sup>2</sup></td>
<td>x<sup>3</sup>y<sup>2</sup>z<sup>2</sup></td>
<td>A<sub>64</sub></td>
<td> </td>
<td>y<sup>3</sup>z<sup>2</sup></td>
<td>xy<sup>3</sup>z<sup>2</sup></td>
<td>A<sub>104</sub></td>
<td> </td>
<td>y<sup>2</sup>z</td>
</tr>
<tr>
<td>A<sub>25</sub></td>
<td> </td>
<td> </td>
<td>y<sup>2</sup>z</td>
<td>xy<sup>2</sup>z</td>
<td>x<sup>2</sup>y<sup>2</sup>z</td>
<td>x<sup>3</sup>y<sup>2</sup>z</td>
<td>A<sub>65</sub></td>
<td> </td>
<td>y<sup>3</sup>z</td>
<td>xy<sup>3</sup>z</td>
<td>A<sub>105</sub></td>
<td> </td>
<td>y<sup>2</sup></td>
</tr>
<tr>
<td>A<sub>26</sub></td>
<td> </td>
<td> </td>
<td>y<sup>2</sup></td>
<td>xy<sup>2</sup></td>
<td>x<sup>2</sup>y<sup>2</sup></td>
<td>x<sup>3</sup>y<sup>2</sup></td>
<td>A<sub>66</sub></td>
<td> </td>
<td>y<sup>3</sup></td>
<td>xy<sup>3</sup></td>
<td>A<sub>106</sub></td>
<td> </td>
<td>yz<sup>6</sup></td>
</tr>
<tr>
<td>A<sub>27</sub></td>
<td> </td>
<td> </td>
<td>yz<sup>3</sup></td>
<td>xyz<sup>3</sup></td>
<td>x<sup>2</sup>yz<sup>3</sup></td>
<td>x<sup>3</sup>yz<sup>3</sup></td>
<td>A<sub>67</sub></td>
<td> </td>
<td>y<sup>2</sup>z<sup>4</sup></td>
<td>xy<sup>2</sup>z<sup>4</sup></td>
<td>A<sub>107</sub></td>
<td> </td>
<td>yz<sup>5</sup></td>
</tr>
<tr>
<td>A<sub>28</sub></td>
<td> </td>
<td> </td>
<td>yz<sup>2</sup></td>
<td>xyz<sup>2</sup></td>
<td>x<sup>2</sup>yz<sup>2</sup></td>
<td>x<sup>3</sup>yz<sup>2</sup></td>
<td>A<sub>68</sub></td>
<td> </td>
<td>y<sup>2</sup>z<sup>3</sup></td>
<td>xy<sup>2</sup>z<sup>3</sup></td>
<td>A<sub>108</sub></td>
<td> </td>
<td>yz<sup>4</sup></td>
</tr>
<tr>
<td>A<sub>29</sub></td>
<td> </td>
<td> </td>
<td>yz</td>
<td>xyz</td>
<td>x<sup>2</sup>yz</td>
<td>x<sup>3</sup>yz</td>
<td>A<sub>69</sub></td>
<td> </td>
<td>y<sup>2</sup>z<sup>2</sup></td>
<td>xy<sup>2</sup>z<sup>2</sup></td>
<td>A<sub>109</sub></td>
<td> </td>
<td>yz<sup>3</sup></td>
</tr>
<tr>
<td>A<sub>30</sub></td>
<td> </td>
<td> </td>
<td>y</td>
<td>xy</td>
<td>x<sup>2</sup>y</td>
<td>x<sup>3</sup>y</td>
<td>A<sub>70</sub></td>
<td> </td>
<td>y<sup>2</sup>z</td>
<td>xy<sup>2</sup>z</td>
<td>A<sub>110</sub></td>
<td> </td>
<td>yz<sup>2</sup></td>
</tr>
<tr>
<td>A<sub>31</sub></td>
<td> </td>
<td> </td>
<td>z<sup>4</sup></td>
<td>xz<sup>4</sup></td>
<td>x<sup>2</sup>z<sup>4</sup></td>
<td>x<sup>3</sup>z<sup>4</sup></td>
<td>A<sub>71</sub></td>
<td> </td>
<td>y<sup>2</sup></td>
<td>xy<sup>2</sup></td>
<td>A<sub>111</sub></td>
<td> </td>
<td>yz</td>
</tr>
<tr>
<td>A<sub>32</sub></td>
<td> </td>
<td> </td>
<td>z<sup>3</sup></td>
<td>xz<sup>3</sup></td>
<td>x<sup>2</sup>z<sup>3</sup></td>
<td>x<sup>3</sup>z<sup>3</sup></td>
<td>A<sub>72</sub></td>
<td> </td>
<td>yz<sup>5</sup></td>
<td>xyz<sup>5</sup></td>
<td>A<sub>112</sub></td>
<td> </td>
<td>y</td>
</tr>
<tr>
<td>A<sub>33</sub></td>
<td> </td>
<td> </td>
<td>z<sup>2</sup></td>
<td>xz<sup>2</sup></td>
<td>x<sup>2</sup>z<sup>2</sup></td>
<td>x<sup>3</sup>z<sup>2</sup></td>
<td>A<sub>73</sub></td>
<td> </td>
<td>yz<sup>4</sup></td>
<td>xyz<sup>4</sup></td>
<td>A<sub>113</sub></td>
<td> </td>
<td>z<sup>7</sup></td>
</tr>
<tr>
<td>A<sub>34</sub></td>
<td> </td>
<td> </td>
<td>z</td>
<td>xz</td>
<td>x<sup>2</sup>z</td>
<td>x<sup>3</sup>z</td>
<td>A<sub>74</sub></td>
<td> </td>
<td>yz<sup>3</sup></td>
<td>xyz<sup>3</sup></td>
<td>A<sub>114</sub></td>
<td> </td>
<td>z<sup>6</sup></td>
</tr>
<tr>
<td>A<sub>35</sub></td>
<td> </td>
<td> </td>
<td>1</td>
<td>x</td>
<td>x<sup>2</sup></td>
<td>x<sup>3</sup></td>
<td>A<sub>75</sub></td>
<td> </td>
<td>yz<sup>2</sup></td>
<td>xyz<sup>2</sup></td>
<td>A<sub>115</sub></td>
<td> </td>
<td>z<sup>5</sup></td>
</tr>
<tr>
<td>A<sub>36</sub></td>
<td> </td>
<td> </td>
<td> </td>
<td>y<sup>5</sup></td>
<td>xy<sup>5</sup></td>
<td>x<sup>2</sup>y<sup>5</sup></td>
<td>A<sub>76</sub></td>
<td> </td>
<td>yz</td>
<td>xyz</td>
<td>A<sub>116</sub></td>
<td> </td>
<td>z<sup>4</sup></td>
</tr>
<tr>
<td>A<sub>37</sub></td>
<td> </td>
<td> </td>
<td> </td>
<td>y<sup>4</sup>z</td>
<td>xy<sup>4</sup>z</td>
<td>x<sup>2</sup>y<sup>4</sup>z</td>
<td>A<sub>77</sub></td>
<td> </td>
<td>y</td>
<td>xy</td>
<td>A<sub>117</sub></td>
<td> </td>
<td>z<sup>3</sup></td>
</tr>
<tr>
<td>A<sub>38</sub></td>
<td> </td>
<td> </td>
<td> </td>
<td>y<sup>4</sup></td>
<td>xy<sup>4</sup></td>
<td>x<sup>2</sup>y<sup>4</sup></td>
<td>A<sub>78</sub></td>
<td> </td>
<td>z<sup>6</sup></td>
<td>xz<sup>6</sup></td>
<td>A<sub>118</sub></td>
<td> </td>
<td>z<sup>2</sup></td>
</tr>
<tr>
<td>A<sub>39</sub></td>
<td> </td>
<td> </td>
<td> </td>
<td>y<sup>3</sup>z<sup>2</sup></td>
<td>xy<sup>3</sup>z<sup>2</sup></td>
<td>x<sup>2</sup>y<sup>3</sup>z<sup>2</sup></td>
<td>A<sub>79</sub></td>
<td> </td>
<td>z<sup>5</sup></td>
<td>xz<sup>5</sup></td>
<td>A<sub>119</sub></td>
<td> </td>
<td>z</td>
</tr>
<tr>
<td>A<sub>40</sub></td>
<td> </td>
<td> </td>
<td> </td>
<td>y<sup>3</sup>z</td>
<td>xy<sup>3</sup>z</td>
<td>x<sup>2</sup>y<sup>3</sup>z</td>
<td>A<sub>80</sub></td>
<td> </td>
<td>z<sup>4</sup></td>
<td>xz<sup>4</sup></td>
<td>A<sub>120</sub></td>
<td> </td>
<td>1</td>
</tr>
</table>
<p>Polynomial shapes can be used to describe a large class of shapes
including the torus, the lemniscate, etc. For example, to declare a quartic
surface requires that each of the coefficients (<em><code>A1 ...
A35</code></em>) be placed in order into a single long vector of 35 terms. As an example let's define a torus the hard way. A Torus can be represented by the equation: <code>x<sup>4</sup> + y<sup>4</sup> + z<sup>4</sup> + 2 x<sup>2</sup> y<sup>2</sup> + 2 x<sup>2</sup> z<sup>2</sup> + 2 y<sup>2</sup> z<sup>2</sup> - 2 (r_02 + r_12)
x<sup>2</sup> + 2 (r_02 - r_12) y<sup>2</sup> - 2 (r_02 + r_12) z<sup>2</sup> + (r_02 - r_12)<sup>2</sup> = 0</code></p>
<p>Where r_0 is the major radius of the torus, the distance from the hole of
the donut to the middle of the ring of the donut, and r_1 is the minor radius
of the torus, the distance from the middle of the ring of the donut to the
outer surface. The following object declaration is for a torus having major
radius 6.3 minor radius 3.5 (Making the maximum width just under 20).</p>
<pre>
// Torus having major radius sqrt(40), minor radius sqrt(12)
quartic {
< 1, 0, 0, 0, 2, 0, 0, 2, 0,
-104, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 0, 2, 0, 56, 0,
0, 0, 0, 1, 0, -104, 0, 784 >
sturm
}
</pre>
<p>
Polynomial surfaces use highly complex computations and will not always render perfectly.
If the surface is not smooth, has dropouts, or extra random pixels, try using
the optional keyword <code>sturm</code> in the definition. This will cause a
slower but more accurate calculation method to be used. Usually, but not
always, this will solve the problem. If sturm does not work, try rotating
or translating the shape by some small amount.</p>
<p>
There are really so many different polynomial shapes, we cannot even
begin to list or describe them all. We suggest you find a good reference
or text book if you want to investigate the subject further.</p></div>
<a name="r3_5_1_3_6"></a>
<div class="content-level-h5" contains="Quadric" id="r3_5_1_3_6">
<h5>3.5.1.3.6 Quadric</h5>
<p>The <code>quadric</code> object can produce shapes like paraboloids (dish
shapes) and hyperboloids (saddle or hourglass shapes). It can also produce
ellipsoids, spheres, cones, and cylinders but you should use the <code>
sphere</code>, <code>cone</code>, and <code>cylinder</code> objects built
into POV-Ray because they are faster than the quadric version.</p>
<p class="Note"><strong>Note:</strong> Do not confuse "quaDRic" with
"quaRTic". A quadric is a 2nd order polynomial while a quartic
is 4th order.</p>
<p>
Quadrics render much
faster and are less error-prone but produce less complex objects. The syntax
is:</p>
<pre>
QUADRIC:
quadric {
<A,B,C>,<D,E,F>,<G,H,I>,J
[OBJECT_MODIFIERS...]
}
</pre>
<p>Although the syntax actually will parse 3 vector expressions followed by a
float, we traditionally have written the syntax as above where <em><code>
A</code></em> through <em><code>J</code></em> are float expressions. These 10
float that define a surface of x, y, z points which satisfy the equation A x<sup>2</sup>
+ B y<sup>2</sup> + C z<sup>2</sup> + D xy + E xz + F yz + G x + H y + I z + J = 0</p>
<p>Different values of <em><code>A, B, C, ... J</code></em> will give
different shapes. If you take any three dimensional point and use its x, y
and z coordinates in the above equation the answer will be 0 if the point is
on the surface of the object. The answer will be negative if the point is
inside the object and positive if the point is outside the object. Here are
some examples:</p>
<table SUMMARY="Some quartic shapes" width="100%">
<tr>
<td width="30%">X<sup>2</sup> + Y<sup>2</sup> + Z<sup>2</sup> - 1 = 0</td>
<td width="70%">Sphere</td>
</tr>
<tr>
<td>X<sup>2</sup> + Y<sup>2</sup> - 1 = 0</td>
<td>Infinite cylinder along the Z axis</td>
</tr>
<tr>
<td>X<sup>2</sup> + Y<sup>2</sup> - Z<sup>2</sup> = 0</td>
<td>Infinite cone along the Z axis</td>
</tr>
</table>
<p>The easiest way to use these shapes is to include the standard file <code>
shapes.inc</code> into your program. It contains several pre-defined quadrics
and you can transform these pre-defined shapes (using translate, rotate and
scale) into the ones you want. For a complete list, see the file <code>
shapes.inc</code>.</p></div>
<a name="r3_5_1_4"></a>
<div class="content-level-h4" contains="Constructive Solid Geometry" id="r3_5_1_4">
<h4>3.5.1.4 Constructive Solid Geometry</h4>
<p>In addition to all of the primitive shapes POV-Ray supports, you can also combine multiple simple shapes into complex shapes using <em> Constructive Solid Geometry</em> (CSG). There are four basic types of CSG operations: <a href="r3_5.html#r3_5_1_4_2">union</a>, <a href="r3_5.html#r3_5_1_4_3">intersection</a>, <a href="r3_5.html#r3_5_1_4_4">difference</a>, and <a href="r3_5.html#r3_5_1_4_5">merge</a>. CSG objects can be composed of primitives or other CSG objects to create more, and more complex shapes.</p>
</div>
<a name="r3_5_1_4_1"></a>
<div class="content-level-h5" contains="Inside and Outside" id="r3_5_1_4_1">
<h5>3.5.1.4.1 Inside and Outside</h5>
<p>Most shape primitives, like spheres, boxes and blobs divide the world into
two regions. One region is inside the object and one is outside. Given any
point in space you can say it is either inside or outside any particular
primitive object. Well, it could be exactly on the surface but this case is
rather hard to determine due to numerical problems.</p>
<p>
Even planes have an inside and an outside. By definition, the surface normal
of the plane points towards the outside of the plane. You should note that
triangles cannot be used as solid objects in CSG since they have no well defined inside and outside. Triangle-based shapes (<code>mesh</code> and <code>mesh2</code>) can only be used in CSG when they are closed objects and have an inside vector specified. </p>
<p class="Note"><strong>Note:</strong> Although the <code>triangle</code>, the <code>bicubic_patch</code> and some other shapes have no well defined inside and outside, they have a front- and backside which makes it possible to use a texture on the front side and an <code>interior_texture</code> on the back side.</p>
<p>
CSG uses the concepts of inside and outside to combine shapes together as
explained in the following sections.</p>
<p>
Imagine you have two objects that partially overlap like shown in the figure
below. Four different areas of points can be distinguished: points that are
neither in object <code>A</code> nor in object <code>B</code>, points that
are in object <code>A</code> but not in object <code>B</code>, points that
are not in object <code>A</code> but in object <code>B</code> and last not
least points that are in object <code>A</code> and object <code>B</code>.
</p>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/8/8f/RefImgObjoverl.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">Two overlapping objects.</p>
</td>
</tr>
</table>
<p>Keeping this in mind it will be quite easy to understand how the CSG
operations work.</p>
<p>
When using CSG it is often useful to invert an object so that it will be
inside-out. The appearance of the object is not changed, just the way that
POV-Ray perceives it. When the <code>inverse</code> keyword is used the <em>
inside</em> of the shape is flipped to become the <em> outside</em> and vice
versa.</p>
<p>
The inside/outside distinction is not important for a <code>union</code>, but is important for <code>intersection</code>, <code>difference</code>, and <code>merge</code>. Therefore any objects may be combined using <code>union</code> but only solid objects, i.e. objects that have a well-defined interior can be used in the other kinds of CSG. The objects described in
<a href="r3_5.html#r3_5_1_2">Finite Patch Primitives</a> have no well defined inside/outside. All objects described in the sections <a href="r3_5.html#r3_5_1_1">Finite Solid Primitives</a> and <a href="r3_5.html#r3_5_1_3">Infinite Solid Primitives</a>.</p></div>
<a name="r3_5_1_4_2"></a>
<div class="content-level-h5" contains="Union" id="r3_5_1_4_2">
<h5>3.5.1.4.2 Union</h5>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/0/0d/RefImgUnionobj.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">The union of two objects.</p>
</td>
</tr>
</table>
<p>The simplest kind of CSG is the <code>union</code>. The syntax is:</p>
<pre>
UNION:
union {
OBJECTS...
[OBJECT_MODIFIERS...]
}
</pre>
<p>Unions are simply glue used to bind two or more shapes into a single
entity that can be manipulated as a single object. The image above shows the
union of <code>A</code> and <code>B</code>. The new object created by the
union operation can be scaled, translated and rotated as a single shape. The
entire union can share a single texture but each object contained in the
union may also have its own texture, which will override any texture
statements in the parent object.</p>
<p>
You should be aware that the surfaces inside the union will not be removed.
As you can see from the figure this may be a problem for transparent unions.
If you want those surfaces to be removed you will have to use the <code>
merge</code> operations explained in a later section.</p>
<p>
The following union will contain a box and a sphere.</p>
<pre>
union {
box { <-1.5, -1, -1>, <0.5, 1, 1> }
cylinder { <0.5, 0, -1>, <0.5, 0, 1>, 1 }
}
</pre>
<p>Earlier versions of POV-Ray placed restrictions on unions so you often had
to combine objects with <code>composite</code> statements. Those earlier
restrictions have been lifted so <code>composite</code> is no longer needed.
It is still supported for backwards compatibility.</p>
</div>
<a name="r3_5_1_4_2_1"></a>
<div class="content-level-h6" contains="Split_Union" id="r3_5_1_4_2_1">
<h6>3.5.1.4.2.1 Split_Union</h6>
<p><code>split_union</code> is a boolean keyword that can be added to a union.
It has two states <code>on</code>/<code>off</code>, its default is <code>on</code>.</p>
<p><code>split_union</code> is used when <a href="r3_4.html#r3_4_3_4_4">photons</a> are shot
at the CSG-object. The object is split up in its compound parts, photons are shot at
each part separately. This is to prevent photons from being shot at 'empty spaces' in the object,
for example the holes in a grid. With compact objects, without 'empty spaces'
<code>split_union off</code> can improve photon
gathering.</p>
<pre>
union {
object {...}
object {...}
split_union off
}
</pre></div>
<a name="r3_5_1_4_3"></a>
<div class="content-level-h5" contains="Intersection" id="r3_5_1_4_3">
<h5>3.5.1.4.3 Intersection</h5>
<p>The <code>intersection</code> object creates a shape containing only those
areas where all components overlap. A point is part of an intersection if it is
inside both objects, <code>A</code> and <code>B</code>, as show in the figure
below.</p>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/7/75/RefImgIsectobj.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">The intersection of two objects.</p>
</td>
</tr>
</table>
<p>The syntax is:</p>
<pre>
INTERSECTION:
intersection {
SOLID_OBJECTS...
[OBJECT_MODIFIERS...]
}
</pre>
<p>The component objects must have well defined inside/outside properties.
Patch objects are not allowed.</p>
<p class="Note"><strong>Note:</strong> If all components do not overlap, the intersection object disappears.</p>
<p>
Here is an example that overlaps:</p>
<pre>
intersection {
box { <-1.5, -1, -1>, <0.5, 1, 1> }
cylinder { <0.5, 0, -1>, <0.5, 0, 1>, 1 }
}
</pre></div>
<a name="r3_5_1_4_4"></a>
<div class="content-level-h5" contains="Difference" id="r3_5_1_4_4">
<h5>3.5.1.4.4 Difference</h5>
<p>The CSG <code>difference</code> operation takes the intersection between
the first object and the inverse of all subsequent objects. Thus only points
inside object <code>A</code> and outside object <code>B</code> belong to the
difference of both objects.</p>
<p>
The result is a subtraction of the 2nd shape from the first shape as shown
in the figure below.</p>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/3/3b/RefImgDiffobj.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">The difference between two objects.</p>
</td>
</tr>
</table>
<p>The syntax is:</p>
<pre>
DIFFERENCE:
difference {
SOLID_OBJECTS...
[OBJECT_MODIFIERS...]
}
</pre>
<p>The component objects must have well defined inside/outside properties.
Patch objects are not allowed. </p>
<p class="Note"><strong>Note:</strong> If the first object is entirely inside the subtracted objects, the difference object disappears.</p>
<p>
Here is an example of a properly formed difference:</p>
<pre>
difference {
box { <-1.5, -1, -1>, <0.5, 1, 1> }
cylinder { <0.5, 0, -1>, <0.5, 0, 1>, 1 }
}
</pre>
<p class="Note"><strong>Note:</strong> Internally, POV-Ray simply adds the <code>inverse</code> keyword
to the second (and subsequent) objects and then performs an intersection.</p>
<p> The
example above is equivalent to:</p>
<pre>
intersection {
box { <-1.5, -1, -1>, <0.5, 1, 1> }
cylinder { <0.5, 0, -1>, <0.5, 0, 1>, 1 inverse }
}
</pre></div>
<a name="r3_5_1_4_5"></a>
<div class="content-level-h5" contains="Merge" id="r3_5_1_4_5">
<h5>3.5.1.4.5 Merge</h5>
<p>The <code>union</code> operation just glues objects together, it does not
remove the objects' surfaces inside the <code>union</code>. Under most
circumstances this does not matter. However if a transparent <code>
union</code> is used, those interior surfaces will be visible. The <code>
merge</code> operations can be used to avoid this problem. It works just like
<code>union</code> but it eliminates the inner surfaces like shown in the
figure below.</p>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/2/25/RefImgMergeobj.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">Merge removes inner surfaces.</p>
</td>
</tr>
</table>
<p>The syntax is:</p>
<pre>
MERGE:
merge {
SOLID_OBJECTS...
[OBJECT_MODIFIERS...]
}
</pre>
<p>The component objects must have well defined inside/outside properties.
Patch objects are not allowed. </p>
<p class="Note"><strong>Note:</strong> In general <code>merge</code> is slower rendering than <code>union</code> when used with non transparent objects. A small test may be needed to determine what is the optimal solution regarding speed and visual result.</p></div>
<a name="r3_5_1_5"></a>
<div class="content-level-h4" contains="Object Modifiers" id="r3_5_1_5">
<h4>3.5.1.5 Object Modifiers</h4>
<p>A variety of modifiers may be attached to objects. The following items may
be applied to any object:</p>
<pre>
OBJECT_MODIFIER:
clipped_by { UNTEXTURED_SOLID_OBJECT... } |
clipped_by { bounded_by } |
bounded_by { UNTEXTURED_SOLID_OBJECT... } |
bounded_by { clipped_by } |
no_shadow |
no_image [ Bool ] |
no_radiosity [ Bool ] |
no_reflection [ Bool ] |
inverse |
sturm [ Bool ] |
hierarchy [ Bool ] |
double_illuminate [ Bool ] |
hollow [ Bool ] |
interior { INTERIOR_ITEMS... } |
material { [MATERIAL_IDENTIFIER][MATERIAL_ITEMS...] } |
texture { TEXTURE_BODY } |
interior_texture { TEXTURE_BODY } |
pigment { PIGMENT_BODY } |
normal { NORMAL_BODY } |
finish { FINISH_ITEMS... } |
photons { PHOTON_ITEMS...}
radiosity { RADIOSITY_ITEMS...}
TRANSFORMATION
</pre>
<p>Transformations such as translate, rotate and scale have already been discussed. The modifiers <em><a href="r3_6.html#r3_6_1">Textures</a></em> and its parts <em><a href="r3_6.html#r3_6_1_1">Pigment</a></em>, <em><a href="r3_6.html#r3_6_1_2">Normal</a></em>, and <em><a href="r3_6.html#r3_6_1_3">Finish</a></em> as well as <em><a href="r3_7.html#r3_7_2_1">Interior</a></em>, and <em><a href="r3_7.html#r3_7_2">Media</a></em> (which is part of interior) are each in major chapters of their own below. In the sub-sections below we cover several other important modifiers: <code><a href="r3_5.html#r3_5_1_5_1">clipped_by</a></code>, <code><a href="r3_5.html#r3_5_1_5_2">bounded_by</a></code>, <code><a href="r3_5.html#r3_5_1_5_3">material</a></code>, <code><a href="r3_5.html#r3_5_1_5_5">inverse</a></code>, <code><a href="r3_5.html#r3_5_1_5_4">hollow</a></code>, <code><a href="r3_5.html#r3_5_1_5_6">no_shadow</a></code>, <code><a href="r3_5.html#r3_5_1_5_7">no_image</a></code>, <code><a href="r3_5.html#r3_5_1_5_8">no_reflection</a></code>, <code><a href="r3_5.html#r3_5_1_5_9">double_illuminate</a></code>, <code><a href="r3_5.html#r3_5_1_5_10">no_radiosity</a></code> and <code><a href="r3_5.html#r3_5_1_5_11">sturm</a></code>. Although the examples below use object statements and object identifiers, these modifiers may be used on any type of object such as sphere, box etc.</p></div>
<a name="r3_5_1_5_1"></a>
<div class="content-level-h5" contains="Clipped By Object Modifier" id="r3_5_1_5_1">
<h5>3.5.1.5.1 Clipped By Object Modifier</h5>
<p>The <code>clipped_by</code> statement is technically an object modifier
but it provides a type of CSG similar to CSG intersection. The syntax is:</p>
<pre>
CLIPPED_BY:
clipped_by { UNTEXTURED_SOLID_OBJECT... } |
clipped_by { bounded_by }
</pre>
<p>Where <em>UNTEXTURED_SOLID_OBJECT</em> is one or more solid objects which
have had no texture applied. For example:</p>
<pre>
object {
My_Thing
clipped_by{plane{y,0}}
}
</pre>
<p>Every part of the object <code>My_Thing</code> that is inside the plane is
retained while the remaining part is clipped off and discarded. In an <code>
intersection</code> object the hole is closed off. With <code>
clipped_by</code> it leaves an opening. For example the following figure
shows object <code>A</code> being clipped by object <code>B</code>.</p>
<table class="centered" width="660x" cellpadding="0" cellspacing="10">
<tr>
<td>
<img class="center" width="640px" src="images/c/c6/RefImgClipobj.gif">
</td>
</tr>
<tr>
<td>
<p class="caption">An object clipped by another object.</p>
</td>
</tr>
</table>
<p>You may use <code>clipped_by</code> to slice off portions of any shape. In
many cases it will also result in faster rendering times than other methods
of altering a shape. Occasionally you will want to use the <code>
clipped_by</code> and <code>bounded_by</code> options with the same object.
The following shortcut saves typing and uses less memory.</p>
<pre>
object {
My_Thing
bounded_by { box { <0,0,0>, <1,1,1> } }
clipped_by { bounded_by }
}
</pre>
<p>This tells POV-Ray to use the same box as a clip that was used as a
bound.</p></div>
<a name="r3_5_1_5_2"></a>
<div class="content-level-h5" contains="Bounded By Object Modifier" id="r3_5_1_5_2">
<h5>3.5.1.5.2 Bounded By Object Modifier</h5>
<p>The calculations necessary to test if a ray hits an object can be quite
time consuming. Each ray has to be tested against every object in the scene.
POV-Ray attempts to speed up the process by building a set of invisible
boxes, called bounding boxes, which cluster the objects together. This way a
ray that travels in one part of the scene does not have to be tested
against objects in another, far away part of the scene. When a large number
of objects are present the boxes are nested inside each other. POV-Ray can
use bounding boxes on any finite object and even some clipped or bounded
quadrics. However infinite objects (such as a planes, quartic, cubic and
poly) cannot be automatically bound. CSG objects are automatically bound if
they contain finite (and in some cases even infinite) objects. This works by
applying the CSG set operations to the bounding boxes of all objects used
inside the CSG object. For difference and intersection operations this will
hardly ever lead to an optimal bounding box. It is sometimes better
(depending on the complexity of the CSG object) to have you place a bounding
shape yourself using a <code>bounded_by</code> statement.</p>
<p>
Normally bounding shapes are not necessary but there are cases where they
can be used to speed up the rendering of complex objects. Bounding shapes
tell the ray-tracer that the object is totally enclosed by a simple shape.
When tracing rays, the ray is first tested against the simple bounding shape.
If it strikes the bounding shape the ray is further tested against the more
complicated object inside. Otherwise the entire complex shape is skipped,
which greatly speeds rendering. The syntax is:</p>
<pre>
BOUNDED_BY:
bounded_by { UNTEXTURED_SOLID_OBJECT... } |
bounded_by { clipped_by }
</pre>
<p>Where <em>UNTEXTURED_SOLID_OBJECT</em> is one or more solid objects which
have had no texture applied. For example:</p>
<pre>
intersection {
sphere { <0,0,0>, 2 }
plane { <0,1,0>, 0 }
plane { <1,0,0>, 0 }
bounded_by { sphere { <0,0,0>, 2 } }
}
</pre>
<p>The best bounding shape is a sphere or a box since these shapes are highly
optimized, although, any shape may be used. If the bounding shape is itself a
finite shape which responds to bounding slabs then the object which it
encloses will also be used in the slab system.</p>
<p>
While it may a good idea to manually add a <code>bounded_by</code> to
intersection, difference and merge, it is best to <em>never</em> bound a
union. If a union has no <code>bounded_by</code> POV-Ray can internally
split apart the components of a union and apply automatic bounding slabs to
any of its finite parts. Note that some utilities such as <code>
raw2pov</code> may be able to generate bounds more efficiently than
POV-Ray's current system. However most unions you create yourself can be
easily bounded by the automatic system. For technical reasons POV-Ray cannot
split a merge object. It is maybe best to hand bound a merge, especially if
it is very complex.</p>
<p class="Note"><strong>Note:</strong> If bounding shape is too small or positioned incorrectly it may
clip the object in undefined ways or the object may not appear at all. To do
true clipping, use <code>clipped_by</code> as explained in the previous
section. Occasionally you will want to use the <code>clipped_by</code> and
<code>bounded_by</code> options with the same object. The following shortcut
saves typing and uses less memory.</p>
<pre>
object {
My_Thing
clipped_by{ box { <0,0,0>,<1,1,1 > }}
bounded_by{ clipped_by }
}
</pre>
<p>This tells POV-Ray to use the same box as a bound that was used as a
clip.</p></div>
<a name="r3_5_1_5_3"></a>
<div class="content-level-h5" contains="Material" id="r3_5_1_5_3">
<h5>3.5.1.5.3 Material</h5>
<p>One of the changes in POV-Ray 3.1 was the removal of several items from <code>
texture { finish{</code>...<code>} }</code> and to move them to the new <code>
interior</code> statement. The <code><a href="r3_6.html#r3_6_1_4">halo</a></code> statement, formerly part of
<code><a href="r3_6.html#r3_6_1">texture</a></code>, is now renamed <code><a href="r3_7.html#r3_7_2">media</a></code> and made a part of
the <code><a href="r3_7.html#r3_7_2_1">interior</a></code>.</p>
<p>
This split was deliberate and purposeful (see
<a href="r3_7.html#r3_7_2_1_1">Why are Interior and Media Necessary?</a>)
however beta testers pointed out that it made it difficult to
entirely describe the surface properties and interior of an object in one
statement that can be referenced by a single identifier in a texture
library.</p>
<p>
The result is that we created a <em>wrapper</em> around <code>texture</code> and <code>interior</code> which we call <code>material</code>.</p>
<p>
The syntax is:</p>
<pre>
MATERIAL:
material { [MATERIAL_IDENTIFIER][MATERIAL_ITEMS...] }
MATERIAL_ITEMS:
TEXTURE | INTERIOR_TEXTURE | INTERIOR | TRANSFORMATIONS
</pre>
<p>For example:</p>
<pre>
#declare MyGlass=material{ texture{ Glass_T } interior{ Glass_I }}
object { MyObject material{ MyGlass}}
</pre>
<p>Internally, the <em>material</em> is not attached to the object. The
material is just a container that brings the texture and interior to the
object. It is the texture and interior itself that is attached to the object.
Users should still consider texture and interior as separate items attached
to the object.</p>
<p>
The material is just a <em>bucket</em> to carry them. If the object
already has a texture, then the material texture is layered over it. If the object
already has an interior, the material interior fully replaces it and the old
interior is destroyed. Transformations inside the material affect only the
textures and interiors which are inside the <code>material{}</code> wrapper
and only those textures or interiors specified are affected. For example:</p>
<pre>
object {
MyObject
material {
texture { MyTexture }
scale 4 //affects texture but not object or interior
interior { MyInterior }
translate 5*x //affects texture and interior, not object
}
}
</pre>
<p class="Note"><strong>Note:</strong> The <code>material</code> statement has nothing to do with the
<code><a href="r3_6.html#r3_6_1_5_3">material_map</a></code> statement. A <code>material_map</code> is <em> not</em> a way to create patterned material. See <a href="r3_6.html#r3_6_1_5_3">Material Maps</a> for explanation of this unrelated, yet similarly named, older feature.</p></div>
<a name="r3_5_1_5_4"></a>
<div class="content-level-h5" contains="Hollow Object Modifier" id="r3_5_1_5_4">
<h5>3.5.1.5.4 Hollow Object Modifier</h5>
<p>POV-Ray by default assumes that objects are made of a solid material that
completely fills the interior of an object. By adding the <code>
hollow</code> keyword to the object you can make it hollow, also see the
<a href="r3_7.html#r3_7_2_1_2">Empty and Solid Objects</a> chapter. That is very
useful if you want atmospheric effects to exist inside an object. It is even
required for objects containing an interior media. The keyword may optionally
be followed by a float expression which is interpreted as a boolean value.
For example <code>hollow off</code> may be used to force it off. When the
keyword is specified alone, it is the same as <code>hollow on</code>.
By default <code>hollow</code> is <code>off</code> when not specified.</p>
<p>
In order to get a hollow CSG object you just have to make the top level
object hollow. All children will assume the same <code>hollow</code> state
except when their state is explicitly set. The following example will set both
spheres inside the union hollow</p>
<pre>
union {
sphere { -0.5*x, 1 }
sphere { 0.5*x, 1 }
hollow
}
</pre>
<p>while the next example will only set the second sphere hollow because the
first sphere was explicitly set to be not hollow.</p>
<pre>
union {
sphere { -0.5*x, 1 hollow off }
sphere { 0.5*x, 1 }
hollow on
}
</pre></div>
<a name="r3_5_1_5_5"></a>
<div class="content-level-h5" contains="Inverse Object Modifier" id="r3_5_1_5_5">
<h5>3.5.1.5.5 Inverse Object Modifier</h5>
<p>When using <a href="r3_5.html#r3_5_1_4">CSG</a> it is often useful to invert an object so that it will be
inside-out. The appearance of the object is not changed, just the way that
POV-Ray perceives it. When the <code>inverse</code> keyword is used the <em>
inside</em> of the shape is flipped to become the <em>outside</em> and vice
versa. For example:</p>
<pre>
object { MyObject inverse }
</pre>
<p>The inside/outside distinction is also important when attaching
<code><a href="r3_7.html#r3_7_2_1">interior</a></code> to an object especially if
<code><a href="r3_7.html#r3_7_1_1">media</a></code> is also used. Atmospheric media
and fog also do not work as expected if your camera is inside an object.
Using <code>inverse</code> is useful to correct that problem.</p></div>
<a name="r3_5_1_5_6"></a>
<div class="content-level-h5" contains="No Shadow Object Modifier" id="r3_5_1_5_6">
<h5>3.5.1.5.6 No Shadow Object Modifier</h5>
<p>You may specify the <code>no_shadow</code> keyword in an object to make
that object cast no shadow. This is useful for special effects and for
creating the illusion that a light source actually is visible. This keyword
was necessary in earlier versions of POV-Ray which did not have the <code>
looks_like</code> statement. Now it is useful for creating things like laser
beams or other unreal effects. During test rendering it speeds things up if
<code>no_shadow</code> is applied.</p>
<p>
Simply attach the keyword as follows:</p>
<pre>
object {
My_Thing
no_shadow
}
</pre></div>
<a name="r3_5_1_5_7"></a>
<div class="content-level-h5" contains="No Image Object Modifier" id="r3_5_1_5_7">
<h5>3.5.1.5.7 No Image Object Modifier</h5>
<p>Syntax:</p>
<pre>
OBJECT {
[OBJECT_ITEMS...]
no_image
}
</pre>
<p>This keyword is very similar in usage and function to the
<code>no_shadow</code> keyword, and control an object's visibility.
<br>You can use any combination of no_image, no_reflection and no_shadow with your object.</p>
<p>When <code>no_image</code> is used, the object will not be seen by
the camera, either directly or through transparent/refractive objects. However,
it will still cast shadows, and show up in reflections (unless <code>no_reflection
</code> and/or <code>no_shadow</code> is used also).</p>
<p>Using these three keywords you can produce interesting effects like a sphere
casting a rectangular shadow, a cube that shows up as a cone in mirrors,
etc.</p></div>
<a name="r3_5_1_5_8"></a>
<div class="content-level-h5" contains="No Reflection Object Modifier" id="r3_5_1_5_8">
<h5>3.5.1.5.8 No Reflection Object Modifier</h5>
<p>Syntax:</p>
<pre>
OBJECT {
[OBJECT_ITEMS...]
no_reflection
}
</pre>
<p>This keyword is very similar in usage and function to the
<code>no_shadow</code> keyword, and control an object's visibility.
<br>You can use any combination of no_reflection, no_image and no_shadow with your object.</p>
<p>When <code>no_reflection</code> is used, the object will not show up in
reflections. It will be seen by the camera (and through transparent/refractive objects)
and cast shadows, unless <code>no_image</code> and/or <code>no_shadow
</code> is used.</p></div>
<a name="r3_5_1_5_9"></a>
<div class="content-level-h5" contains="Double Illuminate Object Modifier" id="r3_5_1_5_9">
<h5>3.5.1.5.9 Double Illuminate Object Modifier</h5>
<p>Syntax:</p>
<pre>
OBJECT {
[OBJECT_ITEMS...]
double_illuminate
}
</pre>
<p>A surface has two sides; usually, only the side facing the light source is illuminated,
the other side remains in shadow. When <code>double_illuminate</code> is used,
the other side is also illuminated.
<br>This is useful for simulating effects like translucency (as in a lamp shade, sheet of paper, etc).</p>
<p class="Note"><strong>Note:</strong> Using <code>double_illuminate</code> only illuminates both sides of the same
surface, so on a sphere, for example, you will not see the effect unless the
sphere is either partially transparent, or if the camera is inside and the light source
outside of the sphere (or vise versa).</p></div>
<a name="r3_5_1_5_10"></a>
<div class="content-level-h5" contains="No Radiosity Object Modifier" id="r3_5_1_5_10">
<h5>3.5.1.5.10 No Radiosity Object Modifier</h5>
<p>Specifying <code>no_radiosity</code> in an object block makes that object invisible to radiosity rays, in the same way as <code>no_image</code>, <code>no_reflection</code> and <code>no_shadow</code> make an object invisible to primary, reflected and shadow test rays, respectively.</p></div>
<a name="r3_5_1_5_11"></a>
<div class="content-level-h5" contains="Sturm Object Modifier" id="r3_5_1_5_11">
<h5>3.5.1.5.11 Sturm Object Modifier</h5>
<p>Some of POV-Ray's objects allow you to choose between a fast but
sometimes inaccurate root solver and a slower but more accurate one. This is
the case for all objects that involve the solution of a cubic or quartic
polynomial. There are analytic mathematical solutions for those polynomials
that can be used.</p>
<p>
Lower order polynomials are trivial to solve while higher order polynomials
require iterative algorithms to solve them. One of those algorithms is the
Sturmian root solver. For example:</p>
<pre>
blob {
threshold .65
sphere { <.5,0,0>, .8, 1 }
sphere { <-.5,0,0>,.8, 1 }
sturm
}
</pre>
<p>The keyword may optionally be followed by a float expression which is
interpreted as a boolean value. For example <code>sturm off</code> may be
used to force it off. When the keyword is specified alone, it is the same as
<code>sturm on</code>. By default <code>sturm</code> is <code>off</code> when not specified.</p>
<p>
The following list shows all objects for which the Sturmian root solver can
be used.</p>
<ul>
<li>blob</li>
<li>cubic</li>
<li>lathe (only with quadratic splines)</li>
<li>lemon</li>
<li>ovus</li>
<li>poly</li>
<li>prism (only with cubic splines)</li>
<li>quartic</li>
<li>sor</li>
<li>torus</li>
</ul></div>
</div>
</div>
</body>
</html>
|
