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
|
<!-- This file copyright Persistence of Vision Raytracer Pty. Ltd. 2003-2004 -->
<html>
<head>
<!-- NOTE: In order to users to help find information about POV-Ray using -->
<!-- web search engines, we ask you to *not* let them index documentation -->
<!-- mirrors because effectively, when searching, users will get hundreds -->
<!-- of results containing the same information! For this reason, the two -->
<!-- meta tags below disable archiving and indexing of this page by all -->
<!-- search engines that support these meta tags. -->
<meta content="noarchive" name="robots">
<meta content="noindex" name="robots">
<meta content="no-cache" http-equiv="Pragma">
<meta content="0" http-equiv="expires">
<title>2.3.3 Other Shapes</title>
<link href="povray35.css" rel="stylesheet" type="text/css">
</head>
<body>
<table class="NavBar" width="100%">
<tr>
<td align="left" nowrap="" valign="middle" width="32">
<a href="s_67.html"><img alt="previous" border="0" src="prev.png"></a>
</td>
<td align="left" valign="middle" width="30%">
<a href="s_67.html">2.3.2 Polygon Based Shapes</a>
</td>
<td align="center" valign="middle">
<strong class="NavBar">POV-Ray 3.6 for UNIX documentation</strong><br> <strong>2.3.3
Other Shapes</strong>
</td>
<td align="right" valign="middle" width="30%">
<a href="s_69.html">2.3.4 Advanced Texture Options</a>
</td>
<td align="right" nowrap="" valign="middle" width="32">
<a href="s_69.html"><img alt="next" border="0" src="next.png"></a>
</td>
</tr>
</table>
<h3><a name="s02_03_03">2.3.3 </a>Other Shapes</h3>
<a name="s02_03_03_i1">
<h4><a name="s02_03_03_01">2.3.3.1 </a>Blob Object</h4>
<p>
Blobs are described as spheres and cylinders covered with "goo" which stretches to smoothly join them
(see section "<a href="s_105.html#s03_04_01_01">Blob</a>").
</p>
<p>
Ideal for modeling atoms and molecules, blobs are also powerful tools for creating many smooth flowing
"organic" shapes.
</p>
<p>
A slightly more mathematical way of describing a blob would be to say that it is one object made up of two or more
component pieces. Each piece is really an invisible field of force which starts out at a particular strength and falls
off smoothly to zero at a given radius. Where ever these components overlap in space, their field strength gets added
together (and yes, we can have negative strength which gets subtracted out of the total as well). We could have just
one component in a blob, but except for seeing what it looks like there is little point, since the real beauty of
blobs is the way the components interact with one another.
</p>
<p>
Let us take a simple example blob to start. Now, in fact there are a couple different types of components but we
will look at them a little later. For the sake of a simple first example, let us just talk about spherical components.
Here is a sample POV-Ray code showing a basic camera, light, and a simple two component blob:
</p>
<pre>
#include "colors.inc"
background{White}
camera {
angle 15
location <0,2,-10>
look_at <0,0,0>
}
light_source { <10, 20, -10> color White }
blob {
threshold .65
sphere { <.5,0,0>, .8, 1 pigment {Blue} }
sphere { <-.5,0,0>,.8, 1 pigment {Pink} }
finish { phong 1 }
}
</pre>
<p>
<img alt="A simple, two-part blob." src="images/tutorial/simpblob.png">
</p>
<p>
The threshold is simply the overall strength value at which the blob becomes visible. Any points within the blob
where the strength matches the threshold exactly form the surface of the blob shape. Those less than the threshold are <em>outside</em>
and those greater than are <em>inside</em> the blob.
</p>
<p>
We note that the spherical component looks a lot like a simple sphere object. We have the sphere keyword, the
vector representing the location of the center of the sphere and the float representing the radius of the sphere. But
what is that last float value? That is the individual strength of that component. In a spherical component, that is
how strong the component's field is at the center of the sphere. It will fall off in a linear progression until it
reaches exactly zero at the radius of the sphere.
</p>
<p>
Before we render this test image, we note that we have given each component a different pigment. POV-Ray allows
blob components to be given separate textures. We have done this here to make it clearer which parts of the blob are
which. We can also texture the whole blob as one, like the finish statement at the end, which applies to all
components since it appears at the end, outside of all the components. We render the scene and get a basic kissing
spheres type blob.
</p>
<p>
The image we see shows the spheres on either side, but they are smoothly joined by that bridge section in the
center. This bridge represents where the two fields overlap, and therefore stay above the threshold for longer than
elsewhere in the blob. If that is not totally clear, we add the following two objects to our scene and re-render. We
note that these are meant to be entered as separate sphere objects, not more components in the blob.
</p>
<pre>
sphere { <.5,0,0>, .8
pigment { Yellow transmit .75 }
}
sphere { <-.5,0,0>, .8
pigment { Green transmit .75 }
}
</pre>
<p>
<img alt="The spherical components made visible." src="images/tutorial/sphblob.png">
</p>
<p>
Now the secrets of the kissing spheres are laid bare. These semi-transparent spheres show where the components of
the blob actually are. If we have not worked with blobs before, we might be surprised to see that the spheres we just
added extend way farther out than the spheres that actually show up on the blobs. That of course is because our
spheres have been assigned a starting strength of one, which gradually fades to zero as we move away from the sphere's
center. When the strength drops below the threshold (in this case 0.65) the rest of the sphere becomes part of the
outside of the blob and therefore is not visible.
</p>
<p>
See the part where the two transparent spheres overlap? We note that it exactly corresponds to the bridge between
the two spheres. That is the region where the two components are both contributing to the overall strength of the blob
at that point. That is why the bridge appears: that region has a high enough strength to stay over the threshold, due
to the fact that the combined strength of two spherical components is overlapping there.
</p>
<h5><a name="s02_03_03_01_01">2.3.3.1.1 </a>Component Types and Other New Features</h5>
<a name="s02_03_03_01_01_i1">
<p>
The shape shown so far is interesting, but limited. POV-Ray has a few extra tricks that extend its range of
usefulness however. For example, as we have seen, we can assign individual textures to blob components, we can also
apply individual transformations (translate, rotate and scale) to stretch, twist, and squash pieces of the blob as we
require. And perhaps most interestingly, the blob code has been extended to allow cylindrical components.
</p>
<p>
Before we move on to cylinders, it should perhaps be mentioned that the old style of components used in previous
versions of POV-Ray still work. Back then, all components were spheres, so it was not necessary to say sphere or
cylinder. An old style component had the form:
</p>
<p>
component Strength, Radius, <Center>
</p>
<p>
This has the same effect as a spherical component, just as we already saw above. This is only useful for backwards
compatibility. If we already have POV-Ray files with blobs from earlier versions, this is when we would need to
recognize these components. We note that the old style components did not put braces around the strength, radius and
center, and of course, we cannot independently transform or texture them. Therefore if we are modifying an older work
into a new version, it may arguably be of benefit to convert old style components into spherical components anyway.
</p>
<p>
Now for something new and different: cylindrical components. It could be argued that all we ever needed to do to
make a roughly cylindrical portion of a blob was string a line of spherical components together along a straight line.
Which is fine, if we like having extra to type, and also assuming that the cylinder was oriented along an axis. If
not, we would have to work out the mathematical position of each component to keep it is a straight line. But no more!
Cylindrical components have arrived.
</p>
<p>
We replace the blob in our last example with the following and re-render. We can get rid of the transparent spheres
too, by the way.
</p>
<pre>
blob {
threshold .65
cylinder { <-.75,-.75,0>, <.75,.75,0>, .5, 1 }
pigment { Blue }
finish { phong 1 }
}
</pre>
<p>
We only have one component so that we can see the basic shape of the cylindrical component. It is not quite a true
cylinder - more of a sausage shape, being a cylinder capped by two hemispheres. We think of it as if it were an array
of spherical components all closely strung along a straight line.
</p>
<p>
As for the component declaration itself: simple, logical, exactly as we would expect it to look (assuming we have
been awake so far): it looks pretty much like the declaration of a cylinder object, with vectors specifying the two
endpoints and a float giving the radius of the cylinder. The last float, of course, is the strength of the component.
Just as with spherical components, the strength will determine the nature and degree of this component's interaction
with its fellow components. In fact, next let us give this fellow something to interact with, shall we?
</p>
<h5><a name="s02_03_03_01_02">2.3.3.1.2 </a>Complex Blob Constructs and Negative Strength</h5>
<a name="s02_03_03_01_02_i1"><a name="s02_03_03_01_02_i2">
<p>
Beginning a new POV-Ray file, we enter this somewhat more complex example:
</p>
<pre>
#include "colors.inc"
background{White}
camera {
angle 20
location<0,2,-10>
look_at<0,0,0>
}
light_source { <10, 20, -10> color White }
blob {
threshold .65
sphere{<-.23,-.32,0>,.43, 1 scale <1.95,1.05,.8>} //palm
sphere{<+.12,-.41,0>,.43, 1 scale <1.95,1.075,.8>} //palm
sphere{<-.23,-.63,0>, .45, .75 scale <1.78, 1.3,1>} //midhand
sphere{<+.19,-.63,0>, .45, .75 scale <1.78, 1.3,1>} //midhand
sphere{<-.22,-.73,0>, .45, .85 scale <1.4, 1.25,1>} //heel
sphere{<+.19,-.73,0>, .45, .85 scale <1.4, 1.25,1>} //heel
cylinder{<-.65,-.28,0>, <-.65,.28,-.05>, .26, 1} //lower pinky
cylinder{<-.65,.28,-.05>, <-.65, .68,-.2>, .26, 1} //upper pinky
cylinder{<-.3,-.28,0>, <-.3,.44,-.05>, .26, 1} //lower ring
cylinder{<-.3,.44,-.05>, <-.3, .9,-.2>, .26, 1} //upper ring
cylinder{<.05,-.28,0>, <.05, .49,-.05>, .26, 1} //lower middle
cylinder{<.05,.49,-.05>, <.05, .95,-.2>, .26, 1} //upper middle
cylinder{<.4,-.4,0>, <.4, .512, -.05>, .26, 1} //lower index
cylinder{<.4,.512,-.05>, <.4, .85, -.2>, .26, 1} //upper index
cylinder{<.41, -.95,0>, <.85, -.68, -.05>, .25, 1} //lower thumb
cylinder{<.85,-.68,-.05>, <1.2, -.4, -.2>, .25, 1} //upper thumb
pigment{ Flesh }
}
</pre>
<p>
<img alt="A hand made with blobs." src="images/tutorial/blobhand.png">
</p>
<p>
As we can guess from the comments, we are building a hand here. After we render this image, we can see there are a
few problems with it. The palm and heel of the hand would look more realistic if we used a couple dozen smaller
components rather than the half dozen larger ones we have used, and each finger should have three segments instead of
two, but for the sake of a simplified demonstration, we can overlook these points. But there is one thing we really
need to address here: This poor fellow appears to have horrible painful swelling of the joints!
</p>
<p>
A review of what we know of blobs will quickly reveal what went wrong. The joints are places where the blob
components overlap, therefore the combined strength of both components at that point causes the surface to extend
further out, since it stays over the threshold longer. To fix this, what we need are components corresponding to the
overlap region which have a negative strength to counteract part of the combined field strength. We add the following
components to our blob.
</p>
<pre>
sphere{<-.65,.28,-.05>, .26, -1} //counteract pinky knucklebulge
sphere{<-.65,-.28,0>, .26, -1} //counteract pinky palm bulge
sphere{<-.3,.44,-.05>, .26, -1} //counteract ring knuckle bulge
sphere{<-.3,-.28,0>, .26, -1} //counteract ring palm bulge
sphere{<.05,.49,-.05>, .26, -1} //counteract middle knuckle bulge
sphere{<.05,-.28,0>, .26, -1} //counteract middle palm bulge
sphere{<.4,.512,-.05>, .26, -1} //counteract index knuckle bulge
sphere{<.4,-.4,0>, .26, -1} //counteract index palm bulge
sphere{<.85,-.68,-.05>, .25, -1} //counteract thumb knuckle bulge
sphere{<.41,-.7,0>, .25, -.89} //counteract thumb heel bulge
</pre>
<p>
<img alt="The hand without the swollen joints." src="images/tutorial/imprhand.png">
</p>
<p>
Much better! The negative strength of the spherical components counteracts approximately half of the field strength
at the points where to components overlap, so the ugly, unrealistic (and painful looking) bulging is cut out making
our hand considerably improved. While we could probably make a yet more realistic hand with a couple dozen additional
components, what we get this time is a considerable improvement. Any by now, we have enough basic knowledge of blob
mechanics to make a wide array of smooth, flowing organic shapes!
</p>
<h4><a name="s02_03_03_02">2.3.3.2 </a>Height Field Object</h4>
<a name="s02_03_03_02_i1">
<p>
A <code>height_field</code> is an object that has a surface that is determined by the color value or palette index
number of an image designed for that purpose. With height fields, realistic mountains and other types of terrain can
easily be made. First, we need an image from which to create the height field. It just so happens that POV-Ray is
ideal for creating such an image.
</p>
<p>
We make a new file called <code>image.pov</code> and edit it to contain the following:
</p>
<pre>
#include "colors.inc"
global_settings {
assumed_gamma 2.2
hf_gray_16
}
</pre>
<p>
The <code><a href="s_102.html#s03_03_03_04">hf_gray_16</a></code> keyword causes the output to be in a special 16
bit grayscale that is perfect for generating height fields. The normal 8 bit output will lead to less smooth surfaces.
</p>
<p>
Now we create a camera positioned so that it points directly down the z-axis at the origin.
</p>
<pre>
camera {
location <0, 0, -10>
look_at 0
}
</pre>
<p>
We then create a plane positioned like a wall at z=0. This plane will completely fill the screen. It will be
colored with white and gray wrinkles.
</p>
<pre>
plane { z, 10
pigment {
wrinkles
color_map {
[0 0.3*White]
[1 White]
}
}
}
</pre>
<p>
Finally, create a light source.
</p>
<pre>
light_source { <0, 20, -100> color White }
</pre>
<p>
We render this scene at 640x480 <code>+A0.1</code> <code>+FT</code>. We will get an image that will produce an
excellent height field. We create a new file called <code>hfdemo.pov</code> and edit it as follows:
</p>
<p class="Note">
<strong>Note:</strong> Windows users, unless you specify <code>+FT</code> as above, you will get a
.BMP file (which is the default Windows version output). In this case you will need to use <code>sys</code> instead of <code>tga</code>
in the <code>height_field</code> statement below.
</p>
<pre>
#include "colors.inc"
</pre>
<p>
We add a camera that is two units above the origin and ten units back ...
</p>
<pre>
camera{
location <0, 2, -10>
look_at 0
angle 30
}
</pre>
<p>
... and a light source.
</p>
<pre>
light_source{ <1000,1000,-1000> White }
</pre>
<p>
Now we add the height field. In the following syntax, a Targa image file is specified, the height field is
smoothed, it is given a simple white pigment, it is translated to center it around the origin and it is scaled so that
it resembles mountains and fills the screen.
</p>
<pre>
height_field {
tga "image.tga"
smooth
pigment { White }
translate <-.5, -.5, -.5>
scale <17, 1.75, 17>
}
</pre>
<p>
We save the file and render it at 320x240 <code>-A</code>. Later, when we are satisfied that the height field is
the way we want it, we render it at a higher resolution with anti-aliasing.
</p>
<p>
<img alt="A height field created completely with POV-Ray." src="images/tutorial/pvhfield.png">
</p>
<p>
Wow! The Himalayas have come to our computer screen!
</p>
<h4><a name="s02_03_03_03">2.3.3.3 </a>Isosurface Object</h4>
<a name="s02_03_03_03_i1"><a name="isosurface, tutorial"></a>
<dl class="famousquote">
<dt>
<em>You know you have been raytracing too long when ...<br> ... You find yourself wishing you'd paid attention in
math class to all those formulae you thought you'd never have any use for in real life.</em>
<dd>
<em>-- Jeff Lee</em>
</dl>
<p>
Isosurfaces are shapes described by mathematical functions.
</p>
<p>
In contrast to the other mathematically based shapes in POV-Ray, isosurfaces are approximated during rendering and
therefore they are sometimes more difficult to handle. However, they offer many interesting possibilities, like real
deformations and surface displacements
</p>
<p>
Some knowledge about mathematical functions and geometry is useful, but not necessarily required to work with
isosurfaces.
</p>
<h5><a name="s02_03_03_03_01">2.3.3.3.1 </a>Simple functions</h5>
<p>
For the start we will choose a most simple function: <code>x</code> The value of this function is exactly the
current x-coordinate.
</p>
<p>
The isosurface object takes this function as a <a href="s_97.html#s03_02_01_06">user defined function</a>:
</p>
<pre>
isosurface {
function { x }
contained_by { box { -2, 2 } }
}
</pre>
<p>
<img alt="Isosurface sample (function { x })" src="images/tutorial/iso_01.png">
</p>
<p>
the resulting shape is fairly simple: a box.
</p>
<p>
The fact that it is a box is only caused by the container object which is required for an isosurface. You can
either use a box or a sphere for this purpose.
</p>
<p>
So only one side of the box is made by the function in fact. This surface is where the x-coordinate is 0 since 0 is
the default threshold. There usually is no reason to change this, since it is the most common and most suggestive
value, but you can specify something different by adding
</p>
<p>
<code> threshold 1</code>
</p>
<p>
to the isosurface definition.
</p>
<p>
<img alt="Isosurface sample (function { x }, threshold 1)" src="images/tutorial/iso_02.png">
</p>
<p>
As you can see, the surface is now at x-coordinate 1.
</p>
<p>
We can also remove the visible surfaces of the container object by adding the word 'open' to the isosurface
definition.
</p>
<p>
<img alt="Isosurface sample (function { x }, open)" src="images/tutorial/iso_03.png">
</p>
<p>
For making it clearer what surfaces are the actual isosurface and what are caused by the container object, the
color will be different in all the following pictures.
</p>
<p>
Now we replace the used function with something different:
</p>
<p>
<code> function { x+y }</code>
</p>
<p>
<img alt="Isosurface sample (plane function)" src="images/tutorial/iso_04.png">
</p>
<p>
<code> function { x+y+z }</code>
</p>
<p>
<img alt="Isosurface sample (plane function)" src="images/tutorial/iso_05.png">
</p>
<p class="Note">
<strong>Note:</strong> 'max_gradient 4' is added to the isosurface definition here, this will be
explained later on.
</p>
<p>
All these functions describe planes going through the origin. The function just describes the normal vector of this
plane.
</p>
<h5><a name="s02_03_03_03_02">2.3.3.3.2 </a>Several surfaces</h5>
<p>
The following two functions lead to identical results:
</p>
<p>
<code> function { abs(x)-1 }</code>
</p>
<p>
<code> function { sqrt(x*x)-1 }</code>
</p>
<p>
<img alt="Isosurface sample (function { abs(x)-1 }, open)" src="images/tutorial/iso_06.png">
</p>
<p>
You can see that there are two planes now. The reason is that both formulas have the same two solutions (where the
function value is 0), namely <code>x=-1</code> and <code>x=1</code>.
</p>
<p>
We can now mix all these elements in different combinations, the results always consist of plane surfaces:
</p>
<p>
<code> function { abs(x)-1+y }</code>
</p>
<p>
<img alt="Isosurface sample (combined linear functions)" src="images/tutorial/iso_07.png">
</p>
<p>
<code> function { abs(x)+abs(y)+abs(z)-2 }</code>
</p>
<p>
<img alt="Isosurface sample (combined linear functions)" src="images/tutorial/iso_08.png">
</p>
<h5><a name="s02_03_03_03_03">2.3.3.3.3 </a>Non-linear functions</h5>
<p>
Curved surfaces of many different kinds can be achieved with non-linear functions.
</p>
<p>
<code> function { pow(x,2) + y }</code>
</p>
<p>
<img alt="Isosurface sample (non-linear function)" src="images/tutorial/iso_09.png">
</p>
<p>
You can see the parabolic shape caused by the square function.
</p>
<p>
To get a cylindrical surface we can use the following function.
</p>
<p>
<code> function { sqrt(pow(x,2) + pow(z,2)) - 1 }</code>
</p>
<p>
In 2 dimensions it describes a circle, since it is constant in the 3rd dimension, we get a cylinder:
</p>
<p>
<img alt="Isosurface sample (cylinder function)" src="images/tutorial/iso_10.png">
</p>
<p>
It is of course not difficult to change this into a cone, we just need to add a linear component in y-direction:
</p>
<p>
<code> function { sqrt(pow(x,2) + pow(z,2)) + y }</code>
</p>
<p>
<img alt="Isosurface sample (cone function)" src="images/tutorial/iso_11.png">
</p>
<p>
And we of course can also make a sphere:
</p>
<p>
<code> function { sqrt(pow(x,2) + pow(y,2) + pow(z,2)) - 2 }</code>
</p>
<p>
<img alt="Isosurface sample (sphere function)" src="images/tutorial/iso_12.png">
</p>
<p>
The <code>2</code> specifies the radius here.
</p>
<h5><a name="s02_03_03_03_04">2.3.3.3.4 </a>Specifying functions</h5>
<p>
As we have seen, the functions used to define the isosurface are written in the <code>function {...}</code> block.
</p>
<p>
Allowed are:
</p>
<p>
User defined functions (like equations). All float expressions and operators (see section "<a href="#l25">User-Defined
Functions</a>") which are legal in POV-Ray, can be used. <br>With the equation of a sphere "<code>x^2+y^2+z^2
= Threshold</code>" we get:
</p>
<pre>
isosurface {
function {pow(x,2) + pow(y,2) + pow(z,2)}
threshold Threshold
...
}
</pre>
<p>
Functions can be declared first (see section "<a href="#l26">Declaring Functions</a>") and then used in
the isosurface.
</p>
<pre>
#declare Sphere = function {pow(x,2) + pow(y,2) + pow(z,2)}
isosurface {
function { Sphere(x,y,z) }
threshold Threshold
...
}
</pre>
<p>
By default a function takes three parameters (x,y,z) and you do not have to explicitly specify the parameter names
when declaring it. <br>When <em>using</em> the identifier, the parameters <em>must</em> be specified. <br>On the other
hand, if you need more or less than three parameters when declaring a function, you also have to explicitly specify
the parameter names.
</p>
<pre>
#declare Sphere = function(x,y,z,Radius) {
pow(x,2) + pow(y,2) + pow(z,2) - pow(Radius,2)
}
isosurface {
function { Sphere(x,y,z,1) }
...
}
</pre>
<h5><a name="s02_03_03_03_05">2.3.3.3.5 </a>Internal functions</h5>
<p>
There are a lot of internal functions available in POV-Ray. For example a sphere could also be generated with <code>function
{ f_sphere(x, y, z, 2) }</code> These functions are declared in the <code>functions.inc</code> include file. Most of
them are more complicated and it is usually faster to use them instead of a hand coded equivalent. See the <a href="s_138.html#s03_07_07_02">complete
list</a> for details.
</p>
<p>
The following makes a torus just like POV-Ray's torus object:
</p>
<pre>
#include "functions.inc"
isosurface {
function { f_torus(x, y, z, 1.6, 0.4) }
contained_by { box { -2, 2 } }
}
</pre>
<p>
<img alt="Isosurface sample (torus function)" src="images/tutorial/iso_13.png">
</p>
<p>
The 4th and 5th parameter are the major and minor radius, just like the corresponding values in the <code>torus{}</code>
object.
</p>
<p>
The parameters x, y and z are required, because it is a declared function. You can also declare functions yourself
like it is explained in the <a href="s_97.html#s03_02_01_06_03">reference section</a>.
</p>
<h5><a name="s02_03_03_03_06">2.3.3.3.6 </a>Combining isosurface functions</h5>
<p>
We can also simulate some Constructive Solid Geometry with isosurface functions. If you do not know about CSG we
suggest you have a look at <a href="s_59.html#s02_02_03_01">"What is CSG?"</a> or the corresponding part of
the <a href="s_110.html#s03_04_06">reference section</a> first.
</p>
<p>
We will take two functions: a cylinder and a rotated box:
</p>
<pre>
#declare fn_A = function { sqrt(pow(y,2) + pow(z,2)) - 0.8 }
#declare fn_B = function { abs(x)+abs(y)-1 }
</pre>
<p>
If we combine them the following way, we get a "merge":
</p>
<p>
<code>function { min(fn_A(x, y, z), fn_B(x, y, z)) }</code>
</p>
<p>
<img alt="Isosurface sample (merge)" src="images/tutorial/iso_14.png">
</p>
<p>
An "intersection" can be obtained by using <code>max()</code> instead of <code>min()</code>:
</p>
<p>
<code>function { max(fn_A(x, y, z), fn_B(x, y, z)) }</code>
</p>
<p>
<img alt="Isosurface sample (intersection)" src="images/tutorial/iso_15.png">
</p>
<p>
Of course also "difference" is possible, we just have to add a minus (-) before the second function:
</p>
<p>
<code>function { max(fn_A(x, y, z), -fn_B(x, y, z)) }</code>
</p>
<p>
<img alt="Isosurface sample (difference)" src="images/tutorial/iso_16.png">
</p>
<p>
Apart from basic CSG you can also obtain smooth transits between the different surfaces (like with the <a href="s_68.html#s02_03_03_01">blob
object</a>)
</p>
<pre>
#declare Blob_threshold=0.01;
isosurface {
function {
(1+Blob_threshold)
-pow(Blob_threshold, fn_A(x,y,z))
-pow(Blob_threshold, fn_B(x,y,z))
}
max_gradient 4
contained_by { box { -2, 2 } }
}
</pre>
<p>
<img alt="Isosurface sample (blob)" src="images/tutorial/iso_17.png">
</p>
<p>
The <code>Blob_threshold</code> value influences the smoothness of the transit between the shapes. a lower value
leads to sharper edges.
</p>
<p>
The function for a negative blob looks like:
</p>
<pre>
function{fn_A(x,y,z) + pow(Blob_threshold,(Fn_B(x,y,z) + Strength))}
</pre>
<h5><a name="s02_03_03_03_07">2.3.3.3.7 </a>Noise and pigment functions</h5>
<p>
Some of the <a href="#l27">internal functions</a> have a random or noise-like structure
</p>
<p>
Together with the pigment functions they are one of the most powerful tools for designing isosurfaces. We can add
real surface displacement to the objects rather than only normal perturbation known from the <a href="#l28">normal{}
statement</a>.
</p>
<p>
The relevant internal functions are:
</p>
<ul>
<li>
<code>f_noise3d(x,y,z)</code><br> uses the <a href="#l29">noise generator</a> specified in <code>global_settings{}</code>
and generates structures like the bozo pattern.
</li>
<li>
<code>f_noise_generator(x, y, z, noise_generator)</code><br> generates noise with a specified noise generator.
</li>
<li>
<code>f_ridged_mf(x, y, z, H, Lacunarity, Octaves, Offset, Gain, noise_generator)</code><br> generates a ridged
multifractal pattern.
</li>
<li>
<code>f_ridge(x, y, z, Lambda, Octaves, Omega, Offset, Ridge, noise_generator)</code><br> generates another noise
with ridges.
</li>
<li>
<code>f_hetero_mf(x, y, z, H, Lacunarity, Octaves, Offset, T, noise_generator)</code><br> generates heterogenic
multifractal noise.
</li>
</ul>
<p>
Using pure noise3d as a function results in the following picture:
</p>
<p>
<code> function { f_noise3d(x, y, z)-0.5 }</code>
</p>
<p>
<img alt="Isosurface sample (noise3d)" src="images/tutorial/iso_18.png">
</p>
<p class="Note">
<strong>Note:</strong> the <code>-0.5</code> is only there to make it match to the used threshold
value of 0, the <code>f_noise3d</code> function returns values between 0 and 1.
</p>
<p>
With this and the other functions you can generate objects similar to heightfields, having the advantage that a
high resolution can be achieved without high memory requirements.
</p>
<p>
<code> function { x + f_noise3d(0, y, z) }</code>
</p>
<p>
<img alt="Isosurface sample (noise3d 'heightfield')" src="images/tutorial/iso_19.png">
</p>
<p>
The noise function can of course also be subtracted which results in an 'inverted' version:
</p>
<p>
<code> function { x - f_noise3d(0, y, z) }</code>
</p>
<p>
<img alt="Isosurface sample (noise3d 'heightfield' inverted)" src="images/tutorial/iso_20.png">
</p>
<p>
In the last two pictures we added the noise function to a plane function. The x-parameter was set to 0 so the noise
function is constant in x-direction. This way we achieve the typical heightfield structure.
</p>
<p>
Of course we can also add noise to any other function. If the noise function is very strong this can result in
several separated surfaces.
</p>
<p>
<code> function { f_sphere(x, y, z, 1.2) - f_noise3d(x, y, z) }</code>
</p>
<p>
<img alt="Isosurface sample (noise3d on sphere)" src="images/tutorial/iso_21.png">
</p>
<p>
This is a noise function applied to a sphere surface, we can influence the intensity of the noise by multiplying it
with a factor and change the scale by multiplying the coordinate parameters:
</p>
<p>
<code> function { f_sphere(x, y, z, 1.6) - f_noise3d(x * 5, y * 5, z * 5) * 0.5 }</code>
</p>
<p>
<img alt="Isosurface sample (noise3d on sphere scaled)" src="images/tutorial/iso_22.png">
</p>
<p>
As alternative to noise functions we can also use any pigment in a function:
</p>
<pre>
#declare fn_Pigm=function {
pigment {
agate
color_map {
[0 color rgb 0]
[1 color rgb 1]
}
}
}
</pre>
<p>
This function is a vector function returning a (color) vector.For use in isosurface functions they <em>must</em> be
declared first. When using the identifier, you have to specify which component of the color vector should be used. To
do this, the dot notation is used: <code>Function(x,y,z).red</code>.
</p>
<p>
A color vector has five components. Supported dot types to access these components are:
</p>
<ul>
<li>
F( ).<code>x</code> | F( ).<code>u</code> | F( ).<code>red</code>
<ul>
<li>
to get the red value of the color vector
</li>
</ul>
</li>
<li>
F( ).<code>y</code> | F( ).<code>v</code> | F( ).<code>green</code>
<ul>
<li>
to get the green value of the color vector
</li>
</ul>
</li>
<li>
F( ).<code>z</code> | F( ).<code>blue</code>
<ul>
<li>
to get the blue value of the color vector
</li>
</ul>
</li>
<li>
F( ).<code>filter</code> | F( ).<code>t</code>
<ul>
<li>
to get the filter value of the color vector
</li>
</ul>
</li>
<li>
F( ).<code>transmit</code>
<ul>
<li>
to get the transmit value of the color vector
</li>
</ul>
</li>
<li>
F( ).<code>gray</code>
<ul>
<li>
to get the gray value of the color vector
</li>
<li>
gray value = Red*29.7% + Green*58.9% + Blue*11.4%
</li>
</ul>
</li>
<li>
F( ).<code>hf</code>
<ul>
<li>
to get the height_field value of the color vector
</li>
<li>
hf value = (Red + Green/255)*0.996093
</li>
<li>
the .hf operator is experimental and will generate a warning.
</li>
</ul>
</li>
</ul>
<p>
<code> function { f_sphere(x, y, z, 1.6)-fn_Pigm(x/2, y/2, z/2).gray*0.5 }</code>
</p>
<p>
<img alt="Isosurface sample (pigment function)" src="images/tutorial/iso_23.png">
</p>
<p>
There are quite a lot of things possible with pigment functions, but you probably have recognized that this renders
quite slow.
</p>
<h5><a name="s02_03_03_03_08">2.3.3.3.8 </a>Conditional directives and loops</h5>
<p>
Conditional directives are allowed in functions:
</p>
<pre>
#declare Rough = yes;
#include "functions.inc"
isosurface {
function { y #if(Rough=1)-f_noise3d(x/0.5,y/0.3,z/0.4)*0.8 #end }
...
}
</pre>
<p>
Loops can also be used in functions:
</p>
<pre>
#include "functions.inc"
#declare Thr = 1/1000;
#declare Ang = radians(45);
#declare Offset = 1.5;
#declare Scale = 1.2;
#declare TrSph = function { f_sphere(x-Offset,y,z,0.7*Scale) }
function {
(1-Thr)
#declare A = 0;
#while (A<8)
-pow(Thr, TrSph(x*cos(A*Ang) + y*sin(A*Ang),
y*cos(A*Ang) -x*sin(A*Ang), z) )
#declare A=A+1;
#end
}
</pre>
<p class="Note">
<strong>Note:</strong> The loops and conditionals are evaluated at parse time, not at render time.
</p>
<h5><a name="s02_03_03_03_09">2.3.3.3.9 </a>Transformations on functions</h5>
<p>
Transforming an isosurface object is done like transforming any POV-Ray object. Simply use the object modifiers
(scale, translate, rotate, ...).
</p>
<p>
However, when you want to transform functions within the contained_by object, you have to substitute parameters in
the functions.
</p>
<p>
The results <em>seem</em> inverted to what you would normally expect. Here is an explanation: <br>Take a
Sphere(x,y,z). We know it sits at the origin because x=0. When we want it at x=2 (translating 2 units to the right) we
need to write the second equation in the same form: x-2=0 <br>Now that both equations equal 0, we can replace
parameter x with x-2 <br>So our Sphere(x-2, y,z) moves two units to the <em>right</em>.
</p>
<p>
Let's scale our Sphere 0.5 in the y direction. Default size is y=1 (one unit). We want y=0.5. <br>To get this
equation in the same form as the first one, we have to multiply both sides by two. y*2 = 0.5*2, which gives y*2=1 <br>Now
we can replace the y parameter in our sphere: Sphere(x, y*2, z). This squishes the y-size of the sphere by half. <br>Well,
this is the general idea of substitutions.
</p>
<p>
Here is an overview of some useful substitutions: <br>Using a declared object P(x,y,z)
</p>
<p>
<strong>Scale</strong><br> scale x : replace "<code>x</code>" with "<code>x/scale</code>" (idem
other parameters)
</p>
<pre>scale x*2 gives P(x/2,y,z)</pre>
<p>
<strong>Scale Infinitely</strong><br> scale x infinitely : replace "<code>x</code>" with "<code>0</code>"
(idem other parameters)
</p>
<pre>scale y infinitely gives P(x,0,z)</pre>
<p>
<strong>Translate</strong><br> translate x : replace "<code>x</code>" with "<code>x - translation</code>"
(idem other parameters)
</p>
<pre>translate z*3 gives P(x,y,z-3)</pre>
<p>
<strong>Shear</strong><br> shear in XY-plane : replace "<code>x</code>" with "<code>x +
y*tan(radians(Angle))</code>" (idem other parameters)
</p>
<pre>shear 45 degrees left gives P(x+y*tan(radians(45)), y, z)</pre>
<p>
<strong>Rotate</strong>
</p>
<p class="Note">
<strong>Note:</strong> these rotation substitutions work like normal POV-rotations: they already
compensate for the inverse working
</p>
<p>
rotate around X <br>: replace "<code>y</code>" with "<code>z*sin(radians(Angle)) +
y*cos(radians(Angle))</code>" <br>: replace "<code>z</code>" with "<code>z*cos(radians(Angle)) -
y*sin(radians(Angle))</code>"
</p>
<p>
rotate around Y <br>: replace "<code>x</code>" with "<code>x*cos(radians(Angle)) -
z*sin(radians(Angle))</code>" <br>: replace "<code>z</code>" with "<code>x*sin(radians(Angle)) +
z*cos(radians(Angle))</code>"
</p>
<p>
rotate around Z <br>: replace "<code>x</code>" with "<code>x*cos(radians(Angle)) +
y*sin(radians(Angle))</code>" <br>: replace "<code>y</code>" with "<code>-x*sin(radians(Angle)) +
y*cos(radians(Angle)) </code>"
</p>
<pre>
rotate z*75 gives:
P(x*cos(radians(75)) + y*sin(radians(75)),
-x*sin(radians(75)) + y*cos(radians(75)), z)
</pre>
<p>
<strong>Flip</strong><br> flip X - Y : replace "<code>x</code>" with "<code>y</code>" and
replace "<code>y</code>" with "<code>-x</code>"
</p>
<p>
flip Y - Z : replace "<code>y</code>" with "<code>z</code>" and replace "<code>z</code>"
with "<code>-y</code>"
</p>
<p>
flip X - Z : replace "<code>x</code>" with "<code>-z</code>" and replace "<code>z</code>"
with "<code>x</code>"
</p>
<pre>flip x and y gives P(y, -x, z)</pre>
<p>
<strong>Twist</strong><br> twist N turns/unit around <code>X</code> <br>: replace "<code>y</code>" with
"<code>z*sin(x*2*pi*N) + y*cos(x*2*pi*N)</code>" <br>: replace "<code>z</code>" with "<code>z*cos(x*2*pi*N)
- y*sin(x*2*pi*N)</code>"
</p>
<h5><a name="s02_03_03_03_10">2.3.3.3.10 </a>Improving Isosurface Speed</h5>
<p>
To optimize the approximation of the isosurface and to get maximum rendering speed it is important to adapt certain
values;
</p>
<p>
<code>accuracy</code>
</p>
<p>
The accuracy value influences how accurate the surface geometry is calculated. Lower values lead to a more precise,
but slower result. The default value of <code>0.001</code> is fairly low. We used this value in all the previous
samples, but often you can raise this quite a lot and thereby make things faster.
</p>
<p>
<code>max_gradient</code>
</p>
<p>
For finding the actual surface it is important for POV-Ray to know the maximum gradient of the function, meaning
how fast the function value changes. We can specify a value with the <code>max_gradient</code> keyword. Lower
max_gradient values lead to faster rendering, but if the specified value is below the actual maximum gradient of the
function, there can be holes or other artefacts in the surface.
</p>
<p>
For the same reason functions with infinite gradient should not be used. This applies for pigment functions with
brick or checker pattern for example. You should also be careful when using <code>select()</code> in isosurface
functions because of this.
</p>
<p>
If the real maximum gradient differs too much from the specified value POV-Ray prints a warning together with the
found maximum gradient. It is usually sufficient to use this number for the <code>max_gradient</code> parameter to get
fast and correct results.
</p>
<p>
POV-Ray can also dynamically change the <code>max_gradient</code> when you specify <code>evaluate</code> with 3
parameters the isosurface definition. Concerning the details on this and other things see the <a href="s_108.html#s03_04_04">evaluate</a>
in the reference section.
</p>
<p>
<code>contained_by</code>
</p>
<p>
Make sure your <code>contained_by</code> 'object' fits as tightly as possible. An oversized container can
sky-rocket the render time. <br>When the container has a lot of empty space around the actual isosurface, POV-Ray has
to do a lot of superfluous sampling: especially with complex functions this can become very time consuming. On top of
this, the <code>max_gradient</code> needed to get a proper surface will also increase rapidly (almost proportional to
the oversize!). <br>You could use a transparent copy of the container (using exactly the same transformations) to
check how it fits. Getting the <code>min_extent</code> and <code>max_extent</code> of the <code>isosurface</code> is
not useful because it only gives the extent of the container and not of the actual isosurface.
</p>
<h4><a name="s02_03_03_04">2.3.3.4 </a>Poly Object</h4>
<a name="s02_03_03_04_i1">
<p>
The polynomial object (and its "shortcut" versions: <code><a href="s_107.html#s03_04_03_02">cubic</a></code>,
<code><a href="s_107.html#s03_04_03_02">quartic</a></code> and <code><a href="s_107.html#s03_04_03_03">quadric</a></code>)
of POV-Ray is one of the most complex and mathematical primitives of the program. One could think that it is seldom
used and more or less obsolete, but we have to remember that for example the torus primitive is just a shortcut for
the equivalent <code>quartic</code>, which is just a shortcut for the equivalent <code>poly</code> object. Polys are,
however, seldom used in scenes due to the fact that they are so difficult to define and it is far from trivial to get
the desired shape with just a polynomial equation. It is mostly used by the most mathematically oriented POV-Ray
users.
</p>
<p>
This tutorial explains the process of making a polynomial object in POV-Ray.
</p>
<p class="Note">
<strong>Note:</strong>Since version 3.5, POV-Ray includes the new <code>isosurface</code> object which
makes the polynomial object more or less obsolete. The isosurface is more versatile (you can specify any mathematical
function, not just polynomials), easier to use. You can write the function as is, without needing to put values in a
gigantic vector. Isosurfaces often render considerably faster than equivalent polys.
</p>
<p>
However, the most mathematically oriented still like polys because isosurfaces are calculated just by approximating
the right value, while the poly is calculated in a mathematically exact way. Usually isosurfaces are more than good
enough for most applications, though.
</p>
<p class="Note">
<strong>Note:</strong> at maximum a 15th degree polynomial can be represented with the poly object. If
a higher degree polynomial or other non-polynomial function has to be represented, then it is necessary to use the
isosurface object.
</p>
<h5><a name="s02_03_03_04_01">2.3.3.4.1 </a>Creating the polynomial function</h5>
<p>
The first step is to create the polynomial function to be represented. You will need some (high-school level)
mathematical knowledge for this.
</p>
<p>
<strong>1)</strong> Let's start with an easy example: A sphere.
</p>
<p>
The sphere function is:
</p>
<p>
<br><center><img alt="sphere function" src="images/tutorial/polyfunc1.png"></center>
</p>
<p>
Now we have to convert this to polynomial form:
</p>
<p>
<br><center><img alt="sphere polynomial" src="images/tutorial/polyfunc2.png"></center>
</p>
<p>
We will need a polynomial of the 2nd degree to represent this.
</p>
<p>
<strong>2)</strong> A more elaborated example:
</p>
<p>
Let's take the function:
</p>
<p>
<br><center><img alt="function" src="images/tutorial/polyfunc3.png"></center>
</p>
<p>
Converting this to polynomial form we get:
</p>
<p>
<br><center><img alt="polynomial" src="images/tutorial/polyfunc4.png"></center>
</p>
<p>
Although the highest power is 4 we will need a 5th order polynomial to represent this function (because we cannot
represent y<sup>4</sup>z with a 4th order polynomial).
</p>
<p>
<strong>3)</strong> And since we talked about the torus, let's also take it as an example.
</p>
<p>
A torus can be represented with the function:
</p>
<p>
<br><center><img alt="torus function" src="images/tutorial/polyfunc5.png"></center>
</p>
<p>
where r<sub>1</sub> is the major radius and r<sub>2</sub> is the minor radius.
</p>
<p>
Now, this is tougher to convert to polynomial form, but finally we get:
</p>
<p>
<br><center><img alt="torus polynomial" src="images/tutorial/polyfunc6.png"></center>
</p>
<p>
A 4th order polynomial is enough to represent this.
</p>
<p class="Note">
<strong>Note:</strong> not every function can be represented in polynomial form. Only functions that
use addition (and substraction), multiplication (and division) and scalar powers (including rational powers, eg. the
square root) can be represented. Also, the poly primitive supports only polynomials of the 7th degree at max.
</p>
<p>
Converting a function to polynomial form may be a very laborious task for certain functions. Some mathematical
programs are very helpful in this matter.
</p>
<h5><a name="s02_03_03_04_02">2.3.3.4.2 </a>Writing the polynomial vector</h5>
<p>
Now that we have the function in polynomial form, we have to write it in POV-Ray syntax. The syntax is specified in
the in the chapters "<a href="s_107.html#s03_04_03_02">Poly, Cubic and Quartic</a>" and "<a href="s_107.html#s03_04_03_03">Quadric</a>"
of the SDL section. There is also a table in this chapter which we will be using to make the polynomial vector. It is
easier to have this table printed on paper.
</p>
<p class="Note">
<strong>Note:</strong> It is also possible to make a little program with your favorite programming
language which will print the poly vector from the polynomial function, but making a program like this is up to you.
</p>
<p>
<strong>1)</strong> Let's start with the easy one, ie. the sphere.
</p>
<p>
Since the sphere can be represented with a polynomial of 2nd degree, we look at the row titled "2nd" in
the table. We see that it has 10 items, ie. we need a vector of size 10. Each item of the vector will be the factor of
the term listed in the table.
</p>
<p>
The polynomial was:
</p>
<p>
<br><center><img alt="sphere polynomial" src="images/tutorial/polyfunc2.png"></center>
</p>
<p>
Writing the poly in this way we get:
</p>
<pre>
#declare Radius=1;
poly
{ 2,
<1,0,0,0,1,
0,0,1,0,-Radius*Radius>
}
</pre>
<p>
Put each group of factors (separated with lines in the table) in their own lines.
</p>
<p>
In the table we see that the first item is the factor for x<sup>2</sup>, which is 1 in the function. The next item
is xy. Since it is not in the function, its factor is 0. Likewise the next item, which is xz. And so on. The last item
is the scalar term, which is in this case -r<sup>2</sup>.
</p>
<p>
If we make a proper scene and render it, we get:
</p>
<pre>
camera { location y*4-z*5 look_at 0 angle 35 }
light_source { <100,200,-50> 1 }
background { rgb <0,.25,.5> }
#declare Radius=1;
poly
{ 2,
<1,0,0,0,1,
0,0,1,0,-Radius*Radius>
pigment { rgb <1,.7,.3> } finish { specular .5 }
}
</pre>
<p>
<img alt="Sphere polynomial" src="images/tutorial/polypic1.png">
</p>
<p class="Note">
<strong>Note:</strong> there is a shortcut for 2nd degree polynomials: The <code><a href="s_107.html#s03_04_03_03">quadric</a></code>
primitive. Using a shortcut version, whenever possible, can lead to faster renderings. We can write the sphere code
described above in the following way:
</p>
<pre>
quadric
{ <1,1,1>, <0,0,0>, <0,0,0>, -Radius*Radius
pigment { rgb <1,.7,.3> } finish { specular .5 }
}
</pre>
<p>
<strong>2)</strong> Now lets try the second one. We do it similarly, but this time we need to look at the row
titled "5th" in the table.
</p>
<p>
The polynomial was:
</p>
<p>
<br><center><img alt="5th order polynomial" src="images/tutorial/polyfunc4.png"></center>
</p>
<p>
Writing the poly primitive we get:
</p>
<pre>
poly
{ 5,
<0,0,0,0,0,
0,0,0,0,0,
0,0,0,0,0,
0,0,0,1,0,
0,0,0,0,0,
-2,0,0,0,0,
0,0,0,0,0,
0,1,0,0,0,
0,0,0,0,0,
0,0,0,0,0,
0,0,0,0,0,0>
}
</pre>
<p>
With the proper scene we get:
</p>
<pre>
camera { location <8,20,-10>*.7 look_at x*.01 angle 35 }
light_source { <100,200,20> 1 }
background { rgb <0,.25,.5> }
poly
{ 5,
<0,0,0,0,0,
0,0,0,0,0,
0,0,0,0,0,
0,0,0,1,0,
0,0,0,0,0,
-2,0,0,0,0,
0,0,0,0,0,
0,1,0,0,0,
0,0,0,0,0,
0,0,0,0,0,
0,0,0,0,0,0>
clipped_by { box { <-4,-4,-1><4,4,1> } }
bounded_by { clipped_by }
pigment { rgb <1,.7,.3> } finish { specular .5 }
rotate <0,90,-90>
}
</pre>
<p>
<img alt="5th order polynomial example" src="images/tutorial/polypic2.png">
</p>
<p>
<strong>3)</strong> And finally the torus:
</p>
<p>
The polynomial was:
</p>
<p>
<br><center><img alt="torus polynomial" src="images/tutorial/polyfunc6.png"></center>
</p>
<p>
And we get the proper 4th degree poly primitive:
</p>
<pre>
camera { location y*4-z*5 look_at 0 angle 35 }
light_source { <100,200,-50> 1 }
background { rgb <0,.25,.5> }
#declare r1=1;
#declare r2=.5;
poly
{ 4,
<1,0,0,0,2,
0,0,2,0,-2*(r1*r1+r2*r2),
0,0,0,0,0,
0,0,0,0,0,
1,0,0,2,0,
2*(r1*r1-r2*r2),0,0,0,0,
1,0,-2*(r1*r1+r2*r2),0,pow(r1,4)+pow(r2,4)-2*r1*r1*r2*r2>
pigment { rgb <1,.7,.3> } finish { specular .5 }
}
</pre>
<p>
When rendered we get:
</p>
<p>
<img alt="Torus polynomial" border="0" src="images/tutorial/polypic3.png">
</p>
<p>
There is a shortcut for 4th order polynomials: The <code><a href="s_107.html#s03_04_03_02">quartic</a></code>
primitive. We can write the torus like this:
</p>
<pre>
quartic
{ <1,0,0,0,2,
0,0,2,0,-2*(r1*r1+r2*r2),
0,0,0,0,0,
0,0,0,0,0,
1,0,0,2,0,
2*(r1*r1-r2*r2),0,0,0,0,
1,0,-2*(r1*r1+r2*r2),0,pow(r1,4)+pow(r2,4)-2*r1*r1*r2*r2>
pigment { rgb <1,.7,.3> } finish { specular .5 }
}
</pre>
<h4><a name="s02_03_03_05">2.3.3.5 </a>Superquadric Ellipsoid Object</h4>
<a name="s02_03_03_05_i1">
<p>
Sometimes we want to make an object that does not have perfectly sharp edges like a box does. Then, the
superquadric ellipsoid shape made by the <code>superellipsoid</code> is a useful object. It is described by the simple
syntax:
</p>
<pre>
superellipsoid { <Value_E, Value_N >}
</pre>
<p>
Where <em>Value_E</em> and <em>Value_N</em> are float values greater than zero and less than or equal to one. Let's
make a superellipsoid and experiment with the values of <em>Value_E</em> and <em>Value_N</em> to see what kind of
shapes we can make. We create a file called <code> supellps.pov</code> and edit it as follows:
</p>
<pre>
#include "colors.inc"
camera {
location <10, 5, -20>
look_at 0
angle 15
}
background { color rgb <.5, .5, .5> }
light_source { <10, 50, -100> White }
</pre>
<p>
The addition of a gray background makes it a little easier to see our object. We now type:
</p>
<pre>
superellipsoid { <.25, .25>
pigment { Red }
}
</pre>
<p>
We save the file and trace it at 200x150 <code>-A</code> to see the shape. It will look like a box, but the edges
will be rounded off. Now let's experiment with different values of <em>Value_E</em> and <em> Value_N</em>. For the
next trace, try <1, 0.2>. The shape now looks like a cylinder, but the top edges are rounded. Now try <0.1,
1>. This shape is an odd one! We do not know exactly what to call it, but it is interesting. Finally, let's try
<1, 1>. Well, this is more familiar... a sphere!
</p>
<p>
There are a couple of facts about superellipsoids we should know. First, we should not use a value of 0 for either <em>
Value_E</em> nor <em> Value_N</em>. This will cause POV-Ray to incorrectly make a black box instead of our desired
shape. Second, very small values of <em>Value_E</em> and <em> Value_N</em> may yield strange results so they should be
avoided. Finally, the Sturmian root solver will not work with superellipsoids.
</p>
<p>
Superellipsoids are finite objects so they respond to auto-bounding and can be used in CSG.
</p>
<p>
Now let's use the superellipsoid to make something that would be useful in a scene. We will make a tiled floor and
place a couple of superellipsoid objects hovering over it. We can start with the file we have already made.
</p>
<p>
We rename it to <code> tiles.pov</code> and edit it so that it reads as follows:
</p>
<pre>
#include "colors.inc"
#include "textures.inc"
camera {
location <10, 5, -20>
look_at 0
angle 15
}
background { color rgb <.5, .5, .5> }
light_source{ <10, 50, -100> White }
</pre>
<p class="Note">
<strong>Note:</strong> we have added <code>#include "textures.inc"</code> so we can use
pre-defined textures. Now we want to define the superellipsoid which will be our tile.
</p>
<pre>
#declare Tile = superellipsoid { <0.5, 0.1>
scale <1, .05, 1>
}
</pre>
<p>
Superellipsoids are roughly 2*2*2 units unless we scale them otherwise. If we wish to lay a bunch of our tiles side
by side, they will have to be offset from each other so they do not overlap. We should select an offset value that is
slightly more than 2 so that we have some space between the tiles to fill with grout. So we now add this:
</p>
<pre>
#declare Offset = 2.1;
</pre>
<p>
We now want to lay down a row of tiles. Each tile will be offset from the original by an ever-increasing amount in
both the +z and -z directions. We refer to our offset and multiply by the tile's rank to determine the position of
each tile in the row. We also union these tiles into a single object called <code>Row</code> like this:
</p>
<pre>
#declare Row = union {
object { Tile }
object { Tile translate z*Offset }
object { Tile translate z*Offset*2 }
object { Tile translate z*Offset*3 }
object { Tile translate z*Offset*4 }
object { Tile translate z*Offset*5 }
object { Tile translate z*Offset*6 }
object { Tile translate z*Offset*7 }
object { Tile translate z*Offset*8 }
object { Tile translate z*Offset*9 }
object { Tile translate z*Offset*10 }
object { Tile translate -z*Offset }
object { Tile translate -z*Offset*2 }
object { Tile translate -z*Offset*3 }
object { Tile translate -z*Offset*4 }
object { Tile translate -z*Offset*5 }
object { Tile translate -z*Offset*6 }
}
</pre>
<p>
This gives us a single row of 17 tiles, more than enough to fill the screen. Now we must make copies of the <code>Row</code>
and translate them, again by the offset value, in both the +x and -x directions in ever increasing amounts in the same
manner.
</p>
<pre>
object { Row }
object { Row translate x*Offset }
object { Row translate x*Offset*2 }
object { Row translate x*Offset*3 }
object { Row translate x*Offset*4 }
object { Row translate x*Offset*5 }
object { Row translate x*Offset*6 }
object { Row translate x*Offset*7 }
object { Row translate -x*Offset }
object { Row translate -x*Offset*2 }
object { Row translate -x*Offset*3 }
object { Row translate -x*Offset*4 }
object { Row translate -x*Offset*5 }
object { Row translate -x*Offset*6 }
object { Row translate -x*Offset*7 }
</pre>
<p>
Finally, our tiles are complete. But we need a texture for them. To do this we union all of the <code>Rows</code>
together and apply a <code>White Marble</code> pigment and a somewhat shiny reflective surface to it:
</p>
<pre>
union{
object { Row }
object { Row translate x*Offset }
object { Row translate x*Offset*2 }
object { Row translate x*Offset*3 }
object { Row translate x*Offset*4 }
object { Row translate x*Offset*5 }
object { Row translate x*Offset*6 }
object { Row translate x*Offset*7 }
object { Row translate -x*Offset }
object { Row translate -x*Offset*2 }
object { Row translate -x*Offset*3 }
object { Row translate -x*Offset*4 }
object { Row translate -x*Offset*5 }
object { Row translate -x*Offset*6 }
object { Row translate -x*Offset*7 }
pigment { White_Marble }
finish { phong 1 phong_size 50 reflection .35 }
}
</pre>
<p>
We now need to add the grout. This can simply be a white plane. We have stepped up the ambient here a little so it
looks whiter.
</p>
<pre>
plane {
y, 0 //this is the grout
pigment { color White }
finish { ambient .4 diffuse .7 }
}
</pre>
<p>
To complete our scene, let's add five different superellipsoids, each a different color, so that they hover over
our tiles and are reflected in them.
</p>
<pre>
superellipsoid {
<0.1, 1>
pigment { Red }
translate <5, 3, 0>
scale .45
}
superellipsoid {
<1, 0.25>
pigment { Blue }
translate <-5, 3, 0>
scale .45
}
superellipsoid {
<0.2, 0.6>
pigment { Green }
translate <0, 3, 5>
scale .45
}
superellipsoid {
<0.25, 0.25>
pigment { Yellow }
translate <0, 3, -5>
scale .45
}
superellipsoid {
<1, 1>
pigment { Pink }
translate y*3
scale .45
}
</pre>
<p>
<img alt="Some superellipsoids hovering above a tiled floor." src="images/tutorial/superell.png">
</p>
<p>
We trace the scene at 320x200 <code>-A</code> to see the result. If we are happy with that, we do a final trace at
640x480 <code>+A0.2</code>.
</p>
<p>
<a name="l25">
<small><strong>More about "User-Defined Functions"</strong></small>
</a>
<ul>
<li><small>
<a href="s_155.html#s03_08_03_04">3.8.3.4 User defined Functions</a> in 3.8.3 Language Basics
</small>
<li><small>
<a href="s_97.html#s03_02_01_06">3.2.1.6 User-Defined Functions</a> in 3.2.1 Language Basics
</small>
</ul>
</p>
<p>
<a name="l26">
<small><strong>More about "Declaring Functions"</strong></small>
</a>
<ul>
<li><small>
<a href="s_155.html#s03_08_03_04">3.8.3.4 User defined Functions</a> in 3.8.3 Language Basics
</small>
<li><small>
<a href="s_97.html#s03_02_01_06">3.2.1.6 User-Defined Functions</a> in 3.2.1 Language Basics
</small>
</ul>
</p>
<p>
<a name="l27">
<small><strong>More about "internal functions"</strong></small>
</a>
<ul>
<li><small>
<a href="s_138.html#s03_07_07_02">3.7.7.2 Internal Functions</a> in 3.7.7 functions.inc
</small>
<li><small>
<a href="s_68.html#s02_03_03_03_05">2.3.3.3.5 Internal functions</a> in 2.3.3.3 Isosurface Object
</small>
</ul>
</p>
<p>
<a name="l28">
<small><strong>More about "normal{} statement"</strong></small>
</a>
<ul>
<li><small>
<a href="s_116.html#s03_05_02">3.5.2 Normal</a> in 3.5 Textures
</small>
<li><small>
<a href="s_162.html#s03_08_10_05">3.8.10.5 Normal</a> in 3.8.10 Texture
</small>
<li><small>
<a href="s_69.html#s02_03_04_02">2.3.4.2 Normals</a> in 2.3.4 Advanced Texture Options
</small>
</ul>
</p>
<p>
<a name="l29">
<small><strong>More about "noise generator"</strong></small>
</a>
<ul>
<li><small>
<a href="s_102.html#s03_03_03_10">3.3.3.10 Noise_generator</a> in 3.3.3 Global Settings
</small>
<li><small>
<a href="s_126.html#s03_05_12">3.5.12 Pattern Modifiers</a> in 3.5 Textures
</small>
<li><small>
<a href="s_126.html#s03_05_12_04">3.5.12.4 Noise Generators</a> in 3.5.12 Pattern Modifiers
</small>
</ul>
</p>
<br>
<table class="NavBar" width="100%">
<tr>
<td align="left" nowrap="" valign="middle" width="32">
<a href="s_67.html"><img alt="previous" border="0" src="prev.png"></a>
</td>
<td align="left" valign="middle" width="30%">
<a href="s_67.html">2.3.2 Polygon Based Shapes</a>
</td>
<td align="center" valign="middle">
<strong>2.3.3 Other Shapes</strong>
</td>
<td align="right" valign="middle" width="30%">
<a href="s_69.html">2.3.4 Advanced Texture Options</a>
</td>
<td align="right" nowrap="" valign="middle" width="32">
<a href="s_69.html"><img alt="next" border="0" src="next.png"></a>
</td>
</tr>
</table>
</body> </html>
|
