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
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<HTML>
<!-- Created on July, 22 2003 by texi2html 1.65 -->
<!--
Written by: Lionel Cons <Lionel.Cons@cern.ch> (original author)
Karl Berry <karl@freefriends.org>
Olaf Bachmann <obachman@mathematik.uni-kl.de>
and many others.
Maintained by: Olaf Bachmann <obachman@mathematik.uni-kl.de>
Send bugs and suggestions to <texi2html@mathematik.uni-kl.de>
-->
<HEAD>
<TITLE>User's Guide to the SOLID Collision Detection Library: </TITLE>
<META NAME="description" CONTENT="User's Guide to the SOLID Collision Detection Library: ">
<META NAME="keywords" CONTENT="User's Guide to the SOLID Collision Detection Library: ">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META NAME="Generator" CONTENT="texi2html 1.65">
</HEAD>
<BODY LANG="en" BGCOLOR="#FFFFFF" TEXT="#000000" LINK="#0000FF" VLINK="#800080" ALINK="#FF0000">
<A NAME="SEC_Top"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H1>User's Guide to the SOLID Collision Detection Library</H1></P><P>
This manual documents how to install and use SOLID.
</P><P>
<BLOCKQUOTE><TABLE BORDER=0 CELLSPACING=0>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="solid3.html#SEC1">1. License</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP">Says how you can copy and share SOLID.</TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="solid3.html#SEC2">2. Introduction</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP">What is it?</TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="solid3.html#SEC12">3. Installing the SOLID SDK</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP">How to configure, compile and install SOLID.</TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="solid3.html#SEC15">4. The SOLID API</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP">How the API is set up.</TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="solid3.html#SEC24">5. Projects and other things left to do</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP">Things still left to do.</TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="solid3.html#SEC26">6. Bug Reports</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP">Where to send bugs reports and questions.</TD></TR>
</TABLE></BLOCKQUOTE>
<P>
<HR SIZE=1>
<A NAME="SEC1"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<A NAME="License"></A>
<H1> 1. License </H1>
<!--docid::SEC1::-->
<P>
This open-source edition of SOLID version 3 is released under the terms
of either the GNU General Public License (GPL) or the Q Public License (QPL).
This means that for software created with SOLID version 3 you
must comply with the terms of either one of these licenses.
<BLOCKQUOTE><TABLE BORDER=0 CELLSPACING=0>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="solid3.html#SEC27">A. GNU GENERAL PUBLIC LICENSE</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP">The GNU General Public License</TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="solid3.html#SEC30">B. THE Q PUBLIC LICENSE</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP">The Q Public License</TD></TR>
</TABLE></BLOCKQUOTE>
<P>
For enquiries about commercial use of SOLID, please contact info@dtecta.com.
</P><P>
<A NAME="Introduction"></A>
<HR SIZE="6">
<A NAME="SEC2"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC3"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC12"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H1> 2. Introduction </H1>
<!--docid::SEC2::-->
<P>
SOLID is a software library containing functions for performing
intersection tests and proximity queries that are useful in the context
of collision detection. Collision detection is the process of detecting
pairs of geometric objects that are intersecting or are within a given
proximity of each other. In particular, SOLID is useful for detecting
collisions between objects that are moving relatively of each other over
time. The motions of objects are controlled by the client application,
and are not determined or affected by SOLID.
</P><P>
Furthermore, SOLID provides functionality for determining geometric data
pertaining to a pair of objects that is used by the client application
for computing the appropriate response to a collision. This data,
referred to as response data, is passed to the client application by
means of a callback mechanism or by direct queries from the client
application.
</P><P>
<HR SIZE="6">
<A NAME="SEC3"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC4"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 2.1 Overview </H2>
<!--docid::SEC3::-->
<P>
SOLID's functionality is subdivided into the following categories:
</P><P>
<OL>
<LI>Shape definition
<LI>Object placement and motion
<LI>Scene management
<LI>Response definition
<LI>Global actions
<LI>Broad phase
<P>
</OL>
<P>
<HR SIZE="6">
<A NAME="SEC4"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC3"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC5"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H3> 2.1.1 Shape Definition </H3>
<!--docid::SEC4::-->
<P>
The shape of a geometric object is defined relative to its local
coordinate system. A shape can be a simple geometric primitive, e.g., a
sphere or a line segment, or a complex shape composed of multiple
primitives. Shapes defined in terms of vertex positions, e.g., a
triangle mesh, may use vertex data that is stored in memory maintained
by the client application.
</P><P>
<HR SIZE="6">
<A NAME="SEC5"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC4"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC6"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H3> 2.1.2 Object Placement and Motion </H3>
<!--docid::SEC5::-->
<P>
A geometric object is defined by a shape, an affine transformation, and
a margin. The shape is placed in the world coordinate system by
specifying the object's local coordinate system represented by an affine
transformation.
The actual object is the set of points in world coordinates whose distance to
the shape is at most the margin. Motion of an object can be defined by
changing the placement of the local coordinate system, the margin, or the
positions of the vertices of the shape.
</P><P>
<HR SIZE="6">
<A NAME="SEC6"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC5"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC7"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H3> 2.1.3 Scene Management </H3>
<!--docid::SEC6::-->
<P>
Collections of objects on which collision detection has to be performed
are placed in a scene. SOLID is capable of maintaining multiple
scenes. Objects can be shared by multiple scenes. A scene maintains
cached data pertaining to the objects in the scene in order to speed-up
consecutive collision queries that are performed on the scene.
</P><P>
<HR SIZE="6">
<A NAME="SEC7"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC6"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC8"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H3> 2.1.4 Response Definition </H3>
<!--docid::SEC7::-->
<P>
In SOLID, collision response is handled by callback functions.
The types of response and the callback functions that needs to be executed
for each pair of intersecting objects is stored in a response
table. Currently, there are three types of response:
<OL>
<LI>Simple response: no response data is returned.
<LI>Witnessed response: a common point of the intersecting objects is
returned.
<LI>Depth response: the penetration depth of the intersecting objects
is returned. The penetration depth is the shortest vector over which one
of the objects needs to be translated in order to bring the objects into
touching contact.
</OL>
Response tables can be shared by multiple scenes, and per scene multiple
response tables can be used. To each object that is considered for
collision detection in a response table, a response class is
assigned. Responses can be defined per pair of response classes, on all
pairs containing a specific response class, or as a default for all
pairs of response classes.
On a pair of response classes multiple responses can be defined.
During the lifetime of an object, it is possible to
reassign the response class associated with the object per response
table.
<P>
<HR SIZE="6">
<A NAME="SEC8"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC7"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC9"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H3> 2.1.5 Global Actions </H3>
<!--docid::SEC8::-->
<P>
The main functionality of SOLID is to perform collision tests. Given a
scene and a response table, a collision test computes for all pairs of
colliding objects on which a response is defined the required response data and
passes these data together with the colliding pair to the callback. The
response actions are defined by the client
application in the callback and fall outside the scope of SOLID.
</P><P>
The client application may also obtain response data for a given object or
pair of objects by direct queries. These direct queries are useful for static
interference checks or for tracking the closest points of a pair of objects.
</P><P>
<HR SIZE="6">
<A NAME="SEC9"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC8"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC10"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H3> 2.1.6 Broad Phase </H3>
<!--docid::SEC9::-->
<P>
The broad phase detects changes in the overlap status of pairs of
axis-aligned boxes. The overlap status of a pair of boxes changes
whenever the boxes start or cease to overlap. The boxes are organized in
scenes similar to the scenes for objects. For each pair of boxes in a
scene whose overlap status is changed a callback is called. The client
defines a callback for pairs that start to overlap and one for pairs
that cease to overlap. The broad phase is actually a sub-layer of the
SOLID library, however the API can be directly accessed by the client
application.
</P><P>
<HR SIZE="6">
<A NAME="SEC10"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC9"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC11"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 2.2 Software Package </H2>
<!--docid::SEC10::-->
<P>
Currently, the SOLID package consists of three separate layers:
<OL>
<LI>MT: The Mathematics Toolkit. This is a set of C++ classes containing
abstract data types for scalars, vectors, points, quaternions, matrices,
coordinate systems, and bounding boxes. Global names in this layer are
prefixed with "MT_".
<P>
<LI>The broad phase: A set of C++ classes wrapped by a C API. The broad
phase detects changes in the overlap status of pairs of axis-aligned
boxes (pairs of boxes that start or cease to overlap). Global names in this
layer are prefixed with "BP_".
<P>
<LI>The narrow phase: A set of C++ classes wrapped by a C API. The narrow
phase performs exact collision tests for pairs of objects, and computes
response data for the colliding pairs of objects. Global names in this layer
are prefixed with "DT_".
<P>
</OL>
<P>
<HR SIZE="6">
<A NAME="SEC11"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC10"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC12"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 2.3 New Features of SOLID version 3 </H2>
<!--docid::SEC11::-->
<P>
Since the previous version 2.0 of SOLID, which was released in June 1998
under the terms of the GNU Library General Public License by the
Department of Mathematics & Computing Science of Eindhoven University of
Technology, SOLID has evolved and matured considerably. New
features as well as improvements for robustness and performance have
been added. The most important changes in SOLID 3 are:
<UL>
<LI>Use of single-precision floating-point numbers. The use of a
32-bit floating-point format is a requirement for games and other
interactive applications that run on current PC graphics hardware and
consoles.
<P>
<LI>Penetration depth computation as a new response type. The
penetration depth of a pair of intersecting objects is the shortest
vector over which one of the objects needs to be translated in order to
bring the objects in touching contact. The penetration depth can be used
as an approximation of the contact point and contact plane, which are
necessary for physics-based simulation. The depth response replaces the
smart response in SOLID 2.0. Smart response uses the configuration of
the previous time frame for finding the contact points and plane, which
could give bad results when objects were interpenetrating over a number
of frames.
<P>
<LI>SOLID 3 maintains multiple scenes. This feature is useful when
collision detection is required for multiple tasks. For instance, it is
possible to maintain at the same time a sound scene, a scene used for
visibility culling, and a scene for physics simulations, without the
objects in different scenes interfering with each other.
<P>
<LI>Stridden vertex arrays can be used for defining complex shapes.
<P>
<LI>Objects can be expanded spherically, i.e., a margin that defines
the radius of the sphere that is 'added' can be set for each object. The
object is the set of points whose distance to the shape is at most the
margin. Margins are useful for creating objects with rounded edges, or
'sensitive' areas around an object.
<P>
<LI>Functions for direct computation of response data have been added
to the API. This is useful, since it allows the client to
check overlap status, distance, etc., without having to perform a global
collision test.
<P>
<LI>A ray cast has been added to SOLID. The ray cast returns the
object in a scene that the ray hits first. Also, the hit spot and
surface normal to this object are returned.
<P>
<LI>Response callbacks are defined in a response table independent of
a scene.
<P>
<LI>Response callbacks are defined per pair of response classes rather
than per object pair. With each object for which a response is defined
in a response table, a response class is associated. The response class
of an object may change over time.
<P>
<LI>It is now possible to define multiple response callbacks per
object pair. This is useful for performing several actions for a single
collision (play sound, apply impulse, update statistics).
<P>
</UL>
<P>
<A NAME="Installation"></A>
<HR SIZE="6">
<A NAME="SEC12"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC11"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC13"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC15"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H1> 3. Installing the SOLID SDK </H1>
<!--docid::SEC12::-->
<P>
<HR SIZE="6">
<A NAME="SEC13"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC12"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC14"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 3.1 Requirements </H2>
<!--docid::SEC13::-->
<P>
The SOLID library (libsolid) and the broad-phase library (libbroad) have a
standard C API and can be linked to both C and C++
applications.
Note that libsolid and libbroad are internally coded in C++ and
thus need the libstdc++ library on Unix platforms (link using g++ rather
than gcc).
The mathematics toolkit (MT) is coded in standard C++ and uses templates
extensively. The source code compiles under GNU g++ version 2.95 and higher
and Microsoft Visual C++ version 6.0 SP4 and higher.
</P><P>
<HR SIZE="6">
<A NAME="SEC14"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC13"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC15"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 3.2 Installation </H2>
<!--docid::SEC14::-->
<P>
All header files that export types and functions are stored in the
`<SAMP>include</SAMP>' directory.
Source files are stored in the `<SAMP>src</SAMP>' directory.
On a Unix-like operating system, a `<SAMP>configure</SAMP>' script generated by the
GNU build tools (automake, autoconf, libtool) is used for creating the
Makefiles.
Simply type `<SAMP>./configure</SAMP>' followed by `<SAMP>make</SAMP>' in the root of the
SOLID distribution directory to build the SOLID lib and example programs.
Typing `<SAMP>make install</SAMP>' installs the SOLID library header files and
binaries as well as this documentation in `<SAMP>/usr/local</SAMP>'.
Note that you usually must have root privileges for adding items to
`<SAMP>/usr/local</SAMP>'.
See `<SAMP>./configure --help</SAMP>' for installing the library in a different
location.
</P><P>
The SOLID library can be built under Microsoft Visual C++ 6.0 and higher using
the workspace (.dsw) and project (.dsp) files found in the `<SAMP>VisualC6</SAMP>'
directory.
For use with Visual C++ 7.0 and higher, the workspace and project files must
first be converted to the newer solution (.sln) and project (.vcproj) formats.
On a Win32 platform, the simplest way to make the
SOLID SDK accessible in your applications is to add the SOLID
`<SAMP>include</SAMP>' and `<SAMP>lib</SAMP>' directories to respectively the include and link
path used by the compiler.
For running executables built using SOLID, the `<SAMP>solid.dll</SAMP>' should be
contained in the executable search path.
This is achieved most easily by copying these DLLs either to the directory
containing the executable or to the `<SAMP>WINxxxx/system32</SAMP>' directory.
</P><P>
SOLID uses the Qhull library for computing convex hulls of sets of points.
In case the Qhull library is not available for your platform, SOLID can still
be built. However, in that case, queries on convex hulls built with
<CODE>DT_NewPolytope</CODE> (see below) fall back to brute force algorithms,
and are therefore much slower.
In order to build a SOLID library without Qhull, make sure that the
preprocessing flag `<SAMP>-DQHULL</SAMP>' is not set.
</P><P>
The core of SOLID may use either single or double precision floating-point
arithmetic.
The default option is single-precision.
In order to build a double-precision SOLID core, use `<SAMP>configure</SAMP>' with
the `<SAMP>--enable-doubles</SAMP>' option.
Under Visual C++, doubles can be enabled using the `<SAMP>-DUSE_DOUBLES</SAMP>'
preprocessing flag.
Note that this flag only affects the floating-point numbers that are used
internally in SOLID.
The API functions always use single-precision floating-point numbers.
In the same way, the SOLID core can be built using a tracer class for
scalars.
See the file `<SAMP>MT_ScalarTracer.h</SAMP>' in the `<SAMP>include</SAMP>' directory.
A scalar tracer is used for tracing rounding errors in results of
floating-point operations.
An object of the type <CODE>MT_ScalarTracer</CODE> has a value field and an error
field. The value field holds the result of an operation, and the error
field multiplied by the machine epsilon gives an estimated upper bound for the
relative rounding error.
The <CODE>MT_ScalarTracer</CODE> class behaves as the primitive scalar types
<CODE>float</CODE> and <CODE>double</CODE>, however, constants of this type have to be
constructed explicitly.
In order to build a SOLID core that uses scalar tracers, create Makefiles using
`<SAMP>configure</SAMP>' with the `<SAMP>--enable-tracer</SAMP>' option.
Under Visual C++, the tracers are enabled using the `<SAMP>-DUSE_TRACER</SAMP>'
preprocessing flag.
This option is very useful for debugging purposes.
Makefiles for building debug binaries of SOLILD are created using the
`<SAMP>--enable-debug</SAMP>' option.
</P><P>
</P><P>
<A NAME="Usage"></A>
<HR SIZE="6">
<A NAME="SEC15"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC14"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC16"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC12"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC24"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H1> 4. The SOLID API </H1>
<!--docid::SEC15::-->
<P>
The SOLID API is a set of C functions. All API functions, also referred to as
commands, use arguments that have primitive types, such as, integers, floats,
and arrays of floats, or handles (type-mangled pointers) to internal objects
of SOLID. The types <CODE>DT_Scalar</CODE>, <CODE>DT_Vector</CODE> and <CODE>DT_Quaternion</CODE> are simply
typedefs:
<TABLE><tr><td> </td><td class=example><pre>
typedef float DT_Scalar;
typedef DT_Scalar DT_Vector3[3];
typedef DT_Scalar DT_Quaternion[4];
</pre></td></tr></table>The MT C++ classes can be used for representing geometric data such as vectors
and quaternions as they are implicitly casted to arrays of floats, however the
use of these classes is not required for calling SOLID functions.
SOLID API functions can be called using your own or third-party 3D geometry
objects if you stick with the following rules:
<UL>
<LI>All used scalar types are of the type <CODE>float</CODE>.
<P>
<LI>Quaternions store their imaginary vector part before the real scalar
part. Thus, for an array <CODE>float q[4]</CODE> that represents a quaternion,
<CODE>q[3]</CODE> must be the scalar part.
<P>
<LI>Transformations are specified using arrays of 16 floating-point numbers
representing a 4x4 column-major matrix as used in OpenGL. This matrix representation is
discussed below.
<P>
</UL>
<P>
<HR SIZE="6">
<A NAME="SEC16"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC15"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC17"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 4.1 Building Shapes </H2>
<!--docid::SEC16::-->
<P>
The commands for creating and destroying shapes are
<TABLE><tr><td> </td><td class=example><pre>
DT_ShapeHandle DT_NewBox(DT_Scalar x, DT_Scalar y, DT_Scalar z);
DT_ShapeHandle DT_NewCone(DT_Scalar radius, DT_Scalar height);
DT_ShapeHandle DT_NewCylinder(DT_Scalar radius, DT_Scalar height);
DT_ShapeHandle DT_NewSphere(DT_Scalar radius);
DT_ShapeHandle DT_NewPoint(const DT_Vector3 point);
DT_ShapeHandle DT_NewLineSegment(const DT_Vector3 source,
const DT_Vector3 target);
DT_ShapeHandle DT_NewMinkowski(DT_ShapeHandle shape1,
DT_ShapeHandle shape2);
DT_ShapeHandle DT_NewHull(DT_ShapeHandle shape1,
DT_ShapeHandle shape2);
void DT_DeleteShape(DT_ShapeHandle shape);
</pre></td></tr></table>Shapes are referred to by values of <CODE>DT_ShapeHandle</CODE>.
The command <CODE>DT_NewBox</CODE> creates a rectangular parallelepiped centered
at the origin and aligned with the axes of the shape's local coordinate system.
The parameters specify its extent along the respective coordinate axes.
The commands <CODE>DT_NewCone</CODE> and <CODE>DT_NewCylinder</CODE> create respectively a
cone and a cylinder centered
at the origin and whose central axis is aligned with the <EM>y</EM>-axis of the
local coordinate system.
The cone's apex is at <EM>y = height / 2</EM>.
The command <CODE>DT_NewSphere</CODE> creates a sphere centered at the origin of
the local coordinate system.
The command <CODE>DT_NewPoint</CODE> creates a single point.
The command <CODE>DT_NewLineSegment</CODE> creates a single line segment in a
similar way.
</P><P>
Any pair of convex shapes (including general polytopes) can be combined to
form compound shapes using the commands <CODE>DT_NewMinkowski</CODE> and
<CODE>DT_NewHull</CODE>.
The command <CODE>DT_NewMinkowski</CODE> `adds' the two shapes by sweeping one
shape along the other. For instance, the Minkowski addition of a sphere and a
line segment creates a hot dog.
The command <CODE>DT_NewHull</CODE> creates a shape that represents the exact convex
hull of the two shapes.
</P><P>
Complex shape types composed of simple polytopes (polytope soups) are
created using the
<CODE>DT_NewComplexShape</CODE> command.
Here, a simple polytope is a <EM>d</EM>-dimensional polytopes, where
<EM>d</EM> is at most 3.
A simple <EM>d</EM>-polytope can be a simplex (point, line segment, triangle,
tetrahedron), a convex polygon, or a convex polyhedron.
</P><P>
There are no topological constraints on the set of vertices of a polytope.
In particular, the vertices of a polytope need not be affinely independent,
and need not be extreme vertices of the convex hull.
However, convex polytopes with many vertices may deteriorate the performance.
Such complex polytopes should be created using the <CODE>DT_NewPolytope</CODE> command.
Make sure that in that case, SOLID is built using Qhull.
</P><P>
For constructing complex shapes the following
commands are used:
<TABLE><tr><td> </td><td class=example><pre>DT_VertexBaseHandle DT_NewVertexBase(const void *pointer,
DT_Size stride);
void DT_DeleteVertexBase(DT_VertexBaseHandle vertexBase);
void DT_ChangeVertexBase(DT_VertexBaseHandle vertexBase,
const void *pointer);
DT_ShapeHandle DT_NewComplexShape(DT_VertexBaseHandle vertexBase);
void DT_EndComplexShape();
DT_ShapeHandle DT_NewPolytope(DT_VertexBaseHandle vertexBase);
void DT_EndPolytope();
void DT_Begin();
void DT_End();
void DT_Vertex(const DT_Vector3 vertex);
void DT_VertexIndex(DT_Index index);
void DT_VertexIndices(DT_Count count, const DT_Index *indices);
void DT_VertexRange(DT_Index first, DT_Count count);
</pre></td></tr></table>A <EM>d</EM>-polytope is specified by enumerating its vertices. This can be done
in two ways. In the first way, the vertices are specified by value,
using the <CODE>DT_Vertex</CODE> command. The following example shows how the
faces of a pyramid are specified.
<TABLE><tr><td> </td><td class=example><pre>
DT_Vector3 float verts[] = {
{ 1.0f, 0.0f, 1.0f },
{ 1.0f, 0.0f, -1.0f },
{ -1.0f, 0.0f, -1.0f },
{ -1.0f, 0.0f, 1.0f },
{ 0.0f, 1.27f, 0.0f }
};
DT_ShapeHandle pyramid = DT_NewComplexShape(NULL);
DT_Begin();
DT_Vertex(verts[0]);
DT_Vertex(verts[1]);
DT_Vertex(verts[2]);
DT_Vertex(verts[3]);
DT_End();
DT_Begin();
DT_Vertex(verts[0]);
DT_Vertex(verts[1]);
DT_Vertex(verts[4]);
DT_End();
...
DT_EndComplexShape();
</pre></td></tr></table>Here, an argument of <CODE>NULL</CODE> in <CODE>DT_NewComplexShape</CODE> denotes that the
complex shape does not use an external vertex array.
</P><P>
In the second method, the vertices are referred to by indices.
For each complex shape, we specify a single array of vertices.
Vertex arrays are maintained by the client application and can be
accessed directly by SOLID.
Vertex arrays are accessed via vertex bases.
De command <CODE>DT_NewVertexBase</CODE> creates a vertex base for
the array given by the argument <CODE>pointer</CODE>.
The client must maintain vertex data in single-precision floating-point
format.
The client is free to store vertex data using arbitrary spacing
in-between the individual array items.
The spacing is specified using the <CODE>DT_Size stride</CODE> parameter.
For instance, the client maintains an array of vertices of the type:
<TABLE><tr><td> </td><td class=example><pre>
struct Vertex {
float xyz[3];
float uv[2];
float normal[3];
};
struct Vertex verts[NUM_VERTICES];
</pre></td></tr></table>When specifying a complex shape you can use this data as follows
<TABLE><tr><td> </td><td class=example><pre>
DT_VertexBaseHandle base = DT_NewVertexBase(verts[0].xyz,
sizeof(struct Vertex));
</pre></td></tr></table>A stride of zero denotes that the vertex coordinate data is packed in a
separate array, thus
<TABLE><tr><td> </td><td class=example><pre>
DT_Vector3 verts[NUM_VERTICES];
DT_VertexBaseHandle base = DT_NewVertexBase(verts[0], 0);
</pre></td></tr></table>Each time the vertices are updated, or a new vertex base is assigned,
to a complex shape, for instance, when using a deformable triangle mesh,
the client needs to notify SOLID of a changed vertex array by calling
<CODE>DT_ChangeVertexBase</CODE>.
We discuss the use of this command further on.
</P><P>
The handle to the vertex base is passed as argument to <CODE>DT_NewComplexShape</CODE>.
The command <CODE>DT_VertexIndex</CODE> is used for
specifying vertices. See the following example:
<TABLE><tr><td> </td><td class=example><pre>
DT_ShapeHandle pyramid = DT_NewComplexShape(base);
DT_Begin();
DT_VertexIndex(0);
DT_VertexIndex(1);
DT_VertexIndex(2);
DT_VertexIndex(3);
DT_End();
DT_Begin();
DT_VertexIndex(0);
DT_VertexIndex(1);
DT_VertexIndex(4);
DT_End();
...
DT_EndComplexShape();
</pre></td></tr></table>Alternatively, the indices can be placed into an array and specified
using the command <CODE>DT_VertexIndices</CODE>, as in the following example:
<TABLE><tr><td> </td><td class=example><pre>
DT_Index face0[4] = { 0, 1, 2, 3 };
DT_Index face1[3] = { 0, 1, 4 };
...
DT_VertexIndices(4, face0);
DT_VertexIndices(3, face1);
</pre></td></tr></table>Finally, a polytope can be specified from a range of vertices using the
command <CODE>DT_VertexRange</CODE>.
The range is specified by the first index and the number of vertices.
In the following example a pyramid is constructed as a convex polyhedron,
which is the convex hull of the vertices in the array.
<TABLE><tr><td> </td><td class=example><pre>
DT_ShapeHandle pyramid = DT_NewComplexShape(base);
DT_VertexRange(0, 5);
DT_EndComplexShape();
</pre></td></tr></table>The same shape can be built using the <CODE>DT_NewPolytope</CODE> command:
<TABLE><tr><td> </td><td class=example><pre>
DT_ShapeHandle pyramid = DT_NewPolytope(base);
DT_VertexRange(0, 5);
DT_EndPolytope();
</pre></td></tr></table>Note that within a <CODE>DT_NewPolytope</CODE> construction all the vertex array
commands can be used to specify vertices.
The commands <CODE>DT_Begin</CODE> and <CODE>DT_End</CODE> are ignored for polytopes.
Convex polytopes constructed using the <CODE>DT_NewPolytope</CODE> command
are preprocessed by SOLID in order to allow for faster testing, and
should be used when the number of vertices is large.
</P><P>
<HR SIZE="6">
<A NAME="SEC17"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC16"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC18"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 4.2 Creating and Moving Objects </H2>
<!--docid::SEC17::-->
<P>
An object is an instance of a shape.
The commands for creating, moving and deleting objects are
<TABLE><tr><td> </td><td class=example><pre>
DT_ObjectHandle DT_CreateObject(void *client_object, DT_ShapeHandle shape);
void DT_DestroyObject(DT_ObjectHandle object);
void DT_SetPosition(DT_ObjectHandle object, const DT_Vector3 position);
void DT_SetOrientation(DT_ObjectHandle object,
const DT_Quaternion orientation);
void DT_SetScaling(DT_ObjectHandle object, const DT_Vector3 scaling);
void DT_SetMatrixf(DT_ObjectHandle object, const float *m);
void DT_SetMatrixd(DT_ObjectHandle object, const double *m);
void DT_SetMargin(DT_ObjectHandle object, DT_Scalar margin);
</pre></td></tr></table>An object is referred to by a <CODE>DT_ObjectHandle</CODE>.
The first parameter <CODE>void *client_object</CODE> is a pointer to an
arbitrary structure in the client application.
This pointer is passed as parameter to the callback function in case of
a collision, and can be used for collision handling.
In general, a pointer to a structure in the client application
associated with the collision object should be used.
</P><P>
An object's motion is specified by changing the placement of the local
coordinate system of the shape.
Initially, the local coordinate system of an object
coincides with the world coordinate system.
</P><P>
The placement of an object is changed, either by setting the position,
orientation, and scaling, or by using an OpenGL 4x4
column-major matrix representing an affine transformation.
Placements are specified relative to the world coordinate system.
Rotations are specified using quaternions.
The object's local coordinate system can be scaled non-uniformly by
specifying a scale factor per coordinate axis.
Following example shows how a pair
of objects are given absolute placements.
<TABLE><tr><td> </td><td class=example><pre>
DT_ObjectHandle objectHandle = DT_CreateObject(&myObject, shapeHandle);
float position = { 0.0f, 1.0f, 1.0f };
float orientation = { 0.0f, 0.0f, 0.0f, 0.1f };
float scaling = { 1.0f, 2.0f, 1.0f };
DT_SetPosition(objectHandle, position);
DT_SetOrientation(objectHandle, orientation);
DT_SetScaling(objectHandle, scaling);
</pre></td></tr></table>For scalings along axes that are not coordinate axes, such as shears,
you should construct a 4x4 column-major matrix representation of the
local coordinate system and use <CODE>DT_SetMatrix</CODE> to specify the
object placement.
</P><P>
The <EM>x</EM>-axis of the local coordinate system relative
to the world coordinate system is the vector (m[0], m[1], m[2]), the
<EM>y</EM>-axis is (m[4], m[5], m[6]), the <EM>z</EM>-axis is (m[8], m[9],
m[10]), and the local origin is (m[12], m[13], m[14]).
The elements m[3], m[7], m[11], and m[15] are ignored.
These values are assumed to be 0, 0, 0, and 1, respectively.
Thus, only affine transformations are allowed.
</P><P>
By setting a positive margin using <CODE>DT_SetMargin</CODE> you can
spherically expand an object. The actual collision object is the set of
points whose distance to the transformed shape is at most the margin.
For instance, a hot dog or capsule can be created using
<TABLE><tr><td> </td><td class=example><pre>
DT_Vector3 source = { 0.0f, 0.0f, 0.0f }
DT_Vector3 target = { 0.0f, -1.5f, 0.0f }
DT_ShapeHandle line = DT_NewLineSegment(source, target);
DT_ObjectHandle object = DT_CreateObject(&myHotDog, line);
DT_SetMargin(object, 0.3f);
</pre></td></tr></table>This object is useful for navigating along walls and over terrains
Positions, orientations, scalings, and margins may all be changed during the
life time of an object.
</P><P>
<HR SIZE="6">
<A NAME="SEC18"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC17"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC19"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H3> 4.2.1 Who's Afraid of Quaternions? </H3>
<!--docid::SEC18::-->
<P>
A quaternion is a four-dimensional vector. The set of quaternions of
length one (points on a four-dimensional sphere) map to the set of
orientations in three-dimensional space. Since in many applications an
orientation defined by either a rotation axis and angle or by a
triple of Euler angles is more convenient, the package includes code for
quaternion operations. The code is found in the mathematics toolkit (MT).
</P><P>
The quaternion class is located in the file `<TT>MT_Quaternion.h</TT>'. The class
has constructors and methods for setting a quaternion. For example
<TABLE><tr><td> </td><td class=example><pre>
MT_Quaternion q1(axis, angle);
MT_Quaternion q2(yaw, pitch, roll);
...
q1.setRotation(axis, angle);
q2.setEuler(yaw, pitch, roll);
...
DT_SetOrientation(objectHandle, q1);
</pre></td></tr></table>Also included is a static method <CODE>MT_Quaternion::random()</CODE>, which
returns a random orientation.
</P><P>
<HR SIZE="6">
<A NAME="SEC19"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC18"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC20"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H3> 4.2.2 Proximity Queries </H3>
<!--docid::SEC19::-->
<P>
Objects can also be queried directly using the commands
<TABLE><tr><td> </td><td class=example><pre>
DT_Scalar DT_GetClosestPair(DT_ObjectHandle object1, DT_ObjectHandle object2,
DT_Vector3 point1, DT_Vector3 point2);
DT_Bool DT_GetCommonPoint(DT_ObjectHandle object1, DT_ObjectHandle object2,
DT_Vector3 point);
DT_Bool DT_GetPenDepth(DT_ObjectHandle object1, DT_ObjectHandle object2,
DT_Vector3 point1, DT_Vector3 point2);
</pre></td></tr></table>The command <CODE>DT_GetClosestPair</CODE> returns the distance between
<CODE>object1</CODE> and <CODE>object2</CODE>, and a pair of closest points
<CODE>point1</CODE> and <CODE>point2</CODE> given in world coordinates
The command <CODE>DT_GetCommonPoint</CODE> returns a boolean that denotes
whether the objects <CODE>object1</CODE> and <CODE>object2</CODE> intersect, and, in
case of an intersection, returns a common point <CODE>point</CODE> in world
coordinates.
The command <CODE>DT_GetPenDepth</CODE> also returns a boolean that denotes
whether the objects <CODE>object1</CODE> and <CODE>object2</CODE> intersect, and, in
case of an intersection, returns a pair of witness points of the
penetration depth <CODE>point1</CODE> and <CODE>point2</CODE> in world coordinates.
</P><P>
The maximum relative error in the closest points and penetration depth
computation can be set using
<TABLE><tr><td> </td><td class=example><pre>
void DT_SetAccuracy(DT_Scalar max_error);
</pre></td></tr></table>The default for <CODE>max_error</CODE> is 1.0e-3. Larger errors result
in better performance. Non-positive error tolerances are ignored.
</P><P>
The maximum tolerance on relative errors due to rounding is set using
<TABLE><tr><td> </td><td class=example><pre>
void DT_SetTolerance(DT_Scalar tol_error);
</pre></td></tr></table>This value is the estimated relative rounding error in complex computations
and is used for determining whether a floating-point number should be regarded
as zero or not. The default value for `tol_error' is the machine epsilon,
which is <CODE>FLT_EPSILON</CODE> when floats are used, and <CODE>DBL_EPSILON</CODE> when
double-precision floating-point numbers are used internally. Very large
tolerances result in false collisions. Setting tol_error too small
results in missed collisions. Non-positive error tolerances are ignored.
</P><P>
Furthermore, objects can be queried to return data maintained internally.
The world-axes aligned bounding box of an object is returned using
<TABLE><tr><td> </td><td class=example><pre>
void DT_GetBBox(DT_ObjectHandle object, DT_Vector3 min, DT_Vector3 max);
</pre></td></tr></table>Here, <CODE>min</CODE> and <CODE>max</CODE> are the vertices of the box with
respectively the least and greatest world coordinates.
The local-to-world transformation of an object can be returned using
<TABLE><tr><td> </td><td class=example><pre>
void DT_GetMatrixf(DT_ObjectHandle object, float *m);
void DT_GetMatrixd(DT_ObjectHandle object, double *m);
</pre></td></tr></table>The arguments of these commands are again arrays of 16 floating-point numbers
that represent a 4x4 column-major matrix as discussed above.
<HR SIZE="6">
<A NAME="SEC20"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC19"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC21"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 4.3 Scenes </H2>
<!--docid::SEC20::-->
<P>
For scenes with many objects the number of pairwise intersection queries can
become quite large.
To overcome this bottleneck, objects are maintained in scenes.
The commands for construction and destroying scenes are:
<TABLE><tr><td> </td><td class=example><pre>
DT_SceneHandle DT_CreateScene();
void DT_DestroyScene(DT_SceneHandle scene);
void DT_AddObject(DT_SceneHandle scene,
DT_ObjectHandle object);
void DT_RemoveObject(DT_SceneHandle scene,
DT_ObjectHandle object);
</pre></td></tr></table>Objects can be shared by multiple scenes.
Each scene tracks the changes of
placement and deformations of its objects, and updates its cached data
accordingly. In this way, global collision queries using <CODE>DT_Test</CODE> (see
below) can be processed much faster.
</P><P>
<HR SIZE="6">
<A NAME="SEC21"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC20"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC22"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 4.4 Response Handling </H2>
<!--docid::SEC21::-->
<P>
Collision response in SOLID is handled by means of callback functions.
The callback functions have the type <CODE>DT_ResponseCallback</CODE> defined by
<TABLE><tr><td> </td><td class=example><pre>
typedef DT_Bool (*DT_ResponseCallback)(void *client_data,
void *client_object1,
void *client_object2,
const DT_CollData *coll_data);
</pre></td></tr></table>Here, <CODE>client_data</CODE> is a pointer to an arbitrary structure in the
client application, <CODE>client_object1</CODE> and <CODE>client_object2</CODE> are
the pointers to structures in the client application specified in
<CODE>DT_CreateObject</CODE>, and <CODE>coll_data</CODE> is the response data computed by
SOLID.
The Boolean value returned by a callback functions indicates whether
further processing of callbacks is needed.
If <CODE>DT_FALSE</CODE> or <CODE>DT_CONTINUE</CODE> is returned, then the remaining
colliding object pairs are processed.
If <CODE>DT_TRUE</CODE> or <CODE>DT_DONE</CODE> is returned, then the call to
<CODE>DT_Test</CODE> is exited without further processing.
We discuss the <CODE>DT_Test</CODE> further on.
Currently, there are three types of response: <EM>simple</EM>,
<EM>depth</EM> and <EM>witnessed</EM> response. For simple response the value
of <CODE>coll_data</CODE> is <CODE>NULL</CODE>.
For depth and witnessed response <CODE>coll_data</CODE> points to the
following structure
<TABLE><tr><td> </td><td class=example><pre>
typedef struct DT_CollData {
DT_Vector3 point1;
DT_Vector3 point2;
DT_Vector3 normal;
} DT_CollData;
</pre></td></tr></table>An object of this type represents a pair of points of the respective
objects. The points <CODE>point1</CODE> and <CODE>point2</CODE> are given relative
to the world coordinate system. The <CODE>normal</CODE> field is used for
depth response only.
</P><P>
For witnessed response, the points represent a witness of the collision.
Both points are contained in the intersection of the colliding objects.
Note that the witness points are not necessarily equal.
For depth response, the <CODE>normal</CODE> represent the penetration depth
vector.
The penetration depth vector is the shortest vector over which one
object needs to be translated in order to bring the two objects into
touching contact.
The <CODE>point1</CODE> and <CODE>point2</CODE> fields contain the witness points of
the penetration depth, thus <CODE>normal = point2 - point1</CODE>.
</P><P>
Response callbacks are managed in <EM>response tables</EM>.
Response tables are defined independent of the scenes in which they are used.
Multiple response tables can be used in one scene, and a response table
can be shared among scenes.
Responses are defined on (pairs of) response classes. Each response table
maintains its set of response classes.
A response table is created and destroyed using the commands
<TABLE><tr><td> </td><td class=example><pre>
DT_RespTableHandle DT_CreateRespTable();
void DT_DestroyRespTable(DT_RespTableHandle respTable);
</pre></td></tr></table>A response class for a response table is generated using
<TABLE><tr><td> </td><td class=example><pre>
DT_ResponseClass DT_GenResponseClass(DT_RespTableHandle respTable);
</pre></td></tr></table>To each object for which a response is defined in the response table a
response class needs to be assigned. This is done using
<TABLE><tr><td> </td><td class=example><pre>
void DT_SetResponseClass(DT_RespTableHandle respTable,
DT_ObjectHandle object,
DT_ResponseClass responseClass);
</pre></td></tr></table>For each pair of objects multiple responses can be defined. A response
is a callback together with its response type and client data.
The <CODE>DT_ResponseType</CODE> is defined by
<TABLE><tr><td> </td><td class=example><pre>
typedef enum DT_ResponseType {
DT_NO_RESPONSE,
DT_SIMPLE_RESPONSE,
DT_WITNESSED_RESPONSE
DT_DEPTH_RESPONSE,
} DT_ResponseType;
</pre></td></tr></table>Responses can be defined for all pairs of response classes...
<TABLE><tr><td> </td><td class=example><pre>
void DT_AddDefaultResponse(DT_RespTableHandle respTable,
DT_ResponseCallback response,
DT_ResponseType type,
void *client_data);
void DT_RemoveDefaultResponse(DT_RespTableHandle respTable,
DT_ResponseCallback response);
</pre></td></tr></table>...per response class...
<TABLE><tr><td> </td><td class=example><pre>
void DT_AddClassResponse(DT_RespTableHandle respTable,
DT_ResponseClass responseClass,
DT_ResponseCallback response,
DT_ResponseType type,
void *client_data);
void DT_RemoveClassResponse(DT_RespTableHandle respTable,
DT_ResponseClass responseClass,
DT_ResponseCallback response);
</pre></td></tr></table>... and per pair of response classes...
<TABLE><tr><td> </td><td class=example><pre>
void DT_AddPairResponse(DT_RespTableHandle respTable,
DT_ResponseClass responseClass1,
DT_ResponseClass responseClass2,
DT_ResponseCallback response,
DT_ResponseType type,
void *client_data);
void DT_RemovePairResponse(DT_RespTableHandle respTable,
DT_ResponseClass responseClass1,
DT_ResponseClass responseClass2,
DT_ResponseCallback response);
</pre></td></tr></table>If for an object pair, one of the objects has a class object response defined,
that needs to be overruled by a pair response, then you should remove
the callback defined in the class response for the pair and add the pair
response, thus
<TABLE><tr><td> </td><td class=example><pre>
DT_AddClassResponse(respTable, class1, classResponse,
DT_SIMPLE_RESPONSE, client_data);
DT_RemovePairResponse(respTable, class1, class2, classResponse);
DT_AddPairResponse(respTable, class1, class2, pairResponse,
DT_DEPTH_RESPONSE, client_data);
</pre></td></tr></table>In the same way, a default response can be overruled by a class or pair
response.
</P><P>
The response callback functions are executed by calling
<TABLE><tr><td> </td><td class=example><pre>
DT_Count DT_Test(DT_SceneHandle scene, DT_RespTableHandle respTable);
</pre></td></tr></table>This command calls for each colliding pair of objects the corresponding
callback function until all pairs are processed or until a callback function
returns <CODE>DT_TRUE</CODE> or <CODE>DT_DONE</CODE>.
It returns the number of object pairs for which callback functions have been
executed.
</P><P>
Note: If the response classes of the objects in a callback differ, then
<CODE>client_object1</CODE> has a `lower' response class than
<CODE>client_object2</CODE>. That is, the response class of <CODE>client_object1</CODE> is
generated before the response class of <CODE>client_object2</CODE>.
</P><P>
<HR SIZE="6">
<A NAME="SEC22"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC21"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC23"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 4.5 Deformable Models </H2>
<!--docid::SEC22::-->
<P>
SOLID handles deformations of complex shapes. In this context
deformations are specified by changes of vertex positions.
Complex shapes that are defined using a vertex array in the client
application may be deformed by changing the array elements, or
specifying a new array.
SOLID is notified of a change of vertices by the command
<TABLE><tr><td> </td><td class=example><pre>
void DT_ChangeVertexBase(DT_VertexBaseHandle vertexBase,
const void *pointer);
</pre></td></tr></table>Note that polytopes constructed from a vertex base using <CODE>DT_NewPolytope</CODE> are not affected
by a change of vertices.
</P><P>
<HR SIZE="6">
<A NAME="SEC23"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC22"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC24"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 4.6 Ray Cast </H2>
<!--docid::SEC23::-->
<P>
NOTE: This feature is currently implemented for spheres, boxes, triangles, and
triangle meshes only.
Also, margins are ignored for ray casts.
</P><P>
The commands for performing ray casts are
<TABLE><tr><td> </td><td class=example><pre>
void *DT_RayCast(DT_SceneHandle scene, void *ignore_client,
const DT_Vector3 source,
const DT_Vector3 target,
DT_Scalar max_param,
DT_Scalar *param, DT_Vector3 normal);
DT_Bool DT_ObjectRayCast(DT_ObjectHandle object,
const DT_Vector3 source,
const DT_Vector3 target,
DT_Scalar max_param,
DT_Scalar *param, DT_Vector3 normal);
</pre></td></tr></table>The ray is given by <CODE>source</CODE>, <CODE>target</CODE>, and <CODE>max_param</CODE>.
It represents the line segment <EM><CODE>source</CODE> + (<CODE>target</CODE> - <CODE>source</CODE>) * t</EM>,
where <EM>t</EM> is a member of the interval <EM>[0, <CODE>max_param</CODE>]</EM>.
So, if <CODE>max_param</CODE> is 1, then the ray is simply the line segment from
<CODE>source</CODE> to <CODE>target</CODE>, whereas if <CODE>max_param</CODE> is equal to
<CODE>FLT_MAX</CODE>, then the ray is `infinite'.
</P><P>
<CODE>DT_RayCast</CODE> returns a pointer to the client object of an object in
<CODE>scene</CODE> that is hit first by the ray, or <CODE>NULL</CODE> if no object is hit.
<CODE>DT_ObjectRayCast</CODE> performs a ray cast on a single object and
returns a Boolean indicating a hit.
In case of a hit, <CODE>param</CODE> points to the <EM>t</EM> of the hit spot,
and <CODE>normal</CODE> is a normal to the object's surface in world coordinates.
The normal always points towards the <CODE>source</CODE>.
An object can be made transparent for the ray cast by specifying the
object's client object as <CODE>ignore_client</CODE>.
This is useful if you need to ignore hits of the ray with the source
object of the ray.
For instance terrain following can be implemented by casting a ray down
and setting the moving object at a distance above the spot.
In this case, you are probably interested in hits with the terrain only,
and do not need reports of hits with the moving object.
</P><P>
<A NAME="Projects"></A>
<HR SIZE="6">
<A NAME="SEC24"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC23"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC25"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC15"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC26"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H1> 5. Projects and other things left to do </H1>
<!--docid::SEC24::-->
<P>
<HR SIZE="6">
<A NAME="SEC25"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC24"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC26"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[ << ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 5.1 Coming Attractions </H2>
<!--docid::SEC25::-->
<P>
SOLID 4 will have the following added features:
<OL>
<LI>Compressed AABB trees for reducing the memory footprint of triangle
meshes to roughly 18 bytes per triangle.
<P>
<LI>A general ray cast for all shape types.
<P>
<LI>Shape casting: returning the first collision of a shape that is translated along
a ray.
<P>
<LI>Scene graphs for managing complex shapes.
<P>
<LI>A binary format for streaming of shapes.
<P>
</OL>
<P>
<A NAME="Bugs"></A>
<HR SIZE="6">
<A NAME="SEC26"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC25"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC27"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC24"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ >> ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H1> 6. Bug Reports </H1>
<!--docid::SEC26::-->
<P>
Please send remarks, questions, and bug reports to gino@dtecta.com.
</P><P>
NOTE: There is no termination after a fixed maximum number of
iterations of GJK in this version of SOLID, since I believe in an
industrial-strength GJK without resorting to tolerance tweaking or forced
termination after a certain number of iterations. I welcome any reports
of SOLID misbehaving.
</P><P>
<A NAME="GPL"></A>
<HR SIZE="6">
<A NAME="SEC27"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC26"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC28"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H1> A. GNU GENERAL PUBLIC LICENSE </H1>
<!--docid::SEC27::-->
<center>
Version 2, June 1991
</center>
<P>
<TABLE><tr><td> </td><td class=display><pre style="font-family: serif">Copyright © 1989, 1991 Free Software Foundation, Inc.
59 Temple Place - Suite 330, Boston, MA 02111-1307, USA
Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
</pre></td></tr></table></P><P>
<HR SIZE="6">
<A NAME="SEC28"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC27"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC29"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> Preamble </H2>
<!--docid::SEC28::-->
<P>
The licenses for most software are designed to take away your
freedom to share and change it. By contrast, the GNU General Public
License is intended to guarantee your freedom to share and change free
software--to make sure the software is free for all its users. This
General Public License applies to most of the Free Software
Foundation's software and to any other program whose authors commit to
using it. (Some other Free Software Foundation software is covered by
the GNU Library General Public License instead.) You can apply it to
your programs, too.
</P><P>
When we speak of free software, we are referring to freedom, not
price. Our General Public Licenses are designed to make sure that you
have the freedom to distribute copies of free software (and charge for
this service if you wish), that you receive source code or can get it
if you want it, that you can change the software or use pieces of it
in new free programs; and that you know you can do these things.
</P><P>
To protect your rights, we need to make restrictions that forbid
anyone to deny you these rights or to ask you to surrender the rights.
These restrictions translate to certain responsibilities for you if you
distribute copies of the software, or if you modify it.
</P><P>
For example, if you distribute copies of such a program, whether
gratis or for a fee, you must give the recipients all the rights that
you have. You must make sure that they, too, receive or can get the
source code. And you must show them these terms so they know their
rights.
</P><P>
We protect your rights with two steps: (1) copyright the software, and
(2) offer you this license which gives you legal permission to copy,
distribute and/or modify the software.
</P><P>
Also, for each author's protection and ours, we want to make certain
that everyone understands that there is no warranty for this free
software. If the software is modified by someone else and passed on, we
want its recipients to know that what they have is not the original, so
that any problems introduced by others will not reflect on the original
authors' reputations.
</P><P>
Finally, any free program is threatened constantly by software
patents. We wish to avoid the danger that redistributors of a free
program will individually obtain patent licenses, in effect making the
program proprietary. To prevent this, we have made it clear that any
patent must be licensed for everyone's free use or not licensed at all.
</P><P>
The precise terms and conditions for copying, distribution and
modification follow.
</P><P>
<center>
TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
</center>
</P><P>
<OL>
<LI>
This License applies to any program or other work which contains
a notice placed by the copyright holder saying it may be distributed
under the terms of this General Public License. The "Program", below,
refers to any such program or work, and a "work based on the Program"
means either the Program or any derivative work under copyright law:
that is to say, a work containing the Program or a portion of it,
either verbatim or with modifications and/or translated into another
language. (Hereinafter, translation is included without limitation in
the term "modification".) Each licensee is addressed as "you".
<P>
Activities other than copying, distribution and modification are not
covered by this License; they are outside its scope. The act of
running the Program is not restricted, and the output from the Program
is covered only if its contents constitute a work based on the
Program (independent of having been made by running the Program).
Whether that is true depends on what the Program does.
</P><P>
<LI>
You may copy and distribute verbatim copies of the Program's
source code as you receive it, in any medium, provided that you
conspicuously and appropriately publish on each copy an appropriate
copyright notice and disclaimer of warranty; keep intact all the
notices that refer to this License and to the absence of any warranty;
and give any other recipients of the Program a copy of this License
along with the Program.
<P>
You may charge a fee for the physical act of transferring a copy, and
you may at your option offer warranty protection in exchange for a fee.
</P><P>
<LI>
You may modify your copy or copies of the Program or any portion
of it, thus forming a work based on the Program, and copy and
distribute such modifications or work under the terms of Section 1
above, provided that you also meet all of these conditions:
<P>
<OL>
<LI>
You must cause the modified files to carry prominent notices
stating that you changed the files and the date of any change.
<P>
<LI>
You must cause any work that you distribute or publish, that in
whole or in part contains or is derived from the Program or any
part thereof, to be licensed as a whole at no charge to all third
parties under the terms of this License.
<P>
<LI>
If the modified program normally reads commands interactively
when run, you must cause it, when started running for such
interactive use in the most ordinary way, to print or display an
announcement including an appropriate copyright notice and a
notice that there is no warranty (or else, saying that you provide
a warranty) and that users may redistribute the program under
these conditions, and telling the user how to view a copy of this
License. (Exception: if the Program itself is interactive but
does not normally print such an announcement, your work based on
the Program is not required to print an announcement.)
</OL>
<P>
These requirements apply to the modified work as a whole. If
identifiable sections of that work are not derived from the Program,
and can be reasonably considered independent and separate works in
themselves, then this License, and its terms, do not apply to those
sections when you distribute them as separate works. But when you
distribute the same sections as part of a whole which is a work based
on the Program, the distribution of the whole must be on the terms of
this License, whose permissions for other licensees extend to the
entire whole, and thus to each and every part regardless of who wrote it.
</P><P>
Thus, it is not the intent of this section to claim rights or contest
your rights to work written entirely by you; rather, the intent is to
exercise the right to control the distribution of derivative or
collective works based on the Program.
</P><P>
In addition, mere aggregation of another work not based on the Program
with the Program (or with a work based on the Program) on a volume of
a storage or distribution medium does not bring the other work under
the scope of this License.
</P><P>
<LI>
You may copy and distribute the Program (or a work based on it,
under Section 2) in object code or executable form under the terms of
Sections 1 and 2 above provided that you also do one of the following:
<P>
<OL>
<LI>
Accompany it with the complete corresponding machine-readable
source code, which must be distributed under the terms of Sections
1 and 2 above on a medium customarily used for software interchange; or,
<P>
<LI>
Accompany it with a written offer, valid for at least three
years, to give any third party, for a charge no more than your
cost of physically performing source distribution, a complete
machine-readable copy of the corresponding source code, to be
distributed under the terms of Sections 1 and 2 above on a medium
customarily used for software interchange; or,
<P>
<LI>
Accompany it with the information you received as to the offer
to distribute corresponding source code. (This alternative is
allowed only for noncommercial distribution and only if you
received the program in object code or executable form with such
an offer, in accord with Subsection b above.)
</OL>
<P>
The source code for a work means the preferred form of the work for
making modifications to it. For an executable work, complete source
code means all the source code for all modules it contains, plus any
associated interface definition files, plus the scripts used to
control compilation and installation of the executable. However, as a
special exception, the source code distributed need not include
anything that is normally distributed (in either source or binary
form) with the major components (compiler, kernel, and so on) of the
operating system on which the executable runs, unless that component
itself accompanies the executable.
</P><P>
If distribution of executable or object code is made by offering
access to copy from a designated place, then offering equivalent
access to copy the source code from the same place counts as
distribution of the source code, even though third parties are not
compelled to copy the source along with the object code.
</P><P>
<LI>
You may not copy, modify, sublicense, or distribute the Program
except as expressly provided under this License. Any attempt
otherwise to copy, modify, sublicense or distribute the Program is
void, and will automatically terminate your rights under this License.
However, parties who have received copies, or rights, from you under
this License will not have their licenses terminated so long as such
parties remain in full compliance.
<P>
<LI>
You are not required to accept this License, since you have not
signed it. However, nothing else grants you permission to modify or
distribute the Program or its derivative works. These actions are
prohibited by law if you do not accept this License. Therefore, by
modifying or distributing the Program (or any work based on the
Program), you indicate your acceptance of this License to do so, and
all its terms and conditions for copying, distributing or modifying
the Program or works based on it.
<P>
<LI>
Each time you redistribute the Program (or any work based on the
Program), the recipient automatically receives a license from the
original licensor to copy, distribute or modify the Program subject to
these terms and conditions. You may not impose any further
restrictions on the recipients' exercise of the rights granted herein.
You are not responsible for enforcing compliance by third parties to
this License.
<P>
<LI>
If, as a consequence of a court judgment or allegation of patent
infringement or for any other reason (not limited to patent issues),
conditions are imposed on you (whether by court order, agreement or
otherwise) that contradict the conditions of this License, they do not
excuse you from the conditions of this License. If you cannot
distribute so as to satisfy simultaneously your obligations under this
License and any other pertinent obligations, then as a consequence you
may not distribute the Program at all. For example, if a patent
license would not permit royalty-free redistribution of the Program by
all those who receive copies directly or indirectly through you, then
the only way you could satisfy both it and this License would be to
refrain entirely from distribution of the Program.
<P>
If any portion of this section is held invalid or unenforceable under
any particular circumstance, the balance of the section is intended to
apply and the section as a whole is intended to apply in other
circumstances.
</P><P>
It is not the purpose of this section to induce you to infringe any
patents or other property right claims or to contest validity of any
such claims; this section has the sole purpose of protecting the
integrity of the free software distribution system, which is
implemented by public license practices. Many people have made
generous contributions to the wide range of software distributed
through that system in reliance on consistent application of that
system; it is up to the author/donor to decide if he or she is willing
to distribute software through any other system and a licensee cannot
impose that choice.
</P><P>
This section is intended to make thoroughly clear what is believed to
be a consequence of the rest of this License.
</P><P>
<LI>
If the distribution and/or use of the Program is restricted in
certain countries either by patents or by copyrighted interfaces, the
original copyright holder who places the Program under this License
may add an explicit geographical distribution limitation excluding
those countries, so that distribution is permitted only in or among
countries not thus excluded. In such case, this License incorporates
the limitation as if written in the body of this License.
<P>
<LI>
The Free Software Foundation may publish revised and/or new versions
of the General Public License from time to time. Such new versions will
be similar in spirit to the present version, but may differ in detail to
address new problems or concerns.
<P>
Each version is given a distinguishing version number. If the Program
specifies a version number of this License which applies to it and "any
later version", you have the option of following the terms and conditions
either of that version or of any later version published by the Free
Software Foundation. If the Program does not specify a version number of
this License, you may choose any version ever published by the Free Software
Foundation.
</P><P>
<LI>
If you wish to incorporate parts of the Program into other free
programs whose distribution conditions are different, write to the author
to ask for permission. For software which is copyrighted by the Free
Software Foundation, write to the Free Software Foundation; we sometimes
make exceptions for this. Our decision will be guided by the two goals
of preserving the free status of all derivatives of our free software and
of promoting the sharing and reuse of software generally.
<P>
<center>
NO WARRANTY
</center>
</P><P>
<LI>
BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY
FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN
OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES
PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED
OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS
TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE
PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING,
REPAIR OR CORRECTION.
<P>
<LI>
IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR
REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES,
INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING
OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED
TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY
YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER
PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE
POSSIBILITY OF SUCH DAMAGES.
</OL>
<P>
<center>
END OF TERMS AND CONDITIONS
</center>
</P><P>
<HR SIZE="6">
<A NAME="SEC29"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC28"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC30"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> Appendix: How to Apply These Terms to Your New Programs </H2>
<!--docid::SEC29::-->
<P>
If you develop a new program, and you want it to be of the greatest
possible use to the public, the best way to achieve this is to make it
free software which everyone can redistribute and change under these terms.
</P><P>
To do so, attach the following notices to the program. It is safest
to attach them to the start of each source file to most effectively
convey the exclusion of warranty; and each file should have at least
the "copyright" line and a pointer to where the full notice is found.
</P><P>
<TABLE><tr><td> </td><td class=smallexample><FONT SIZE=-1><pre><VAR>one line to give the program's name and a brief idea of what it does.</VAR>
Copyright (C) <VAR>yyyy</VAR> <VAR>name of author</VAR>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
</FONT></pre></td></tr></table></P><P>
Also add information on how to contact you by electronic and paper mail.
</P><P>
If the program is interactive, make it output a short notice like this
when it starts in an interactive mode:
</P><P>
<TABLE><tr><td> </td><td class=smallexample><FONT SIZE=-1><pre>Gnomovision version 69, Copyright (C) 19<VAR>yy</VAR> <VAR>name of author</VAR>
Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
This is free software, and you are welcome to redistribute it
under certain conditions; type `show c' for details.
</FONT></pre></td></tr></table></P><P>
The hypothetical commands `<SAMP>show w</SAMP>' and `<SAMP>show c</SAMP>' should show
the appropriate parts of the General Public License. Of course, the
commands you use may be called something other than `<SAMP>show w</SAMP>' and
`<SAMP>show c</SAMP>'; they could even be mouse-clicks or menu items--whatever
suits your program.
</P><P>
You should also get your employer (if you work as a programmer) or your
school, if any, to sign a "copyright disclaimer" for the program, if
necessary. Here is a sample; alter the names:
</P><P>
<TABLE><tr><td> </td><td class=example><pre>Yoyodyne, Inc., hereby disclaims all copyright interest in the program
`Gnomovision' (which makes passes at compilers) written by James Hacker.
<VAR>signature of Ty Coon</VAR>, 1 April 1989
Ty Coon, President of Vice
</pre></td></tr></table></P><P>
This General Public License does not permit incorporating your program into
proprietary programs. If your program is a subroutine library, you may
consider it more useful to permit linking proprietary applications with the
library. If this is what you want to do, use the GNU Library General
Public License instead of this License.
</P><P>
<A NAME="QPL"></A>
<HR SIZE="6">
<A NAME="SEC30"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC29"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC31"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H1> B. THE Q PUBLIC LICENSE </H1>
<!--docid::SEC30::-->
<center>
version 1.0
</center>
<P>
<TABLE><tr><td> </td><td class=display><pre style="font-family: serif">Copyright © 1999 Troll Tech AS, Norway.
Everyone is permitted to copy and
distribute this license document.
</pre></td></tr></table></P><P>
The intent of this license is to establish freedom to share and change the
software regulated by this license under the open source model.
</P><P>
This license applies to any software containing a notice placed by the
copyright holder saying that it may be distributed under the terms of
the Q Public License version 1.0. Such software is herein referred to as
the Software. This license covers modification and distribution of the
Software, use of third-party application programs based on the Software,
and development of free software which uses the Software.
</P><P>
<HR SIZE="6">
<A NAME="SEC31"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC30"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC32"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> Granted Rights </H2>
<!--docid::SEC31::-->
<P>
<OL>
<LI>
You are granted the non-exclusive rights set forth in this license
provided you agree to and comply with any and all conditions in this
license. Whole or partial distribution of the Software, or software
items that link with the Software, in any form signifies acceptance of this license.
<P>
<LI>
You may copy and distribute the Software in unmodified form provided
that the entire package, including - but not restricted to - copyright,
trademark notices and disclaimers, as released by the initial developer of the Software, is distributed.
<P>
<LI>
You may make modifications to the Software and distribute your modifications, in a form that is separate from the Software, such as patches. The following restrictions apply to modifications:
<P>
<OL>
<LI>
Modifications must not alter or remove any copyright notices in the Software.
<P>
<LI>
When modifications to the Software are released under this license, a non-exclusive royalty-free right is granted to the initial developer of the Software to distribute your modification in future versions of the Software provided such versions remain available under these terms in addition to any other license(s) of the initial developer.
<P>
</OL>
<P>
<LI>
You may distribute machine-executable forms of the Software or machine-executable forms of modified versions of the Software, provided that you meet these restrictions:
<P>
<OL>
<LI>
You must include this license document in the distribution.
<P>
<LI>
You must ensure that all recipients of the machine-executable forms are
also able to receive the complete machine-readable source code to the
distributed Software, including all modifications, without any charge beyond the costs of data transfer, and place prominent notices in the distribution explaining this.
<P>
<LI>
You must ensure that all modifications included in the machine-executable forms are available under the terms of this license.
<P>
</OL>
<P>
<LI>
You may use the original or modified versions of the Software to compile, link and run application programs legally developed by you or by others.
<P>
<LI>
You may develop application programs, reusable components and other software items that link with the original or modified versions of the Software. These items, when distributed, are subject to the following requirements:
<P>
<OL>
<LI>
You must ensure that all recipients of machine-executable forms of these items are also able to receive and use the complete machine-readable source code to the items without any charge beyond the costs of data transfer.
<P>
<LI>
You must explicitly license all recipients of your items to use and re-distribute original and modified versions of the items in both machine-executable and source code forms. The recipients must be able to do so without any charges whatsoever, and they must be able to re-distribute to anyone they choose.
<P>
<LI>
If the items are not available to the general public, and the initial developer of the Software requests a copy of the items, then you must supply one.
<P>
</OL>
</OL>
<P>
<HR SIZE="6">
<A NAME="SEC32"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC31"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC33"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> Limitations of Liability </H2>
<!--docid::SEC32::-->
In no event shall the initial developers or copyright holders be liable
for any damages whatsoever, including - but not restricted to - lost
revenue or profits or other direct, indirect, special, incidental or
consequential damages, even if they have been advised of the possibility
of such damages, except to the extent invariable law, if any, provides
otherwise.
<P>
<HR SIZE="6">
<A NAME="SEC33"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC32"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC34"> > </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> No Warranty </H2>
<!--docid::SEC33::-->
<P>
The Software and this license document are provided AS IS with NO WARRANTY
OF ANY KIND, INCLUDING THE WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS
FOR A PARTICULAR PURPOSE.
</P><P>
<HR SIZE="6">
<A NAME="SEC34"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC33"> < </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[ > ]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> << </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC1"> Up </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC2"> >> </A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT"> <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> Choice of Law </H2>
<!--docid::SEC34::-->
<P>
This license is governed by the Laws of Norway. Disputes shall be settled
by Oslo City Court.
</P><P>
<HR SIZE="6">
<A NAME="SEC_Contents"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H1>Table of Contents</H1>
<BLOCKQUOTE>
<A NAME="TOC1" HREF="solid3.html#SEC1">1. License</A>
<BR>
<A NAME="TOC2" HREF="solid3.html#SEC2">2. Introduction</A>
<BR>
<BLOCKQUOTE>
<A NAME="TOC3" HREF="solid3.html#SEC3">2.1 Overview</A>
<BR>
<BLOCKQUOTE>
<A NAME="TOC4" HREF="solid3.html#SEC4">2.1.1 Shape Definition</A>
<BR>
<A NAME="TOC5" HREF="solid3.html#SEC5">2.1.2 Object Placement and Motion</A>
<BR>
<A NAME="TOC6" HREF="solid3.html#SEC6">2.1.3 Scene Management</A>
<BR>
<A NAME="TOC7" HREF="solid3.html#SEC7">2.1.4 Response Definition</A>
<BR>
<A NAME="TOC8" HREF="solid3.html#SEC8">2.1.5 Global Actions</A>
<BR>
<A NAME="TOC9" HREF="solid3.html#SEC9">2.1.6 Broad Phase</A>
<BR>
</BLOCKQUOTE>
<A NAME="TOC10" HREF="solid3.html#SEC10">2.2 Software Package</A>
<BR>
<A NAME="TOC11" HREF="solid3.html#SEC11">2.3 New Features of SOLID version 3</A>
<BR>
</BLOCKQUOTE>
<A NAME="TOC12" HREF="solid3.html#SEC12">3. Installing the SOLID SDK</A>
<BR>
<BLOCKQUOTE>
<A NAME="TOC13" HREF="solid3.html#SEC13">3.1 Requirements</A>
<BR>
<A NAME="TOC14" HREF="solid3.html#SEC14">3.2 Installation</A>
<BR>
</BLOCKQUOTE>
<A NAME="TOC15" HREF="solid3.html#SEC15">4. The SOLID API</A>
<BR>
<BLOCKQUOTE>
<A NAME="TOC16" HREF="solid3.html#SEC16">4.1 Building Shapes</A>
<BR>
<A NAME="TOC17" HREF="solid3.html#SEC17">4.2 Creating and Moving Objects</A>
<BR>
<BLOCKQUOTE>
<A NAME="TOC18" HREF="solid3.html#SEC18">4.2.1 Who's Afraid of Quaternions?</A>
<BR>
<A NAME="TOC19" HREF="solid3.html#SEC19">4.2.2 Proximity Queries</A>
<BR>
</BLOCKQUOTE>
<A NAME="TOC20" HREF="solid3.html#SEC20">4.3 Scenes</A>
<BR>
<A NAME="TOC21" HREF="solid3.html#SEC21">4.4 Response Handling</A>
<BR>
<A NAME="TOC22" HREF="solid3.html#SEC22">4.5 Deformable Models</A>
<BR>
<A NAME="TOC23" HREF="solid3.html#SEC23">4.6 Ray Cast</A>
<BR>
</BLOCKQUOTE>
<A NAME="TOC24" HREF="solid3.html#SEC24">5. Projects and other things left to do</A>
<BR>
<BLOCKQUOTE>
<A NAME="TOC25" HREF="solid3.html#SEC25">5.1 Coming Attractions</A>
<BR>
</BLOCKQUOTE>
<A NAME="TOC26" HREF="solid3.html#SEC26">6. Bug Reports</A>
<BR>
<A NAME="TOC27" HREF="solid3.html#SEC27">A. GNU GENERAL PUBLIC LICENSE</A>
<BR>
<BLOCKQUOTE>
<A NAME="TOC28" HREF="solid3.html#SEC28">Preamble</A>
<BR>
<A NAME="TOC29" HREF="solid3.html#SEC29">Appendix: How to Apply These Terms to Your New Programs</A>
<BR>
</BLOCKQUOTE>
<A NAME="TOC30" HREF="solid3.html#SEC30">B. THE Q PUBLIC LICENSE</A>
<BR>
<BLOCKQUOTE>
<A NAME="TOC31" HREF="solid3.html#SEC31">Granted Rights</A>
<BR>
<A NAME="TOC32" HREF="solid3.html#SEC32">Limitations of Liability</A>
<BR>
<A NAME="TOC33" HREF="solid3.html#SEC33">No Warranty</A>
<BR>
<A NAME="TOC34" HREF="solid3.html#SEC34">Choice of Law</A>
<BR>
</BLOCKQUOTE>
</BLOCKQUOTE>
<HR SIZE=1>
<A NAME="SEC_OVERVIEW"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H1>Short Table of Contents</H1>
<BLOCKQUOTE>
<A NAME="TOC1" HREF="solid3.html#SEC1">1. License</A>
<BR>
<A NAME="TOC2" HREF="solid3.html#SEC2">2. Introduction</A>
<BR>
<A NAME="TOC12" HREF="solid3.html#SEC12">3. Installing the SOLID SDK</A>
<BR>
<A NAME="TOC15" HREF="solid3.html#SEC15">4. The SOLID API</A>
<BR>
<A NAME="TOC24" HREF="solid3.html#SEC24">5. Projects and other things left to do</A>
<BR>
<A NAME="TOC26" HREF="solid3.html#SEC26">6. Bug Reports</A>
<BR>
<A NAME="TOC27" HREF="solid3.html#SEC27">A. GNU GENERAL PUBLIC LICENSE</A>
<BR>
<A NAME="TOC30" HREF="solid3.html#SEC30">B. THE Q PUBLIC LICENSE</A>
<BR>
</BLOCKQUOTE>
<HR SIZE=1>
<A NAME="SEC_About"></A>
<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Top">Top</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[Index]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="solid3.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H1>About this document</H1>
This document was generated
using <A HREF="http://www.mathematik.uni-kl.de/~obachman/Texi2html
"><I>texi2html</I></A>
<P></P>
The buttons in the navigation panels have the following meaning:
<P></P>
<table border = "1">
<TR>
<TH> Button </TH>
<TH> Name </TH>
<TH> Go to </TH>
<TH> From 1.2.3 go to</TH>
</TR>
<TR>
<TD ALIGN="CENTER">
[ < ] </TD>
<TD ALIGN="CENTER">
Back
</TD>
<TD>
previous section in reading order
</TD>
<TD>
1.2.2
</TD>
</TR>
<TR>
<TD ALIGN="CENTER">
[ > ] </TD>
<TD ALIGN="CENTER">
Forward
</TD>
<TD>
next section in reading order
</TD>
<TD>
1.2.4
</TD>
</TR>
<TR>
<TD ALIGN="CENTER">
[ << ] </TD>
<TD ALIGN="CENTER">
FastBack
</TD>
<TD>
previous or up-and-previous section
</TD>
<TD>
1.1
</TD>
</TR>
<TR>
<TD ALIGN="CENTER">
[ Up ] </TD>
<TD ALIGN="CENTER">
Up
</TD>
<TD>
up section
</TD>
<TD>
1.2
</TD>
</TR>
<TR>
<TD ALIGN="CENTER">
[ >> ] </TD>
<TD ALIGN="CENTER">
FastForward
</TD>
<TD>
next or up-and-next section
</TD>
<TD>
1.3
</TD>
</TR>
<TR>
<TD ALIGN="CENTER">
[Top] </TD>
<TD ALIGN="CENTER">
Top
</TD>
<TD>
cover (top) of document
</TD>
<TD>
</TD>
</TR>
<TR>
<TD ALIGN="CENTER">
[Contents] </TD>
<TD ALIGN="CENTER">
Contents
</TD>
<TD>
table of contents
</TD>
<TD>
</TD>
</TR>
<TR>
<TD ALIGN="CENTER">
[Index] </TD>
<TD ALIGN="CENTER">
Index
</TD>
<TD>
concept index
</TD>
<TD>
</TD>
</TR>
<TR>
<TD ALIGN="CENTER">
[ ? ] </TD>
<TD ALIGN="CENTER">
About
</TD>
<TD>
this page
</TD>
<TD>
</TD>
</TR>
</TABLE>
<P></P>
where the <STRONG> Example </STRONG> assumes that the current position
is at <STRONG> Subsubsection One-Two-Three </STRONG> of a document of
the following structure:
<UL>
<LI> 1. Section One </LI>
<UL>
<LI>1.1 Subsection One-One</LI>
<UL>
<LI> ... </LI>
</UL>
<LI>1.2 Subsection One-Two</LI>
<UL>
<LI>1.2.1 Subsubsection One-Two-One
</LI><LI>1.2.2 Subsubsection One-Two-Two
</LI><LI>1.2.3 Subsubsection One-Two-Three <STRONG>
<== Current Position </STRONG>
</LI><LI>1.2.4 Subsubsection One-Two-Four
</LI></UL>
<LI>1.3 Subsection One-Three</LI>
<UL>
<LI> ... </LI>
</UL>
<LI>1.4 Subsection One-Four</LI>
</UL>
</UL>
<HR SIZE=1>
<BR>
<FONT SIZE="-1">
This document was generated
by <I>Gino van den Bergen</I> on <I>July, 22 2003</I>
using <A HREF="http://www.mathematik.uni-kl.de/~obachman/Texi2html
"><I>texi2html</I></A>
</FONT>
</BODY>
</HTML>
|
