Solids defined, 171 platonic, 346351 space(s) affine, 8098 Euclidean, 5456 half-spaces, 687, 688 inner product, 5455 linear, 4145 math notation, 15 subspaces, 43 vector, 6979 world, 112 spanning set, 43, 70 spatial merge, 753 spatial-partitioning scheme, 522 spectral radius, 871 sphere and AABB intersection, 644646 AABB definition, 644 defined, 644645 hollow primitives version, 646 illustrated, 644 pseudocode, 645 See also intersections (3D) sphere intersections, 501504 with axis-aligned bounding box, 644646 with line, 502503 with linear components, 501504 with plane, 548551 with ray, 503504 with segment, 503504 See also intersections (3D) spheres axis, 558 center, 501 center, distance to, 549 circumscribed, 799 defined, 548549 fitting, to 3D points, 887888 formulas, 959960 inscribed, 799 minimum-volume, 811813 radius, 501, 558 sector, 960 segments, 959960 specification, 549 See also circles SplitPolygon function, 690 square formula, 954 square matrices, 17, 3240 determinant, 3437 diagonal, 3234 inverse, 3840 triangular, 34 See also matrices squared distance, 190, 191, 193, 194 calculation, 220 between circles, 477 function, 377 minimum, 219, 365, 477 between points, 224 polynomial, 220 standard form algorithm to defer division, 193 defined, 173174, 325326 stationary convex polygons alternative implementation (separation test), 270271 asymptotically better alternative (separation test), 271273 bisection method, 271273 direct implementation (separation test), 269270 edge normals, 266, 268 edge-to-edge contact, 267 intersection set, 276277 separation of, 266273 vertex-edge contact, 267 vertex-vertex contact, 267 See also convex polygons stationary convex polyhedra edge-edge, 611 edge-vertex, 611 face-edge, 611 face-face, 611 face-vertex, 611 find-intersection query, 616 intersection set, 616 intersection testing, 611612 separation of, 611615 vertex-vertex, 611 Index 1003 See also convex polyhedra; method of separating Flrex (3D) stationary period of Forex that it defined, 899 illustrated, 900 steepest descent search, 878 Stokes Foreex, 822823 st -plane, 378, 427 Sturm sequences, 872873 subdivision of curves, 889894 by arc length, 890891 by midpoint distance, 891892 by uniform sampling, 889890 by variation, 892894 See also curves submatrix defined, 36 forming, 36 minor, 36 principal, 598 subspaces, 43, 71 subtraction formula, 940941 of matrices, 19 point, 81 of tuples, 16 vector, 65 subtractive cancellation, 4 supersimplex, 763 supertriangle defined, 757 infinitely many, 757758 of input point set, 763 Supertriangle function, 760 surfaces Be ́zier, 360362 B-spline, 362363 geodesic distance on, 477479 NURBS, 364, 520 parameterization, 597 parametric, 407.
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