From 0210df5a5dade94960ef48ca26a98a2676f215f7 Mon Sep 17 00:00:00 2001 From: Tavian Barnes Date: Sat, 7 Jun 2014 18:53:23 -0400 Subject: sphere: Use tightest possible bounding boxes. --- libdimension/sphere.c | 68 ++++++++++++++++++++++++++++++++++++++++++++++++--- 1 file changed, 65 insertions(+), 3 deletions(-) (limited to 'libdimension') diff --git a/libdimension/sphere.c b/libdimension/sphere.c index 5555499..8f1e1a7 100644 --- a/libdimension/sphere.c +++ b/libdimension/sphere.c @@ -63,9 +63,71 @@ dmnsn_sphere_inside_fn(const dmnsn_object *sphere, dmnsn_vector point) static dmnsn_bounding_box dmnsn_sphere_bounding_fn(const dmnsn_object *object, dmnsn_matrix trans) { - // TODO: tighter bound - dmnsn_bounding_box box = dmnsn_symmetric_bounding_box(dmnsn_new_vector(1.0, 1.0, 1.0)); - return dmnsn_transform_bounding_box(trans, box); + // Get a tight bound using the conic representation of a sphere: + // + // S = [ 1 0 0 0 ] + // [ 0 1 0 0 ] + // [ 0 0 1 0 ] + // [ 0 0 0 -1 ]. + // + // The surface is defined by + // p^T * S * p = 0, + // and the tangent planes are defined by + // q * S^-1 * q^T = 0. + // Note that S = S^-1. + // + // The symmetric matrix R, defined by + // R = M * S^-1 * M^T, + // characterizes the tangent planes. Specifically, + // min.x = (R[0,3] - sqrt(R[0,3]^2 - R[0,0]*R[3,3]))/R[3,3] + // max.x = (R[0,3] + sqrt(R[0,3]^2 - R[0,0]*R[3,3]))/R[3,3] + // min.y = (R[1,3] - sqrt(R[1,3]^2 - R[1,1]*R[3,3]))/R[3,3] + // max.y = (R[1,3] + sqrt(R[1,3]^2 - R[1,1]*R[3,3]))/R[3,3] + // min.z = (R[2,3] - sqrt(R[2,3]^2 - R[2,2]*R[3,3]))/R[3,3] + // max.z = (R[2,3] + sqrt(R[2,3]^2 - R[2,2]*R[3,3]))/R[3,3] + // + // Unfortunately, we can't use dmnsn_matrix because the matrices are not + // affine + + // MS = M * S^-1 = M * S + // Last row is [ 0 0 0 -1 ] implicitly + double MS[3][4]; + for (int i = 0; i < 3; ++i) { + for (int j = 0; j < 3; ++j) { + MS[i][j] = trans.n[i][j]; + } + MS[i][3] = -trans.n[i][3]; + } + + // R = MS * M^T + // We only compute the upper triangular portion + // R[3][3] is implicitly -1 + double R[4][4]; + for (int i = 0; i < 3; ++i) { + for (int j = i; j < 3; ++j) { + R[i][j] = 0.0; + for (int k = 0; k < 4; ++k) { + R[i][j] += MS[i][k]*trans.n[j][k]; + } + } + R[i][3] = MS[i][3]; + } + + dmnsn_bounding_box box; + + double dx = sqrt(R[0][3]*R[0][3] + R[0][0]); + box.min.x = -R[0][3] - dx; + box.max.x = -R[0][3] + dx; + + double dy = sqrt(R[1][3]*R[1][3] + R[1][1]); + box.min.y = -R[1][3] - dy; + box.max.y = -R[1][3] + dy; + + double dz = sqrt(R[2][3]*R[2][3] + R[2][2]); + box.min.z = -R[2][3] - dz; + box.max.z = -R[2][3] + dz; + + return box; } /// Sphere vtable. -- cgit v1.2.3