sphere: Use tightest possible bounding boxes.

1 files changed, 65 insertions, 3 deletions

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.