2D Polyhedral Bounds of a Clipped, Perspective-Projected 3D Sphere

We show how to efficiently compute 2D polyhedral bounds of the (elliptic) perspective projection of a 3D sphere that has been clipped to the near plane. For the special case of a 2D axis-aligned bounding box, the algorithm is especially efficient.

This has applications for bounding the screen-space effect of an emitter under deferred shading, bounding the kernel in density estimation during image space photon mapping, and bounding the pixel extent of objects for ray casting.

Our solution is designed to elegantly handle the case where the sphere crosses the near clipping plane and efficiently handle all cases on vector processors. In addition to the algorithm, we provide implementations of two common applications: light-tile classification in C++ and expanding an attribute array of spheres (encoded as points and radii) into polygons that cover their silhouettes as a GLSL geometry shader.