Robust, Generic and Efficient Construction of Envelopes of Surfaces in Three-Dimensional Space

Michal Meyerovitch


Lower envelopes are fundamental structures in computational geometry that have many applications, such as computing general Voronoi diagrams and performing hidden surface removal in computer graphics. We present a generic, robust and efficient implementation for computing the envelopes of surfaces in $\reals^3$. To the best of our knowledge, this is the first exact implementation that computes envelopes in three-dimensional space. Our implementation is based on \cgal\ and is designated as a \cgal\ package. The separation of topology and geometry in our solution allows the reuse of the algorithm with different families of surfaces, provided that a small set of geometric objects and operations on them is supplied. We used our algorithm to compute the lower and upper envelope for several types of surfaces. Our implementation follows the exact geometric computation paradigm. Since exact arithmetic is typically slower than floating-point arithmetic, especially when higher order surfaces are involved, we minimize the number of such operations, to gain better performance in practice. Our experiments show interesting phenomena in the behavior of the divide-and-conquer algorithm and the combinatorics of lower envelopes of random surfaces.

   Code for computing 3D envelopes

   Extended experimental results

   Some input sets (for triangles and spheres)

Examples of envelopes computed by our program

Envelope of approximately 16000 triangles:

Envelope of ~16000 triangles

Envelope of 50 spheres:

Envelope of 50 spheres