Homotopy Preserving Approximate Voronoi Diagram of 3D Polyhedron

We present a novel algorithm to compute a homotopy preserving bounded-error approximate Voronoi diagram of a 3D polyhedron. Our approach uses spatial subdivision to generate an adaptive volumetric grid and computes an approximate Voronoi diagram within each grid cell. Moreover, we ensure each grid cell satisfies a homotopy preserving criterion by computing an arrangement of 2D conics within a plane. Homotopy equivalence implies a one-to-one correspondence between various topological components of the approximate Voronoi diagram and the exact Voronoi diagram. Our algorithm also satisfies Hausdorff distance bounds between the approximate and the exact Voronoi diagrams. We use distance based culling techniques to reduce number of non-linear arrangement computations and accelerate the computation. In practice, our algorithm can compute an approximate Voronoi diagram of complex models with thousands of primitives in tens of seconds.