Unions of Balls
A sample from the surface of a foot model and the union of the corresponding set of interior polar balls, forming a good approximation of the shape.
The poles are a subset of the vertices of the Voronoi
diagram of S which ought to lie near the medial axis of
F.
The Voronoi balls centered at poles
are called polar balls.
The finite union of the polar balls forms a good discrete approximation to the
MAT, which is a representation of the object as an infinite
union of balls.
For each sample s in S, the interior pole of s is the farthest vertex of the Voronoi cell of s that is inside F and the exterior pole is the farthest vertex of the Voronoi cell of s that is outside F.
In 3D, there are lots of Voronoi vertices which lie near the surface, and not near the medial axis. For instance, on the right we see the union of all of the Voronoi balls from the input sample above. The union of polar balls forms a much better approximation of the MAT.
We prove a number of theorems about the quality of the approximation in our papers [2,3].