Goal is to show: Cone carving cannot fail if the sampling is sufficiently good 

1. x lfs(x) Medial axis Goal is to show: Cone carving cannot fail if the sampling is sufficiently good  • Assume that the sampling of the surface is “good”. • A generally accepted notion of a “good” sampling of a surface is due to Amenta and Bern, related to what is called the local feature size (lfs) • The lfs(x) at a point x is the shortest distance from x to the medial axis of the shape.

2. x A sample A good sampling • An r-sampling is one where if x is a point on the surface, then a sphere centered at x with a radius r * lfs(x) must contain a sample. • A good sampling requires r to be small, to get a correct surface. Our analysis is independent of r.

3. The kind of result we want • If we have an r-sampling and the cone radius is sufficiently large, then our empty cones cannot “pierce through” the surface. • What I can show now is that: If we have an r-sampling and the cone radius is sufficiently large, then the center axis of our empty cones cannot intersect the medial axis of the shape (bound by the surface).

4. Very simple proof • If the center axis of an empty cone had intersected the medial axis, which lies inside the shape, then the axis must intersect the surface somewhere. Let us consider the “outermost” of such surface intersection. x surface  Medial axis L

5. Very simple proof • Since the cone is empty, the circle/sphere centered at x and tangential to the cone sides must be empty too. • Its radius is R = L*sin and it must be less than r*lfs(x) since we have an r-sample x surface  Medial axis L

6. Very simple proof • R = L*sin < r*lfs(x) • Clearly, L > lfs(x). So it follows that sin < r. That is, if the cone opening angle is > 2*arcsin(r), then the cone axis cannot intersect the medial axis. x surface  Medial axis L