Differential and Topological Properties of Medial Axis Transforms

Themedial axis transformis a representation of an object which has been shown to be useful in design, interrogation, animation, finite element mesh generation, performance analysis, manufacturing simulation, path planning, and tolerance specification. In this paper, the theory of the medial axis transform for 3-D objects is developed. For objects with piecewiseC²boundaries, relationships between the curvature of the boundary and the position of the medial axis are developed. Forn-dimensional submanifolds of $\text{R}$ⁿwith boundaries which are piecewiseC²and completelyG¹, a deformation retract is set up between each object and its medial axis, which demonstrates that if the object is path connected, then so is its medial axis. Finally, it is proven that path connected polyhedral solids without cavities have path connected medial axes.