A formal classification of 3D medial axis points and their local geometry

This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we show that, genetically, the medial axis consists of five types of points, which are then organized into sheets, curves, and points: 1) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency A₁² (A_kⁿ notation means n distinct k-fold tangencies of the sphere of contact, as explained in the text); two types of curves, 2) the intersection curve of three sheets and the locus of centers of tritangent spheres, A₁³, and 3) the boundary of sheets, which are the locus of centers of spheres whose radius equals the larger principal curvature, i.e., higher order contact A₃ points; and two types of points, 4) centers of quad-tangent spheres, A₁⁴, and 5) centers of spheres with one regular tangency and one higher order tangency, A₁A₃. The geometry of the 3D medial axis thus consists of sheets (A₁²) bounded by one type of curve (A₃) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of curve (A₁³), which support a generalized cylinder description. The A₃ curves can only end in A₁ A₃ points where they must meet an A₁³ curve. The A₁³ curves meet together in fours at an A₁⁴ point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A₁⁴ and A₁ A₃ points), links between pairs of nodes (A₁³ and A₃ curves) and hyperlinks between groups of links (A₁² sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function. Thus, this information completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design, and man- ipulation of shapes.