Farouki and Johnstone [99,100] have studied bisector problems between a planar curve and a point on its plane, and between two co-planar curves. They introduce a natural method for tracing the bisector between two curves by using the exact representation of the bisectors of the first curve and successive points on the second curve.
Chiang [54] and Brandt [38,39] studied many of the mathematical properties of the MAT, primarily for the domain of 2-D regions. Chiang [54] provides a proof that for two-dimensional regions with piecewise boundaries, the MA is connected, the MAT is invertible, and, provided the 2-D solid is homeomorphic to the closed unit disk, the maximal disc of a given MA point (not an end point of a MA arc) divides the MA and the solid into two disjoint trees and two disjoint 2-D solids, respectively (which provides justification for divide-and-conquer approaches). Brandt [38,39] computes first order and second order differential properties of the planar skeleton, and explores the determination of the skeleton under different metrics (which may be useful for determining the skeleton of binary images). He also explores the notion of skeleton point classification [39], classifying skeleton points according to the number of footpoints.
The three-dimensional problem is studied in some detail by Nackman [282] and Nackman and Pizer [283], who also derive relationships between curvatures of the boundary, the skeleton, and the associated radius function. Curvature relationships in the planar case are considered by Blum [29]. Anoshkina et al. [9,8] consider properties of the Medial Axis in the context of an investigation of singularities of the distance function to the bounding surface.
Sherbrooke et al. [395,391] further developed the theory of the MAT of 3-D objects. They established the relationships between the curvature of the boundary and the position of the medial axis and also set up a deformation retract between each object and its medial axis for n-dimensional submanifolds of with boundaries which are piecewise and completely . They demonstrated that if the object is path connected, then so is the medial axis. Specifically, they proved that path connected polyhedral solids without cavities have path connected medial axes.
Stifter [407,408] considers the Voronoi Diagram of any subset of 3-D space characterized by certain axioms and analyzes various properties of the subset with applications to robotics.