Local Forms and Transitions of the Medial Axis

Abstract

We define the concept of contact between a plane curve and a circle, and between a surface in 3D and a sphere, and describe all the contacts which generically occur for curves and surfaces, or 1-parameter families of such. We then list the local forms of the symmetry set and medial axis in 2D and 3D and briefly describe the local reconstruction of a boundary surface from its symmetry set or medial axis. Next, we describe the transitions which occur on the medial axis and symmetry set in a 1-parameter family of plane curves and which occur on the medial axis in a 1-parameter family of surfaces in 3D. Finally we make some remarks on the constraints which the medial axis must obey at special points such as Y-junctions. Some applications are given to illustrate these ideas.