Two-dimensional MAT to boundary conversion

Abstract

The medial axis transform (MAT) has potential as a powerful representation for a conceptual design tool for objects with inherent symmetry or near-symmetry. The medial axis of two-dimensional objects or medial surface of three-dimensional objects provides a conceptual design base, with transition to a detailed design occuring when the radius function is added to the medial axis or surface, since this additional information completely specifies a particular object. To make such a design tool practicable, however, it is essential to be able to convert from an MAT format to a boundary representation of an object. Such a conversion is possible because the MAT is an informationally complete solid representation. In this paper, we provide the details for the conversion of the MAT of a set of twodimensional objects to a boundary representation. We demonstrate certain smoothness properties of the MAT and show the relationship between the tangent to the MAT at a point and the boundary points related to that MAT point. For each of the three general types of MAT points (end points, normal points, and branch points) we detail the method for obtaining the boundary points related to it and for determining whether finite contact occurs at that point. We discuss requirements for an MAT to be valid in the sense that the given curves could actually be the MAT of an allowable twodimensional object. We also provide a theoretical error bound on the computation. Finally, we discuss an implementation of our algorithm and show some results we have obtained.