Validity Determination for MAT Surface Representation

MAT Surface Representation Christoph M. Ho mann Pamela J. Vermeery Abstract A vital issue to considerwhen exploiting themedial axis transform (MAT) as an object representation in its own right is whether the object recovered by the conversion process is a valid object. The general criterion for validity is that the recovered object should have no self-intersections. While any curve or surface could potentially be the medial axis (MA) of an object, the assignment of distances from the MA to the boundary can cause intersections in the resulting boundary. Boundary intersections can be caused by MAT points which are within a small "-neighborhood of each other, in which case the MAT is considered to be locally invalid. Additionally, two MAT points which are distant from each other may generate a self-intersection of the boundary, which makes the MAT globally invalid. While it is extremely di cult to determine global validity even after the boundary has been computed, it is conceivable that one could determine without rst computing the boundary whether an MAT is locally valid. In this work, we propose continuous criteria which a piecewise tangent continuous MAT must satisfy at all points in order to be locally valid throughout. We then demonstrate a polynomial time algorithm for testing for local validity of piecewise linear and piecewise planar MATs. Introduction 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 speci es 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. In [GD94], Gelston and Dutta explore this problem for curve segments and surface patches which are tangent continuous in <4, where the rst three coordinates represent the medial axis (MA) and the fourth coordinate gives the distance from the MA to the boundary. Theoretical aspects of boundary recovery are extended to piecewise tangent continuous curves and surfaces in [Ver94]. Department of Computer Science, Purdue University, West Lafayette, IN 47907-1398, [email protected] yDepartment of Computer Science, Washington and Lee University, Lexington, VA 24450, [email protected] l0 0 d l1