Constructing Medial Axis Transform for Free-form Trimmed Surfaces

Abstract

Indian Institute of Science, bgm@mecheng.iisc.ernet.in ABSTRACT This paper presents an algorithm for constructing Medial Axis Transform for free-form surfaces. The algorithm uses a combination of tracing and bisector based techniques. This algorithm finds bisector curves by a tracing method and then trims these bisector curves for finding the branch points and constructing the MAT. Results on typical free-form domains are presented. Keywords: Medial Axis Transform, Free-form surface, geodesic. 1. INTRODUCTION One of the fundamental issues in CAD has been to represent objects in a way that not only allows the user to create and modify their shape and other attributes but also enables reasoning about the shape for various tasks in the product design cycle. Objects are normally described by their boundaries, but they can also be described by shifting the description to the interior. This is done by representing the skeleton/symmetric axis of the shape. Various skeleton/symmetric axis representation of shapes like Medial Axis Transform (MAT), Voronoi diagrams, box skeletons, mid surfaces and 2.5D skeletons are possible. Of these MAT has shown to be useful in various applications such as finite element mesh generation, pattern analysis and image analysis, path generation for pocket machining, robot motion planning and more Work on MAT up to now has been restricted to planar 2D and 3D objects. Very little work has been reported on construction of the MAT for 2D free form surface domains. An algorithm to develop MAT on free form surfaces (exact representation) is presented in this paper. 1.1 Medial Axis Transform (MAT) The medial axis transform was first introduced by Blum [1] to describe biological shapes. It can be viewed as the locus of the center of a maximal disk as it rolls inside an object. 1.2 Definition of 2D MAT for Planar Domains The medial axis (MA) of the set D, denoted M(D), is defined as the locus of points inside D which lie at the centers of all closed disks which are maximal in D, together with the limit points of this locus. A closed disk is said to be maximal in a subset D of the 2D space if it is contained in D but is not a proper subset of any other disk contained in D. The radius function of the MA of D is a continuous, real-valued function defined on M(D) whose value at each point on the MA is equal to the radius of the associated maximal disk. The medial axis transform (MAT) of D is the MA along with its associated radius function [8]. Fig. 1. illustrates the MA segments corresponding to a simple convex domain.