1 Polygon-to-Function Conversion for Sweeping ALEXANDER A. PASKO Department of Computer Software, The Univ. of Aizu, Aizu-Wakamatsu City, Fukushima, Japan ANDREI V. SAVCHENKO Fraunhofer Center for Research in Computer Graphics, Inc., Providence, RI, USA VLADIMIR V. SAVCHENKO Department of Computer Software, The Univ. of Aizu, Aizu-Wakamatsu City, Fukushima, Japan ABSTRACT This paper describes an approach to the representation of polygons by real functions and its application to sweeping. We combine an algorithm employing a monotone function of a boolean operation, with R-functions. Application of this method results in a continuous function F(x,y) with zero value at polygon edges. We discuss and illustrate different sweeping techniques with the use of functionally defined generators. 1. INTRODUCTION Recently the representation of geometric objects by real continuous functions of several variables (so-called implicit representation or F-rep) has attracted a great deal of attention in many technical applications and animation. The exact conversion from the boundary representation to a real function is an open problem. This paper deals with such conversion of a two-dimensional polygon. We discuss an algorithm for a simple polygon bounded by straight line segments. We combine an algorithm employing a monotone function of a boolean operation, with R-functions. Although these techniques are known in computational geometry and geometric modeling, the combination of them gives a basis for new applications. We discuss and illustrate in this paper the application of the conversion for sweeping by a planar contour. The resulting swept solid is represented by a single real function of three variables which is defined analytically or procedurally. This functionally defined solid can be a subject for further operations such as set- theoretic, blending, generalized offsetting, metamorphosis and others [10]. The polygon-to-function conversion problem is stated as follows. A two-dimensional simple polygon is defined by a finite set of segments. The segments are the edges and their extremes are the vertices of the polygon. A polygon is simple if there is no pair of nonadjacent edges sharing a point and convex if its interior is a convex set. We consider the problem of representation of polygons by continuous real functions F(x,y) taking zero values at polygon edges. The algorithm should satisfy the following requirements: It should provide exact polygon description as a zero set of a real function; No points with zero function value should be inside or outside a polygon; It should allow processing a simple arbitrary polygon without any additional information. We give here a brief overview of the works dealing with this problem. 1.1 Distance based methods Singh and Parent [19] generalize the skeleton based functions [2, 24] for objects with explicitly