Exact Signed Distance Function Representation of Polygons

Abstract

Introduction: Signed distance functions (SDF) are applied from high-quality text rendering [3] to geometric representation for collision detection [5], 3D printing, additive manufacturing [1], or advanced real-time graphics e ects [7]. The SDF is usually stored as a regular grid of samples for high-performance applications, but various spatial subdivision or interpolation schemes have been proposed for storage, such as octrees [2] or hierarchical T-meshes [6]. In complex shapes, applications mainly focus on storing a discrete approximation to the exact SDF in conjunction with various interpolation techniques. We propose a conservative but exact SDF representation for planar polygons. The exact SDF is composed of two classes of regions, separated by parabolic and linear boundaries. We construct conservative polygonal bounds to these regions. Our algorithm performs a series of cuts to determine the bounding polygons that represent the distance function on the region. The exact SDF can be evaluated using these polygons. Such a formulation is closely related to point and segment Voronoi diagrams [4]; however, our goal is to preserve the inside-outside partitioning of the plane as well.