Operations on Signed Distance Function Estimates

Abstract

Introduction: Our paper presents a general theoretical framework to investigate the quantitative aspects of bounding distance functions. We propose a precision de nition that quanti es the accuracy of the min/max representation of set-theoretic operations [5] in the entire space and demonstrate how the precision and the geometric con guration of the arguments determine the accuracy of the resulting approximation. Our theorems can be applied in an arbitrary geometrical context, e.g., for objects with or without volumes, implicit curves, non-di erentiable or non-manifold surfaces, fractals, and any combination of these. We identify a subset of Hart's signed distance lower bounds [3] called signed distance function estimates (SDFE) and show that the sphere tracing algorithm retains convergence under set-theoretic union and intersection operations, a result for which a general derivation has not yet been presented. Most so-called distance estimates used by the industry and the online creative coding communities such as ShaderToy are SDFEs, placing no practical restrictions on the applicability of our results. This paper builds upon the theoretical results of Luo et al. [4], Bálint et al.[1], and Valasek et al. [6].