Closest Points, Moving Surfaces, and Algebraic Geometry

Abstract. This paper presents a method for computing closest points to a given parametric surface based on “moving surfaces”. Moving surfaces are implicitly defined objects that allow us to for-mulate the problem in terms of two univariate polynomial equa-tions. The idea is to obtain a faster and more reliable method of computing closest points, compared to conventional methods based on Newton iterations. We also describe an implementation of our algorithm which – although not being fast – is very reliable. In this paper, we present a new method for calculating closest points to a parametric surface. The method is based on algebraic techniques, in particular on moving surfaces. Moving surfaces are objects that have previously been used for implicitization [6], but the closest point proble