Distance Functions

Abstract

The computation of maximal and minimal distance between points sets in Euclidean space is a basic problem in computational geometry and geometric modeling. It is useful in surface intersections [210], numerical control machining, tolerance region and access space representation in solid modeling, robotics, inspection of manufactured objects [353, 297, 186, 3], and in feature recognition through the construction of medial axis transforms [298,81, 450]. For this purpose, it is important to have computational methods which are efficient and reliable to compute extrema for the distance between two varible points where each of those variable points assumes all possible positions in a given set. In practical situations, this set can be a surface, a curve, or a single point. In this chapter, we examine the computation of the stationary points of the squared distance function in five cases [461]: 1. Between a given point and a variable point on a 3-D space parametric curve. 2. Between a given point and a variable point on a parametric surface patch. 3. Between two variable points located on two given 3-D space parametric curves. 4. Between two variable points, one of which is located on a 3-D space parametric curve and the other is location on a parametric surface patch. 5. Between two variable points, located on two different given parametric surface patches.