NP-hardness in geometric construction problems with one interval parameter

Abstract

One of the tasks of Computer-Aided Design/Computer-Aided Manufacturing is to transform an object's technical drawing - in which geometry is described by distances and angles - into precise coordinates of the object's points and lines. One of the best ways of finding these coordinates is by using step-by-step geometric constructions (e.g., ruler-and-compass constructions). These constructions are easy to implement if we start with exact geometric information (distances and angles). In practice, however, all technical drawings have tolerances, so instead of requiring the exact values of distance and angles, we only require that they stay within prescribed intervals. Different values within these intervals lead to different coordinates of the corresponding points. In this paper, we show that the problem of computing the intervals of possible values of these coordinates is computationally complicated (NP-hard). Crudely speaking, this means that it is not possible to have an algorithm for constructing the endpoints of the corresponding coordinate intervals in feasible time.