Linear time algorithms for testing approximate congruence in the plane

Abstract

Let A, B be two sets of n points in the Euclidean plane. We want to test if they are congruent. Unfortunately, in many practical applications the input data are not given precisely, but only within a small tolerance factor ɛ. With this notion in mind, we ask if A and B are approximately congruent, i.e. if there are sets A′ and B′ consisting of points in the ɛ-neighborhoods of the points of A and B respectively that are exactly congruent. In this paper we give optimal algorithms for some problems of the labelled case, i.e. we assume that we already know which point of A should be transformed to which point of B. First, we give a linear time algorithm for the test if two planar point sets are approximately congruent by a reflection in a line. The algorithm presented in this paper uses a generalization of the linear programming algorithm by Megiddo which is interesting in its own right. It solves the problem of finding a feasible solution for a general system of algebraic inequalities of bounded degree. Next, we derive a linear time algorithm for the test on congruence by a rotation around a fixed center. Finally, if we allow an arbitrarily small but fixed range of uncertainty, we obtain a linear time algorithm for the test on arbitrary congruence.