Approximate motion planning and the complexity of the boundary of the union of simple geometric figures
Helmut Alt, Rudolf Fleischer, Michael Kaufmann, Kurt Mehlhorn, Stefan Näher, Stefan Schirra, Christian Uhrig · Algorithmica · 1990
We study rigid motions of a rectangle amidst polygonal obstacles. The best known algorithms for this problem have running time Ω(n2) where n is the number of obstacle corners. We introduce the tightness of a motion planning problem as a measure of the difficulty of a planning problem in an intuitive sense and describe an algorithm with running time ο((a/b · 1/ε crit + 1)n(log n)2), where a ≥ b are the lengths of the sides of a rectangle and εcrit is the tightness of the problem. We show further that the complexity (= number of vertices) of the boundary of n bow-ties (c.f. Figure 1.1) is Ο(n). Similar results for the union of other simple geometric figures such as triangles and wedges are also presented.