COMPUTER VISION, GRAPHICS, AND IMAGE PROCESSING 36, 31-41 (1986) Computational-Geometric Methods for Polygonal Approximations of a Curve HIROSHI IMAI Department of Computer Science and Communication Engineering, Kyushu University, Hakozaki, Fukuoka 812, Japan AND MASAO IRI Department of Mathematical Engineeringand Instrumentation Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received May 1985; revised May 13, 1986 In cartography, computer graphics, pattern recognition, etc., we often encounter the problem of approximating a given finer piecewise linear curve by another coarser piecewise linear curve consisting of fewer line segments. In connection with this problem, a number of papers have been published, but it seems that the problem itself has not been well modelled from the standpoint of specific applications, nor has a nice algorithm, nice from the computa- tional-geometric viewpoint, been proposed. In the present paper, we first consider (i) the problem of approximating a piecewise linear curve by another whose vertices are a subset of the vertices of the former, and show that an optimum solution of this problem can be found in a polynomial time. We also mention recent results on related problems by several researchers including the authors themselves. We then pose (ii) a problem of covering a sequence of n points by a minimum number of rectangles with a given width, and present an O(n log n)-time algorithm by making use of some fundamental established techniques in computational geometry. Furthermore, an O(mn(log n)2)-time algorithm is presented for finding the mini- mum width w such that a sequence of n points can be covered by at most m rectangles with width w. Finally, (iii) several related problems are discussed. 9 1986 AcademicPress, Inc. 1. INTRODUCTION Piecewise linear or polygonal curves are often used to approximate complex boundaries of figures in cartography and other geographical data processing, com- puter graphics, pattern recognition, etc. In particular, when we want to reduce the size of a map, or compress the amount of cartographic data, the problem of approximating a given finer piecewise linear curve by another coarser piecewise linear curve such that the maximum distance of the former from the latter is bounded by a constant is of fundamental importance. The general problem for finding an approximate piecewise linear curve with the minimum number of segments seems quite difficult. We here formulate two types of approximation problems with additional constraints and present efficient algorithms for them, which will be practically useful enough. A number of papers have been published for various kinds of piecewise linear approximation problems. Ichida and Kiyono [4] considered the problem of ap- proximating a piecewise linear curve by a sequence of segments which may not be continuous. Their approach is simple, but the algorithm given in [4] is rather complicated. Kurozumi and Davis [9] generalized the result of [4] and gave an algorithm that finds an approximate piecewise linear curve whose maximum dis- 31 0734-189X/86 $3.00 Copyright 9 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.