# On approximating polygonal curves in two and three dimensions

David Eu, Godfried T. Toussaint

Research output: Contribution to journalConference articlepeer-review

## Abstract

Given a polygonal curve P = [p1, p2, · · ·, pn], the polygonal approximation problem considered in this paper calls for determining a new curve P' = [p1, p2> · · ·, pm] such that (i) m is significantly smaller than n, (ii) the vertices of P are a subset of the vertices of P and (iii) any line segment [pA, pA+1] of P' that substitutes a chain [pB,..., pC] in P is such that for all i where B ≤ i ≤ C, the approximation error of pi with respect to [pA, PA+1], according to some specified criterion and metric, is less than a predetermined error tolerance. Using the parallel-strip error criterion, we study the following problems for a curve P in Rd, where d ≥ 2: (i) minimize m for a given error tolerance and (ii) given m, find the curve P' that has the minimum approximation error over all curves that have at most m vertices. These problems are called the min-# and min-∈ problems, respectively. For R2 and with any one of the L1, L2 or L distance metrics, we give algorithms to solve the min-# problem in 0(n2) time and the min-∈ problem in 0(n2 log n) time, improving the best known algorithms to date by a factor of log n. When P is a polygonal curve in R3 that is strictly monotone with respect to one of the three axes, we show that if the L1 and L metrics are used then the min-# problem can be solved in O(n2) time and the min-∈ problem can be solved in O(n3) time. If distances are computed using the L2 metric then the min-# and min-∈ problems can be solved in O(n3) and O(n3 log n) time respectively. All our algorithms exhibit O(n2) space complexity.

Original language English (US) 97-110 14 Proceedings of SPIE - The International Society for Optical Engineering 1832 https://doi.org/10.1117/12.142160 Published - Apr 9 1993 Vision Geometry 1992 - Boston, United StatesDuration: Nov 16 1992 → …

## ASJC Scopus subject areas

• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
• Computer Science Applications
• Applied Mathematics
• Electrical and Electronic Engineering

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