ICTCS 2005: Theoretical Computer Science pp 23-35

# Efficient Algorithms for Detecting Regular Point Configurations

• Luzi Anderegg
• Mark Cieliebak
• Giuseppe Prencipe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3701)

## Abstract

A set of n points in the plane is in equiangular configuration if there exist a center and an ordering of the points such that the angle of each two adjacent points w.r.t. the center is $$\frac{360^{\circ}}{n}$$, i.e., if all angles between adjacent points are equal. We show that there is at most one center of equiangularity, and we give a linear time algorithm that decides whether a given point set is in equiangular configuration, and if so, the algorithm outputs the center. A generalization of equiangularity is σ-angularity, where we are given a string σ of n angles and we ask for a center such that the sequence of angles between adjacent points is σ. We show that σ-angular configurations can be detected in time O(n 4 log n).

## Keywords

Weber point equiangularity σ-angularity design of algorithms computational geometry

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