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Efficient Computation of the Directional Extremal Boundary of a Union of Equal-Radius Circles

Alexander Gribov

TL;DR

This work addresses the upper directional boundary of the union of equal-radius circles by presenting an $O(n \log n)$ algorithm. It sorts circle centers by horizontal position and incrementally builds a piecewise circular-arc boundary, maintaining a sequence of contributing circles and their horizontal transition positions. A key contribution is the Uniqueness of Horizontal Transition Lemma, enabling a linear-time boundary update via an inductive proof, and a segment-based representation that supports efficient boundary evaluation through binary search. The approach yields precise upper envelopes suitable for applications in robotics, infrastructure monitoring, and spatial analysis, with potential for parallelization in the sorting step and divide-and-conquer merging in boundary construction.

Abstract

This paper focuses on computing the directional extremal boundary of a union of equal-radius circles. We introduce an efficient algorithm that accurately determines this boundary by analyzing the intersections and dominant relationships among the circles. The algorithm has time complexity of O(n log n).

Efficient Computation of the Directional Extremal Boundary of a Union of Equal-Radius Circles

TL;DR

This work addresses the upper directional boundary of the union of equal-radius circles by presenting an algorithm. It sorts circle centers by horizontal position and incrementally builds a piecewise circular-arc boundary, maintaining a sequence of contributing circles and their horizontal transition positions. A key contribution is the Uniqueness of Horizontal Transition Lemma, enabling a linear-time boundary update via an inductive proof, and a segment-based representation that supports efficient boundary evaluation through binary search. The approach yields precise upper envelopes suitable for applications in robotics, infrastructure monitoring, and spatial analysis, with potential for parallelization in the sorting step and divide-and-conquer merging in boundary construction.

Abstract

This paper focuses on computing the directional extremal boundary of a union of equal-radius circles. We introduce an efficient algorithm that accurately determines this boundary by analyzing the intersections and dominant relationships among the circles. The algorithm has time complexity of O(n log n).

Paper Structure

This paper contains 8 sections, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Algorithm Description
  4. Proof of Algorithm Correctness
  5. Complexity Analysis
  6. Binary Search for Finding the Upper Value
  7. Conclusion
  8. Horizontal Transition Position of Two Unit Circles

Figures (2)

  • Figure 1: Visualization of the upper boundary of unit circles. The circles are shown as blue and red arcs, with the red arcs forming the upper boundary, which defines the highest points along the $y$-axis.
  • Figure 2: Comparison of the upper boundaries of two unit circles. Intersecting circles with a continuous upper boundary.A discontinuity (in orange) when one circle overshadows the other.