Arc Spline Approximation of Envelopes of Evolving Planar Domains

TL;DR

This paper introduces a novel method for approximating the envelope of evolving planar domains using circular arc splines, leveraging the medial axis transform and Minkowski space to improve accuracy and computational efficiency.

Contribution

It presents a new approach combining medial axis transforms and Minkowski space representations to efficiently compute high-quality arc spline approximations of evolving domain envelopes.

Findings

Achieves high approximation accuracy with circular arc splines.

Ensures computational efficiency with a sweep line algorithm.

Provides a comparative analysis of different approximation methods.

Abstract

Computing the envelope of deforming planar domains is a significant and challenging problem with a wide range of potential applications. We approximate the envelope using circular arc splines, curves that balance geometric flexibility and computational simplicity. Our approach combines two concepts to achieve these benefits. First, we represent a planar domain by its medial axis transform (MAT), which is a geometric graph in Minkowski space $\mathbb R^{2,1}$ (possibly with degenerate branches). We observe that circular arcs in the Minkowski space correspond to MATs of arc spline domains. Furthermore, as a planar domain evolves over time, each branch of its MAT evolves and forms a surface in the Minkowski space. This allows us to reformulate the problem of envelope computation as a problem of computing cyclographic images of finite sets of curves on these surfaces. We propose and compare two pairs of methods for approximating the curves and boundaries of their cyclographic images. All of these methods result in an arc spline approximation of the envelope of the evolving domain. Second, we exploit the geometric flexibility of circular arcs in both the plane and Minkowski space to achieve a high approximation rate. The computational simplicity ensures the efficient trimming of redundant branches of the generated envelope using a sweep line algorithm with optimal computational complexity.