Arc Spline Approximation of Envelopes of Evolving Planar Domains

TL;DR

The paper tackles the challenging problem of computing envelopes for evolving planar domains. It introduces a framework that represents domains via medial axis transforms in Minkowski space $\mathbb{R}^{2,1}$ and uses cyclographic mappings to convert envelope computation into interpolation of curves on surfaces, enabling arc-spline boundaries. Four interpolation schemes (direct/indirect Minkowski arcs/biarcs) are developed and compared, with adaptive sampling to improve efficiency, and a sweep-line approach to trimming to the true envelope. The method is extended from worm-like domains to free-form domains, demonstrated through examples, and shown to yield high-accuracy, efficiently computable envelopes suitable for applications in design and analysis of evolving shapes.

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.

Figures (17)

• Figure 1: The cyclographic image of a curve segment in $\mathbb R^{2,1}$ is a planar domain (left). Its pre--boundary consists of the envelope curves (green and dark green) and cyclographic images of the endpoints of the curve segment (blue). The boundary of the domain is a subset of the pre--boundary (right).
• Figure 2: Affine types of Minkowski arcs and circles (red) and envelopes of the corresponding one parameter families of oriented circles (green).
• Figure 3: A worm is a cyclographic image of a curve segment in Minkowski space $\mathbb R^{2,1}$.
• Figure 4: The singular (red) and the boundary (dark and light green) curves on a surface in the Minkowski space (left) and their cyclographic images (right).
• Figure 5: Comparison of methods for approximation of curves in $\mathbb R^{2,1}$ by Minkowski arcs for $N=4$.

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Remark
• Proposition 1
• proof
• Proposition 2
• proof
• Definition 3
• Definition 4
• Definition 5