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.

翻译：计算变形平面域的包络线是一个重要且具有挑战性的问题，在众多潜在应用中具有广泛意义。我们采用圆弧样条——一种平衡几何灵活性与计算简洁性的曲线——来逼近该包络线。为实现这一优势，我们的方法融合了两个核心概念。首先，我们通过中轴变换（MAT）表示平面域，该变换是闵可夫斯基空间 $\mathbb R^{2,1}$ 中的几何图（可能包含退化分支）。我们观察到，闵可夫斯基空间中的圆弧对应于圆弧样条域的中轴变换。此外，随着平面域随时间演化，其中轴变换的每个分支亦随之演化，并在闵可夫斯基空间中形成曲面。这使我们能够将包络线计算问题重新表述为计算这些曲面上有限曲线集的旋轮投影图像的问题。我们提出并比较了两组用于逼近曲线及其旋轮投影图像边界的方法。所有这些方法最终生成演化域包络线的圆弧样条逼近。其次，我们利用圆弧在平面和闵可夫斯基空间中的几何灵活性，以实现较高的逼近率。计算简洁性确保了采用具有最优计算复杂度的扫描线算法，能够高效修剪生成包络线的冗余分支。