🤖 AI Summary
To address the low accuracy and poor efficiency of envelope computation during dynamic deformation of planar regions, this paper proposes an arc-spline approximation method based on medial axis transform and Minkowski-space embedding. We innovatively map the medial axis into Minkowski space, establishing a geometric correspondence between the deforming domain’s boundary and a closed curve in the cyclic image space; consequently, envelope computation is reformulated as an optimal arc-spline approximation problem on this closed curve. A scan-line algorithm is integrated to enable efficient sampling and redundant-branch pruning. The method achieves high geometric fidelity while reducing time complexity to the theoretical optimum, significantly enhancing both computational stability and efficiency. It is broadly applicable to CAD/CAM, motion planning, and deformation analysis.
📝 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.