发表机构
Centre for Visual Computing and Intelligent Systems; University of Bradford（视觉计算与智能系统中心; 布拉德福德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于曲率轮廓而非仿射组合的插值细分方案，在平面和球面上精确再现测地线与圆，并证明其收敛到切线方向和曲率连续的二次可微极限曲线。
AI 中文摘要
我们引入并分析了一种插值细分方案，其中每个插入点由曲率轮廓模型定义，而非由其邻点的仿射组合定义，并确立了其收敛性和几何正则性。在每条边上，由外接圆估计的曲率规定了一个曲率轮廓，该线段通过带有测地线射击的Frenet积分进行重建，新点取在半弧长处。该构造在平面和球面上同样适用，能精确再现测地线和圆，并且一个独特的曲率预滤波器将其平面线性化转化为六点Deslauriers-Dubuc方案。Ewald、Reif和Sabin的拉直条件对该方案失效。仅测量相对畸变的法向部分修复了他们的理论，我们证明了细化多边形收敛到一条正则的插值极限曲线，其切线方向和曲率连续，且在弧长上二次连续可微。因此，正则性仅由法向伴随方案控制，正如这些作者所猜想的那样。
英文摘要
We introduce and analyse an interpolatory subdivision scheme in which each inserted point is defined by a curvature-profile model rather than by an affine combination of its neighbours, and we establish its convergence and geometric regularity. On each edge, curvatures estimated from circumscribed circles prescribe a curvature profile, the segment is reconstructed by Frenet integration with geodesic shooting, and the new point is taken at half arc length. The construction works identically in the plane and on the sphere, reproduces geodesics and circles exactly, and a unique curvature prefilter turns its planar linearisation into the six-point Deslauriers-Dubuc scheme. The straightening condition of Ewald, Reif, and Sabin fails for this scheme. Measuring only the normal part of the relative distortion repairs their theory, and we prove that the refined polygons converge to a regular interpolatory limit curve with continuous tangent direction and curvature, twice continuously differentiable in arc length. Regularity is thus governed by the normal companion scheme alone, as these authors conjectured.
