摘要:针对现有网格曲面曲线设计方法鲁棒性差、收敛慢、适用范围窄等不足，提出一种基于距离约束的新方法.该方法将复杂的流形约束转化为距离约束，并与光滑、插值（逼近）约束共同描述成优化问题.求解时，用切平面逼近局部曲面，并将距离约束松弛成用点到切平面的距离.由于计算距离所用的曲线上的点与其对应的切点相互依赖，采用“整体-局部”交替迭代的策略，并运用Gauss-Newton法的思想控制其收敛行为：整体阶段，通过距离近似将其松弛成凸优化问题求解迭代步长；局部阶段，采用鲁棒高效的投影法将优化后的曲线映射到曲面以更新切平面；最后，利用切割平面法将所有处于松弛状态的折线映射到网格曲面.实验结果表明：该方法与现有方法相比，在效率、鲁棒性、可控性、应用范围等方面均表现出优势.



Abstract:Existing work of designing curves on mesh surface suffers from issues such as weak robustness, slow convergence, and narrow application ranges. To address these issues, a distance constrained approach is proposed, which converts the complicated manifold constraint into distance constraint, and formulates the problem as a constrained optimization combining with smoothness and interpolation (approximation) constraints. To solve the optimization, the curve is discretized into a poly-line, and the distance constraint is relaxed to point-to-plane distance by approximating the local surface patch with tangent plane. Since the curve points and the corresponding tangent points involved in the distance calculation are interdependence, a “local/global” alternating iteration scheme is adopted and the idea of Gauss-Newton method is used to control the convergence behavior. In the global stage, the iterative step is solved by relaxingthe problem into a convex optimization via distance approximation. In the local stage, a robust and efficient projection method is applied to update tangent planes. Finally, each segment of the poly-line is projected onto the surface by cutting planes. Experiments exhibit that the proposed method outperforms existing work on various aspects, including effectiveness, robustness, controllability, and practicability.



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