Curvature of planar curves serves as a key regularization term for computing second-order minimal paths, due to its tight relevance to desirable geometric properties such as smoothness, rigidity, and elasticity. In this paper, we tackle a more challenging problem in computational physics and geometry problem: tracking minimal paths whose curvature is constrained by arbitrary upper and lower bounds. For that purpose, we propose a new curvature-bounded geodesic model, developed under the Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) framework. It provides strong geometric control over minimal paths by enforcing curvature range constraints, whose paths are smooth and of bounded curvature limitation. We also present a discretization scheme for the Hamiltonian and the HJB PDE incorporating curvature bounds, allowing efficient solver for estimating numerical solutions to the model. Finally, we illustrate the capability of the proposed curvature-bounded geodesic model in applications of robot path planning and curvilinear structures tracking from images. Numerical experiments demonstrate that the proposed curvature-bounded geodesic model serves as a powerful and robust tool for finding satisfactory paths.

翻译：平面曲线的曲率作为计算二阶最小路径的关键正则化项，因其与光滑性、刚性和弹性等理想几何特性密切相关。本文针对计算物理与几何中的更具挑战性问题：追踪曲率受任意上下界约束的最小路径。为此，我们提出了一种基于Hamilton-Jacobi-Bellman（HJB）偏微分方程（PDE）框架的新型曲率约束测地线模型。该模型通过强制执行曲率范围约束，对最小路径提供强几何控制，使得路径光滑且曲率受限。同时，我们提出了哈密顿量及含曲率约束的HJB PDE离散化方案，使得模型数值解的高效求解成为可能。最后，我们展示了所提曲率约束测地线模型在机器人路径规划与图像曲线结构追踪中的应用能力。数值实验表明，该曲率约束测地线模型是求解满意路径的有效且稳健的工具。