通用三角覆盖曲线与多边形链：逃逸森林与拟合蠕虫
Universal Triangle Covering Curve and Polygonal Chain: Escaping Forest and Fitting Worm
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中文总结 AI 辅助
本文提出通用公式解决曲线与多边形链的三角覆盖、拟合问题，将其归为Bellman逃逸森林问题和Moser蠕虫问题的特例，推导带支撑函数约束的泛函极小化公式并证明等价性与收敛性，结合数值方法给出相关结果。
中文摘要 AI 辅助
本文提出一种通用公式，用于解决用三角形覆盖曲线和多边形链、以及将这些曲线拟合入三角形的问题。这些问题可分别作为Bellman的“迷失在森林”问题（逃逸三角形森林）和Moser的“蠕虫问题”（被三角形覆盖）的特例。我们通过保持曲线静止、允许三角形平移和旋转来建模和重构问题，随后推导带支撑函数约束的泛函极小化公式以求解，还证明了公式的等价性与收敛性。最后，我们采用数值方法，给出用不同角度的任意三角形覆盖曲线的结果，同时呈现一些推论与变体结果，包括闭合曲线和闭合多边形链。
英文摘要
In this paper, we present a general formulation to address the problems of covering curves and polygonal chains with triangle, and fitting these curves into triangle. These problems can be formulated as special cases of Bellman's lost-in-a-forest problem (escaping triangular forest) and Moser's worm problem (covered by triangle). We model and reformulate the problem by keeping the curve stationary while allowing the triangle to translate and rotate. Subsequently, we derive the functional minimization formulation with support function constraints to solve. We also prove the equivalence and convergence of the formulas. Finally, we employ numerical methods and present results for covering curves with arbitrary triangles of various angles. We also present some corollaries and variant results, including closed curves and closed polygonal chains.
