关于点-回旋曲线距离算法的一个注记
A Note on the Point-Clothoid Distance Algorithm
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中文总结 AI 辅助
本文针对回旋曲线最近点计算，利用渐屈线几何证明平方距离函数至多三个驻点且极值顺序为极小-极大-极小，从而完善了Frego-Bertolazzi算法的候选点选择逻辑，并允许省略不必要的中间搜索，数值实验显示效率提升。
中文摘要 AI 辅助
计算回旋曲线上最近点是几何设计、道路与铁路线形设计以及路径规划中的常见任务。Frego和Bertolazzi提出的高效算法解决了这一问题，但其候选点选择分析假设每个搜索区间至多有一个局部最小值。我们展示了存在具有两个局部最小值的合法配置，这引发了一个问题：现有策略是否考虑了所有可能的最小值。利用回旋曲线渐屈线的几何性质，我们证明对于任意查询点和任意切线角变化至多$2\pi$的适当无拐点平面回旋曲线段，平方距离函数至多有三个驻点；若三个驻点均为局部极值，则其顺序为极小-极大-极小。这确立了原始候选点选择逻辑在单最小值前提之外的完备性。同时表明，当两个端点导数检验均不活跃时，无需进行内部搜索，从而可以在保留数值回退的同时省略不必要的中间点搜索。数值实验展示了迭代次数和评估时间的减少。
英文摘要
Computing the closest point on a clothoid is a recurring task in geometric design, road and railway alignment, and path planning. The efficient algorithm of Frego and Bertolazzi addresses this problem, but its candidate-selection analysis assumes at most one local minimum per search interval. We exhibit admissible configurations with two local minima, raising the question of whether the existing strategy accounts for every possible minimum. Using the geometry of the clothoid evolute, we prove that, for any query point and any proper no-inflection planar clothoid segment with tangent-angle variation at most $2π$, the squared-distance function has at most three stationary points; if all three are local extrema, their order is min-max-min. This establishes the completeness of the original candidate-selection logic beyond the one-minimum premise. It also shows that no interior search is needed when neither endpoint derivative test is active, allowing unnecessary midpoint searches to be omitted while retaining numerical fallback. Numerical experiments demonstrate reductions in iteration count and evaluation time.
