粒子在具有线性边界的分段均匀二维曲面上运动的斯涅耳定律
Snell's law for particles moving on piecewise homogeneous two dimensional surface with linear boundaries
摘要：
本文研究了质点在由分段均匀区域组成的二维曲面上运动的最短时间轨迹的性质。假设粒子在不同的区域和区域之间的边界上以不同的恒定速度移动。假设粒子沿两个区域边界形成的边运动时速度最快。我们得到一个类似于光折射的斯涅耳定律的行为，但形式更一般化。该模型可用于研究动物和昆虫的痕迹，往往形成主要沿边缘的属性。该模型预测了三种类型的行为的轨迹附近的角落形成的边缘：完全边缘以下，部分边缘以下和完全避免的边缘，这确实是观察到的自然蚂蚁踪迹。
英文摘要：
We study the properties of least time trajectories for particles moving on a two dimensional surface which consists of piecewise homogeneous regions. The particles are assumed to move with different constant speeds on different regions and on the boundary between regions. The speed of the particle is assumed to be highest when it moves along the edges formed by the boundary of two regions. We get an analogous behavior to Snell's Law of light refraction, but in a more generalized form. The model could be used for studying properties of animal and insect trails which tend to form predominantly along edges. The model predicts three types of behavior for the trajectories near a corner forming edge: fully edge following, partial edge following and complete avoidance of the edge, which are indeed observed in natural ant trails.
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