We study the problem of connecting two points in a simple polygon with a self-approaching path. A self-approaching path is a directed curve such that the Euclidean distance between a point moving along the path and any future position does not increase, that is, for all points a, b, and c that appear in that order along the curve, $|ac|\ge |bc|$. We analyze properties of self-approaching paths inside simple polygons, and characterize shortest self-approaching paths. In particular, we show that a shortest self-approaching path connecting two points in a simple polygon can be forced to follow a general class of non-algebraic curves. While this makes it difficult to design an exact algorithm, we show how to find the shortest self-approaching path under a model of computation which assumes that we can compute involute curves of high order. Lastly, we provide an efficient algorithm to test if a given simple polygon is self-approaching, that is, if there exists a self-approaching path for any two points inside the polygon.

我们研究了在具有自逼近路径的简单多边形中连接两个点的问题。自接近路径是一条有向曲线，因此沿该路径移动的点与任何将来位置之间的欧几里得距离不会增加，也就是说，对于沿该曲线按此顺序出现的所有点a，b和c，$|\mathrm{一种}C|\ge |bC|$。我们分析了简单多边形内部的自逼路径的属性，并描述了最短的自逼路径。特别是，我们表明，连接简单多边形中两个点的最短自逼路径可能会被迫遵循一般的非代数曲线。尽管这使设计精确的算法变得困难，但我们展示了如何在计算模型下找到最短的自逼路径，该计算模型假设我们可以计算高阶渐开线曲线。最后，我们提供了一种有效的算法来测试给定的简单多边形是否是自逼近的，也就是说，对于多边形内的任何两个点是否都存在自逼近路径。