计算两个给定圆弧之间的最小距离 [英] Calculate the minimum distance between two given circular arcs

问题描述

我试图找到一种方法来计算两个给定圆弧之间的最小距离。

I'm trying to find a way to calculate the minimum distance between two given circular arcs.

我找到了一个解决方法。似乎是对的，但我不知道为什么这是正确的！有人可以证明它正确吗？

I found a solution in this link. It seems true but I don't know why that is correct!Can anyone prove it right?

推荐答案

您所引用的解决方案之所以起作用是因为它基于方舟的属性：

The solution you refer to works because it is based on the properties of arks:

1. 圆弧是圆的一部分


2. 始终在端点或垂直方向都达到最小距离，因为它使距离最小（目标函数）。想想两个圆-最小值总是垂直于两个圆。


3. 垂直于圆弧总是穿过圆弧的中心，因为半径始终垂直于圆


4. 垂直情况是当连接中心的直线彼此相交时两条圆弧都相交


5. 端点情况是上一条直线处的端点。项不能同时穿过两个弧线-那么在最靠近中心线的端点上达到了最小距离。

1. An arc is a part of the circle
2. Minimum distance is always reached either at the endpoints or at the perpendicular because it minimizes the distance (objective function). Think of two circles - minimal is always perpendicular to both.
3. Perpendicular to the arc always crosses the center of the arc because radius is always perpendicular to the circle
4. Perpendicular case is when straight line that connects centers crosses both of arcs when they are convex to each other
5. Endpoint case is when the line from the prev. item does not cross both arcs - then minimum of distance is reached on the endpoints closest to the line between the centers.

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