查找带加权顶点的多边形的质心 [英] Find the centroid of a polygon with weighted vertices

问题描述

我知道如何找到正多边形的质心（质心）。这假设多边形的每个部分重量相同。
但我该如何计算一个没有重量的多边形（由气凝胶制成）的质心，其中每个顶点都有一个重量？

意思是用直线表示：

  5kg ----------------- 5kg 
 ^重心
 
 10kg --------------- 5kg 
 ^重心偏移du对顶点的权重
  

当然，我知道如何在具有加权顶点的直线上计算重心，但我该如何操作在具有加权顶点的多边形上？

感谢您的时间！

解决方案

你想对所有顶点进行加权平均。所以说你的顶点是质量为m1，m2 ... mn的v1，v2，v3 .... vn，并且有x和y坐标v1x，v1y，v2x，v2y等，然后得到质心（cx，cy）你想：

pre code> cx =（v1x * m1 + v2x * m2 + ... vnx * mn）/（m1 + m2 ... mn）
cy =（v1y * m1 + v2y * m2 + ... vny * mn）/（m1 + m2 .... mn）

它基本上与您为一条线做同样的原则。

I know how to find the centroid (center of mass) of a regular polygon. This assumes that every part of the polygon weighs the same. But how do I calculate the centroid of a weightless polygon (made from aerogel perhaps :), where each vertex has a weight?

Simplified illustration of what I mean using straight line:

5kg-----------------5kg
           ^center of gravity

10kg---------------5kg
        ^center of gravity offset du to weight of vertices

Of course, I know how to calculate the center of gravity on a straight line with weighted vertices, but how do I do it on a polygon with weighted vertices?

Thanks for your time!

解决方案

You want take a weighted average over all the vertices. So say your vertices are v1, v2, v3 .... vn with masses m1, m2 ...mn and have x and y coordinates v1x, v1y, v2x, v2y etc then to get the center of mass (cx, cy) you want:

cx = (v1x*m1 + v2x*m2 + ... vnx*mn) / (m1 + m2 .... mn) 
cy = (v1y*m1 + v2y*m2 + ... vny*mn) / (m1 + m2 .... mn)

It's essentially the same principle as when you do it for a line.

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