找到多边形的质心?
为了获得中心，我尝试将每个顶点添加到总数中，然后除以顶点数.
To get the center, I have tried, for each vertex, to add to the total, divide by the number of vertices.
我也试过找到最上面、最下面 -> 找到中点...找到最左边、最右边，找到中点.
I've also tried to find the topmost, bottommost -> get midpoint... find leftmost, rightmost, find the midpoint.
这两个都没有返回完美的中心，因为我依靠中心来缩放多边形.
Both of these did not return the perfect center because I'm relying on the center to scale a polygon.
我想缩放我的多边形，所以我可以在它们周围放置一个边框.
I want to scale my polygons, so I may put a border around them.
考虑到多边形可能是凹的、凸的并且有许多不同长度的边，找到多边形质心的最佳方法是什么?
What is the best way to find the centroid of a polygon given that the polygon may be concave, convex and have many many sides of various lengths?
推荐答案
给出公式这里 表示顶点按照它们沿多边形周长的出现次数排序.
对于那些难以理解这些公式中的 sigma 符号的人，这里有一些 C++ 代码展示了如何进行计算:
For those having difficulty understanding the sigma notation in those formulas, here is some C++ code showing how to do the computation:
#include <iostream>
struct Point2D
{
double x;
double y;
};
Point2D compute2DPolygonCentroid(const Point2D* vertices, int vertexCount)
{
Point2D centroid = {0, 0};
double signedArea = 0.0;
double x0 = 0.0; // Current vertex X
double y0 = 0.0; // Current vertex Y
double x1 = 0.0; // Next vertex X
double y1 = 0.0; // Next vertex Y
double a = 0.0; // Partial signed area
// For all vertices except last
int i=0;
for (i=0; i<vertexCount-1; ++i)
{
x0 = vertices[i].x;
y0 = vertices[i].y;
x1 = vertices[i+1].x;
y1 = vertices[i+1].y;
a = x0*y1 - x1*y0;
signedArea += a;
centroid.x += (x0 + x1)*a;
centroid.y += (y0 + y1)*a;
}
// Do last vertex separately to avoid performing an expensive
// modulus operation in each iteration.
x0 = vertices[i].x;
y0 = vertices[i].y;
x1 = vertices[0].x;
y1 = vertices[0].y;
a = x0*y1 - x1*y0;
signedArea += a;
centroid.x += (x0 + x1)*a;
centroid.y += (y0 + y1)*a;
signedArea *= 0.5;
centroid.x /= (6.0*signedArea);
centroid.y /= (6.0*signedArea);
return centroid;
}
int main()
{
Point2D polygon[] = {{0.0,0.0}, {0.0,10.0}, {10.0,10.0}, {10.0,0.0}};
size_t vertexCount = sizeof(polygon) / sizeof(polygon[0]);
Point2D centroid = compute2DPolygonCentroid(polygon, vertexCount);
std::cout << "Centroid is (" << centroid.x << ", " << centroid.y << ")
";
}
我只针对右上 x/y 象限中的方形多边形进行了测试.
I've only tested this for a square polygon in the upper-right x/y quadrant.
如果您不介意在每次迭代中执行两个(可能很昂贵)额外的模数运算，那么您可以将之前的 compute2DPolygonCentroid 函数简化为以下内容:
If you don't mind performing two (potentially expensive) extra modulus operations in each iteration, then you can simplify the previous compute2DPolygonCentroid function to the following:
Point2D compute2DPolygonCentroid(const Point2D* vertices, int vertexCount)
{
Point2D centroid = {0, 0};
double signedArea = 0.0;
double x0 = 0.0; // Current vertex X
double y0 = 0.0; // Current vertex Y
double x1 = 0.0; // Next vertex X
double y1 = 0.0; // Next vertex Y
double a = 0.0; // Partial signed area
// For all vertices
int i=0;
for (i=0; i<vertexCount; ++i)
{
x0 = vertices[i].x;
y0 = vertices[i].y;
x1 = vertices[(i+1) % vertexCount].x;
y1 = vertices[(i+1) % vertexCount].y;
a = x0*y1 - x1*y0;
signedArea += a;
centroid.x += (x0 + x1)*a;
centroid.y += (y0 + y1)*a;
}
signedArea *= 0.5;
centroid.x /= (6.0*signedArea);
centroid.y /= (6.0*signedArea);
return centroid;
}
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