Finding the centroid of a polygon?

Question

To get the center, I have tried, for each vertex, to add to the total, divide by the number of vertices.

I've also tried to find the topmost, bottommost -> get midpoint... find leftmost, rightmost, find the midpoint.

Both of these did not return the perfect center because I'm relying on the center to scale a polygon.

I want to scale my polygons, so I may put a border around them.

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?

Answer 1

The formula is given here for vertices sorted by their occurance along the polygon's perimeter.

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 << ")\n"; } 

I've only tested this for a square polygon in the upper-right x/y quadrant.

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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; } 

Answer 2

The centroid can be calculated as the weighted sum of the centroids of the triangles it can be partitioned to.

Here is the C source code for such an algorithm:

/*     Written by Joseph O'Rourke     [email protected]     October 27, 1995      Computes the centroid (center of gravity) of an arbitrary     simple polygon via a weighted sum of signed triangle areas,     weighted by the centroid of each triangle.     Reads x,y coordinates from stdin.       NB: Assumes points are entered in ccw order!       E.g., input for square:         0   0         10  0         10  10         0   10     This solves Exercise 12, p.47, of my text,     Computational Geometry in C.  See the book for an explanation     of why this works. Follow links from         http://cs.smith.edu/~orourke/  */ #include <stdio.h>  #define DIM     2               /* Dimension of points */ typedef int     tPointi[DIM];   /* type integer point */ typedef double  tPointd[DIM];   /* type double point */  #define PMAX    1000            /* Max # of pts in polygon */ typedef tPointi tPolygoni[PMAX];/* type integer polygon */  int     Area2( tPointi a, tPointi b, tPointi c ); void    FindCG( int n, tPolygoni P, tPointd CG ); int     ReadPoints( tPolygoni P ); void    Centroid3( tPointi p1, tPointi p2, tPointi p3, tPointi c ); void    PrintPoint( tPointd p );  int main() {     int n;     tPolygoni   P;     tPointd CG;      n = ReadPoints( P );     FindCG( n, P ,CG);     printf("The cg is ");     PrintPoint( CG ); }  /*      Returns twice the signed area of the triangle determined by a,b,c,     positive if a,b,c are oriented ccw, and negative if cw. */ int Area2( tPointi a, tPointi b, tPointi c ) {     return         (b[0] - a[0]) * (c[1] - a[1]) -         (c[0] - a[0]) * (b[1] - a[1]); }  /*           Returns the cg in CG.  Computes the weighted sum of     each triangle's area times its centroid.  Twice area     and three times centroid is used to avoid division     until the last moment. */ void FindCG( int n, tPolygoni P, tPointd CG ) {     int     i;     double  A2, Areasum2 = 0;        /* Partial area sum */         tPointi Cent3;      CG[0] = 0;     CG[1] = 0;     for (i = 1; i < n-1; i++) {         Centroid3( P[0], P[i], P[i+1], Cent3 );         A2 =  Area2( P[0], P[i], P[i+1]);         CG[0] += A2 * Cent3[0];         CG[1] += A2 * Cent3[1];         Areasum2 += A2;     }     CG[0] /= 3 * Areasum2;     CG[1] /= 3 * Areasum2;     return; }  /*     Returns three times the centroid.  The factor of 3 is     left in to permit division to be avoided until later. */ void Centroid3( tPointi p1, tPointi p2, tPointi p3, tPointi c ) {     c[0] = p1[0] + p2[0] + p3[0];     c[1] = p1[1] + p2[1] + p3[1];     return; }  void PrintPoint( tPointd p ) {     int i;      putchar('(');     for ( i=0; i<DIM; i++) {         printf("%f",p[i]);         if (i != DIM - 1) putchar(',');     }     putchar(')');     putchar('\n'); }  /*     Reads in the coordinates of the vertices of a polygon from stdin,     puts them into P, and returns n, the number of vertices.     The input is assumed to be pairs of whitespace-separated coordinates,     one pair per line.  The number of points is not part of the input. */ int ReadPoints( tPolygoni P ) {     int n = 0;      printf("Polygon:\n");     printf("  i   x   y\n");           while ( (n < PMAX) && (scanf("%d %d",&P[n][0],&P[n][1]) != EOF) ) {         printf("%3d%4d%4d\n", n, P[n][0], P[n][1]);         ++n;     }     if (n < PMAX)         printf("n = %3d vertices read\n",n);     else         printf("Error in ReadPoints:\too many points; max is %d\n", PMAX);     putchar('\n');      return  n; } 

There's a polygon centroid article on the CGAFaq (comp.graphics.algorithms FAQ) wiki that explains it.