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多角形の重心を見つけますか?

中心を取得するために、各頂点について、合計に追加し、頂点の数で除算することを試みました。

また、最上部、最下部を見つけようとしました->中間点を取得します...左端、右端、中間点を見つけます。

どちらもポリゴンをスケーリングするために中心に依存しているため、どちらも完璧な中心を返しませんでした。

ポリゴンをスケーリングしたいので、ポリゴンの周りに境界線を付けます。

多角形が凹面、凸面であり、さまざまな長さの多くの側面を持っている場合、多角形の重心を見つける最良の方法は何ですか?

43
jmasterx

式は here で与えられます。

これらの式のシグマ表記法を理解するのが難しい人のために、計算方法を示すC++コードをいくつか示します。

#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";
}

これは、右上のx/y象限の正方形ポリゴンに対してのみテストしました。


各反復で2つの(潜在的に高価な)余分なモジュラス演算を実行してもかまわない場合は、以前のcompute2DPolygonCentroid次の機能:

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;
}
65
Emile Cormier

重心は、分割可能な三角形の重心の加重和として計算できます。

このようなアルゴリズムの Cソースコード は次のとおりです。

/*
    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;
}

polygon centroid CGAFaq(comp.graphics.algorithms FAQ)ウィキには、それを説明する記事があります。

10
Firas Assaad
boost::geometry::centroid(your_polygon, p);
7
Arlen

それを三角形に分割し、それぞれの面積と重心を見つけて、部分面積を重みとして使用して、すべての部分重心の平均を計算します。凹面では、一部の領域がマイナスになる可能性があります。

0
Ben Voigt