B样条曲线

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参考文献

B样条曲线
二次B样条曲线代码

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B样条曲线

给定n+m+1个平面或空间顶点Pi(i=0，1，…，n+m)，称为n次参数曲线段：

Pk,n(t)=∑i=0nPi+kGi,n(t) ,    t∈[0,1]

为第k段n次B样条曲线段 (k=0，1，…，m)，这些曲线段的全体称为n次B样条曲线，其顶点Pi(i=0，1，…，n+m)所组成的多边形称为B样条曲线的特征多边形。 其中，基函数Gi,n(t)定义为：

Gi,n(t)=1n!∑j=0n−i(−1)jCjn+1(t+n−i−j)nt∈[0,1] ,   i=0，1，…，n

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二次B样条曲线

取n=2，则有二次B样条曲线的基函数如下：

⎧⎩⎨⎪⎪⎪⎪G0,2(t)=12(t−1)2G1,2(t)=12(−2t2+2t+1)G2,2(t)=12t2

二次B样条曲线段P0,2(t)=∑2i=0PiGi,2(t)是一段抛物线。二次B样条曲线的矩阵表示为：

P0,2(t)=12[1tt2]⎡⎣⎢1−2112−2001⎤⎦⎥⎡⎣⎢P0P1P2⎤⎦⎥   t∈[0,1]

端点位置：

P0,2(0)=12(P0+P1)P0,2(1)=12(P1+P2)

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代码实现

#include <stdio.h>#include <opencv2\opencv.hpp>#include <time.h>using namespace cv;void generatePointArray(int * x, int * y, int size, int minX, int maxX, int minY, int maxY) {    srand((unsigned int) time(0));    int bandX = maxX - minX;    int bandY = maxY - minY;    int i;    for (i = 0; i < size; i++) {        x[i] = rand() % bandX + minX;        y[i] = rand() % bandY + minY;    }}void drawPointWithLine(Mat & board, int * x, int * y, int size){    if (size < 1)        return;    circle(board, Point(x[0], y[0]), 2, Scalar(255, 0, 0), 1, CV_AA);    int i;    for (i = 1; i < size; i++) {        line(board, Point(x[i-1], y[i-1]), Point(x[i], y[i]), Scalar(0, 255, 0), 1, CV_AA);        circle(board, Point(x[i], y[i]), 2, Scalar(255, 0, 0), 1, CV_AA);    }}void BSpline(Mat & board, int * x, int * y, int size) {    if (size != 3)        return;    double t, delta = 1/100.0;    int px = 0, py = 0;    for (t = 0; t <= 1.0; t += delta) {        /*** core code ***/        double g0 = 0.5*(t - 1)*(t - 1);        double g1 = 0.5*(-2*t*t + 2*t + 1);        double g2 = 0.5*t*t;        int _x = (int) (x[0]*g0 + x[1]*g1 + x[2]*g2);        int _y = (int) (y[0]*g0 + y[1]*g1 + y[2]*g2);        /*****************/        if (t == 0)            circle(board, Point(_x, _y), 2, Scalar(0, 0, 255), 1, CV_AA);        else            line(board, Point(px, py), Point(_x, _y), Scalar(0, 128, 255), 1, CV_AA);        px = _x;        py = _y;    }    if (t != 1.0)        circle(board, Point((x[1] + x[2]) / 2, (y[1] + y[2]) / 2), 2, Scalar(0, 0, 255), 1, CV_AA);}int main(int argc, char* argv[]){    int boardWidth = 400, boardHeight = 300;    Mat board = Mat(boardHeight, boardWidth, CV_8UC3);    memset(board.data, 255, boardHeight*boardWidth*3);    int x[5], y[5];    generatePointArray(x, y, 5, 0, boardWidth, 0, boardHeight);    drawPointWithLine(board, x, y, 5);    int i;    for (i = 0; i < 3; i++)        BSpline(board, x+i, y+i, 3);    imshow("B-Spline", board);    waitKey(0);    return 0;}

结果1：

结果2：