穿过已知点画平滑曲线（3次贝塞尔曲线）_C/C++_编程开发_程序员俱乐部

为了把一串点连成光滑的曲线，先研究贝塞尔曲线，又搞B样条插值。。。。都没有成功(数学没那么强)。

后来在

“[翻译] AGG 之贝塞尔插值”http://liyiwen.javaeye.com/blog/705489?。看到一种比较好的方法：

运用贝塞尔曲线的光滑性来穿过这些点。

大致思路就是 先算出相邻原始点的中点，在把相邻中点连成的线段平移到对应的原始点，以平移后的中点作为控制点，相邻原始点为起始点画贝塞尔曲线，这样就保证了连接处的光滑。而贝塞尔曲线本身是光滑的，所以就把这些原始点用光滑曲线连起来了。

我封装了一个函数，留着以后用。

（c++版，其它语言只要把数组和可变数组稍微变一下就能用）

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void createCurve(CvPoint *originPoint,int originCount,vector<CvPoint> &curvePoint){
    //控制点收缩系数 ，经调试0.6较好，CvPoint是opencv的，可自行定义结构体(x,y)
    float scale = 0.6;
    CvPoint midpoints[originCount];
    //生成中点     
    for(int i = 0 ;i < originCount ; i++){    
        int nexti = (i + 1) % originCount;
        midpoints[i].x = (originPoint[i].x + originPoint[nexti].x)/2.0;
        midpoints[i].y = (originPoint[i].y + originPoint[nexti].y)/2.0;
    }    
    
    //平移中点
    CvPoint extrapoints[2 * originCount]; 
    for(int i = 0 ;i < originCount ; i++){
         int nexti = (i + 1) % originCount;
         int backi = (i + originCount - 1) % originCount;
         CvPoint midinmid;
         midinmid.x = (midpoints[i].x + midpoints[backi].x)/2.0;
         midinmid.y = (midpoints[i].y + midpoints[backi].y)/2.0;
         int offsetx = originPoint[i].x - midinmid.x;
         int offsety = originPoint[i].y - midinmid.y;
         int extraindex = 2 * i;
         extrapoints[extraindex].x = midpoints[backi].x + offsetx;
         extrapoints[extraindex].y = midpoints[backi].y + offsety;
         //朝 originPoint[i]方向收缩 
         int addx = (extrapoints[extraindex].x - originPoint[i].x) * scale;
         int addy = (extrapoints[extraindex].y - originPoint[i].y) * scale;
         extrapoints[extraindex].x = originPoint[i].x + addx;
         extrapoints[extraindex].y = originPoint[i].y + addy;
         
         int extranexti = (extraindex + 1)%(2 * originCount);
         extrapoints[extranexti].x = midpoints[i].x + offsetx;
         extrapoints[extranexti].y = midpoints[i].y + offsety;
         //朝 originPoint[i]方向收缩 
         addx = (extrapoints[extranexti].x - originPoint[i].x) * scale;
         addy = (extrapoints[extranexti].y - originPoint[i].y) * scale;
         extrapoints[extranexti].x = originPoint[i].x + addx;
         extrapoints[extranexti].y = originPoint[i].y + addy;
         
    }    
    
    CvPoint controlPoint[4];
    //生成4控制点，产生贝塞尔曲线
    for(int i = 0 ;i < originCount ; i++){
           controlPoint[0] = originPoint[i];
           int extraindex = 2 * i;
           controlPoint[1] = extrapoints[extraindex + 1];
           int extranexti = (extraindex + 2) % (2 * originCount);
           controlPoint[2] = extrapoints[extranexti];
           int nexti = (i + 1) % originCount;
           controlPoint[3] = originPoint[nexti];    
           float u = 1;
           while(u >= 0){
               int px = bezier3funcX(u,controlPoint);
               int py = bezier3funcY(u,controlPoint);
               //u的步长决定曲线的疏密
               u -= 0.005;
               CvPoint tempP = cvPoint(px,py);
               //存入曲线点 
               curvePoint.push_back(tempP);
           }    
    }
}

//三次贝塞尔曲线
float bezier3funcX(float uu,CvPoint *controlP){
   float part0 = controlP[0].x * uu * uu * uu;
   float part1 = 3 * controlP[1].x * uu * uu * (1 - uu);
   float part2 = 3 * controlP[2].x * uu * (1 - uu) * (1 - uu);
   float part3 = controlP[3].x * (1 - uu) * (1 - uu) * (1 - uu);   
   return part0 + part1 + part2 + part3; 
}    

float bezier3funcY(float uu,CvPoint *controlP){
   float part0 = controlP[0].y * uu * uu * uu;
   float part1 = 3 * controlP[1].y * uu * uu * (1 - uu);
   float part2 = 3 * controlP[2].y * uu * (1 - uu) * (1 - uu);
   float part3 = controlP[3].y * (1 - uu) * (1 - uu) * (1 - uu);   
   return part0 + part1 + part2 + part3; 
} 

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我用opencv画的效果图：

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