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“贝塞尔曲线 (/bɛz.i.eɪ/ BEH-zee-ay) 是一种用于计算机图形学和相关领域的参数化曲线。一组离散的控制点通过公式定义平滑、连续的曲线。
本文首先叙述一阶/二阶/三阶贝塞尔曲线的基本原理，然后针对路径不平滑问题给出C++解决方案。”
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一次Bézier曲线
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二次Bézier曲线
画出图如下：
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三次Bézier曲线
画出来：
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C++实现
//以下是一个使用C++实现的贝塞尔曲线平滑路径曲线的示例代码：#include<iostream>#include<vector>#include<fstream>#include<iomanip>#include<string>
structPoint {float x;float y;};
Point calculateBezierPoint(float t, const Point& p0, const Point& p1, const Point& p2, const Point& p3){ Point point;float u = 1 - t;float tt = t * t;float uu = u * u;float uuu = uu * u;float ttt = tt * t;
point.x = uuu * p0.x + 3 * uu * t * p1.x + 3 * u * tt * p2.x + ttt * p3.x; point.y = uuu * p0.y + 3 * uu * t * p1.y + 3 * u * tt * p2.y + ttt * p3.y;
return point;}
std::vector<Point> smoothPath(conststd::vector<Point>& path, int numSegments){std::vector<Point> smoothedPath; smoothedPath.reserve((path.size() - 2) * numSegments + 2);
for (int i = 0; i < path.size() - 3; i++) {const Point& p0 = path;const Point& p1 = path[i + 1];const Point& p2 = path[i + 2];const Point& p3 = path[i + 3];
for (int j = 0; j < numSegments; j++) {float t = static_cast<float>(j) / numSegments; Point point = calculateBezierPoint(t, p0, p1, p2, p3); smoothedPath.push_back(point); } i = i + 3; }
smoothedPath.push_back(path[path.size() - 2]); smoothedPath.push_back(path[path.size() - 1]);
return smoothedPath;}
intmain(){double x[69] = {-3.7961,-3.7461,-3.6961,-3.6461,-3.5961,-3.5461,-3.4961,-3.4461,-3.3961,-3.3461,-3.2961,-3.2461,-3.1961,-3.1461,-3.0961,-3.0461,-2.9961,-2.9461,-2.8961,-2.8461,-2.7961,-2.7461,-2.6961,-2.6461,-2.5961,-2.5461,-2.4961,-2.4461,-2.3961,-2.3461,-2.2961,-2.2461,-2.1961,-2.1461,-2.0961,-2.0461,-1.9961,-1.9461,-1.8961,-1.8461,-1.7961,-1.7461,-1.6961,-1.6461,-1.5961,-1.5461,-1.4961,-1.4461,-1.3961,-1.3461,-1.2961,-1.2461,-1.1961,-1.1461,-1.0961,-1.0461,-0.9961,-0.9461,-0.8961,-0.8461,-0.7961,-0.7461,-0.6961,-0.6461,-0.5961,-0.5461,-0.4961,-0.4961,-0.4961}; double y[69]= {-0.5775,-0.5775,-0.5775,-0.5775,-0.5775,-0.5775,-0.5775,-0.5775,-0.5775,-0.5775,-0.5775,-0.6275,-0.6775,-0.7275,-0.7775,-0.8275,-0.8775,-0.9275,-0.9775,-1.0275,-1.0775,-1.1275,-1.1775,-1.2275,-1.2275,-1.2775,-1.2775,-1.2775,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.3275,-1.2775,-1.2775,-1.2775,-1.2275,-1.2275,-1.2275,-1.2275,-1.2275,-1.1775,-1.1275,-1.0775,-1.0275,-0.9775,-0.9275,-0.8775,-0.8275,-0.8275,-0.7775,-0.7775,-0.7275,-0.6775,-0.6275,-0.5775,-0.5775,-0.5775};
std::vector<Point> path;// 添加30个路径点的x和y坐标for(int i = 0;i < 69;i++){ path.push_back({ x, y });}//path.push_back({ 1, 2 });// path.push_back({ 3, 5 });// 添加更多路径点...// path.push_back({ 10, 10 });
int numSegments = 10; // 控制平滑路径的细分数
std::vector<Point> smoothedPath = smoothPath(path, numSegments);
// 输出平滑路径的坐标for (const Point& point : smoothedPath) {std::cout << point.x << " " << point.y << std::endl;//std::ofstream file;// file.open("/home/juchunyu/20231013/240218/log.txt",std::ios::app);// if(file.is_open()){// file<< point.x << " " << point.y << std::endl;// }// file.close(); }
return0;}
以上代码实现的贝塞尔曲线平滑路径曲线的原理如下：
定义了一个Point结构体，包含了一个点的x和y坐标。
实现了calculateBezierPoint函数，用于计算贝塞尔曲线上的某个点的坐标。该函数使用了三阶贝塞尔曲线的公式，根据参数t（范围为0到1）、起始点p0、控制点p1和终止点p2 p3 计算出贝塞尔曲线上的相应点的坐标。
实现了smoothPath函数，用于平滑给定的路径曲线。该函数通过遍历路径上的相邻三个点，将每两个点之间的线段使用贝塞尔曲线进行平滑。在每个线段上，等间隔地取若干个t值，然后通过调用calculateBezierPoint函数计算出每个t值对应的贝塞尔曲线上的点，并将这些点存入一个新的向量中。
在main函数中，定义了一个包含30个路径点的path向量。你可以自行修改和添加路径点的坐标。
调用smoothPath函数，将给定的路径曲线进行平滑处理。numSegments参数用于控制每两个路径点之间贝塞尔曲线的细分数，即将每段线段平滑成多少个小曲线段。
输出smoothedPath向量中的路径点，即平滑后的路径曲线的坐标。
通过以上步骤，代码实现了使用贝塞尔曲线对给定的路径曲线进行平滑处理，得到了一条平滑的路径曲线。
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效果
蓝色代表原始路径，红色代表平滑后路径。 |
