贝塞尔曲线的一些资料收集

一本免费的在线书籍，供你在非常需要了解如何处理贝塞尔相关的事情。
https://pomax.github.io/bezierinfo/zh-CN/index.html

An algorithm to find bounding box of closed bezier curves? - Stack Overflow
https://stackoverflow.com/questions/2587751/an-algorithm-to-find-bounding-box-of-closed-bezier-curves

前端动画之贝塞尔曲线推导及应用-CSDN博客
https://blog.csdn.net/frontend_frank/article/details/123437040

贝塞尔曲线在线绘制🚀
https://www.bezier-curve.com/

Calculating / Computing the Bounding Box of Cubic Bezier https://floris.briolas.nl/floris/2009/10/bounding-box-of-cubic-bezier/

获取三阶贝塞尔曲线的最小包围盒

c# https://floris.briolas.nl/floris/2009/10/bounding-box-of-cubic-bezier/

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Drawing;namespace BoundingBoxTestProject
{
public static class BezierOp
{public delegate TResult Func<T1, T2, T3, T4, T5, TResult>(T1 arg1, T2 arg2, T3 arg3, T4 arg4, T5 arg5);//cubic polinomal//x = At^3 + Bt^2 + Ct + D//where A,B,C,D://A = p3 -3 * p2 + 3 * p1 - p0//B = 3 * p2 - 6 * p1 +3 * p0//C = 3 * p1 - 3 * p0//D = p0            public static Func<double, double, double, double, double, double> bezierSpline = (p0, p1, p2, p3, t) =>(p3 - 3 * p2 + 3 * p1 - p0) * Math.Pow(t, 3)+ (3 * p2 - 6 * p1 + 3 * p0) * Math.Pow(t, 2)+ (3 * p1 - 3 * p0) * t+ (p0);//X = At^3 + Bt^2 + Ct + D//where A,B,C,D://A = (p3 -3 * p2 + 3 * p1 - p0)//B = (3 * p2 - 6 * p1 +3 * p0)//C = (3 * p1 - 3 * p0)//D = (p0)//We would like to know the values of t where X = 0//X  = (p3-3*p2+3*p1-p0)t^3 + (3*p2-6*p1+3*p0)t^2 + (3*p1-3*p0)t + (p0)//Derivetive ://X' = 3(p3-3*p2+3*p1-p0)t^(3-1) + 2(6*p2-12*p1+6*p0)t^(2-1) + 1(3*p1-3*p0)t^(1-1)              [f(x)=aX^n => f'(x)=a*n*X^(n-1)  remember?]//simplified://X' = (3*p3-9*p2+9*p1-3*p0)t^2 + (6*p2-12*p1+6*p0)t + (3*p1-3*p0)//**!!reusing a,b,and c!!!***//taken as aX^2 + bX + c  a,b and c are:          public static Func<double, double, double, double, double> A = (p0, p1, p2, p3) => 3 * p3 - 9 * p2 + 9 * p1 - 3 * p0;//ommitting power 2 for now            public static Func<double, double, double, double> B = (p0, p1, p2) => 6 * p2 - 12 * p1 + 6 * p0;public static Func<double, double, double> C = (p0, p1) => 3 * p1 - 3 * p0;//b^2 - 4ac = Determinantpublic static Func<double, double, double, double> Determinant = (a, b, c) => Math.Pow(b, 2) - 4d * a * c;public static Func<double, double, double, double[]> Solve = (a, b, c) =>{Func<double, double, double, bool, double> _Solve =(a_, b_, c_, s) =>(-b_ +(Math.Sqrt((b_ * b_) - (4d * a_ * c_)) *((s) ? 1d : -1d))) / (2d * a_);double d = Determinant(a, b, c);if (d < 0)return new double[] { };if (a == 0)//aX^2 + bX + c well then then this is a simple line//x= -c / breturn new double[] { -c / b };if (d == 0){return new double[] { _Solve(a, b, c, true) };}elsereturn new double[]{_Solve(a, b, c, true),_Solve(a, b, c, false)};};public static RectangleF GetRect(PointF p1, PointF c1, PointF c2, PointF p2){double aX = A(p1.X, c1.X, c2.X, p2.X);double bX = B(p1.X, c1.X, c2.X);double cX = C(p1.X, c1.X);double aY = A(p1.Y, c1.Y, c2.Y, p2.Y);double bY = B(p1.Y, c1.Y, c2.Y);double cY = C(p1.Y, c1.Y);var resX = Solve(aX, bX, cX).Where(t => (t >= 0) && (t <= 1)); //solve for t ==0 & filtervar resY = Solve(aY, bY, cY).Where(t => (t >= 0) && (t <= 1)); //solve for t ==0 & filter//Draw min and max;List<PointF> _BBox = new List<PointF>();_BBox.Add(p1); //Add Begin and end point not the control points!_BBox.Add(p2);foreach (var e in resX.Union(resY)){double x = bezierSpline(p1.X, c1.X, c2.X, p2.X, e);double y = bezierSpline(p1.Y, c1.Y, c2.Y, p2.Y, e);PointF p = new PointF((float)x, (float)y);_BBox.Add(p);}float minX = float.MaxValue;float minY = float.MaxValue;float maxX = float.MinValue;float maxY = float.MinValue;foreach (var e in _BBox) //find the bounding box.{minX = Math.Min(e.X, minX);minY = Math.Min(e.Y, minY);maxX = Math.Max(e.X, maxX);maxY = Math.Max(e.Y, maxY);}return new RectangleF(minX, minY, maxX - minX, maxY - minY);}}}

c++ 由上述c#代码改写而来,未做优化

#include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>// 表示一个边界框的结构体
struct BoundingBox {double x;      // 左上角 x 坐标double y;      // 左上角 y 坐标double width;  // 宽度double height; // 高度BoundingBox(double _x, double _y, double _width, double _height) : x(_x), y(_y), width(_width), height(_height) {}
};// 表示一个二维点的结构体
struct Point {double x; // x 坐标double y; // y 坐标Point(double _x, double _y) : x(_x), y(_y) {}
};// 计算三次贝塞尔曲线的值
double bezierSpline(double p0, double p1, double p2, double p3, double t) {double mt = 1 - t;return mt * mt * mt * p0 + 3 * mt * mt * t * p1 + 3 * mt * t * t * p2 + t * t * t * p3;
}// 计算贝塞尔曲线的导数系数
void calculateCoefficients(double p0, double p1, double p2, double p3, double &a, double &b, double &c) {a = 3 * (p3 - 3 * p2 + 3 * p1 - p0);b = 6 * (p2 - 2 * p1 + p0);c = 3 * (p1 - p0);
}// 解二次方程 ax^2 + bx + c = 0，并返回在区间 [0, 1] 内的实数解
std::vector<double> solveQuadratic(double a, double b, double c) {std::vector<double> solutions;// 计算判别式double discriminant = b * b - 4 * a * c;if (discriminant >= 0) {double sqrtDiscriminant = std::sqrt(discriminant);double t1 = (-b + sqrtDiscriminant) / (2 * a);double t2 = (-b - sqrtDiscriminant) / (2 * a);if (t1 >= 0 && t1 <= 1) {solutions.push_back(t1);}if (t2 >= 0 && t2 <= 1) {solutions.push_back(t2);}}return solutions;
}// 计算贝塞尔曲线的边界框
BoundingBox getBoundsOfCurve(double x0, double y0, double x1, double y1, double x2, double y2, double x3, double y3) {double aX, bX, cX, aY, bY, cY;// 计算 x 和 y 方向的导数系数calculateCoefficients(x0, x1, x2, x3, aX, bX, cX);calculateCoefficients(y0, y1, y2, y3, aY, bY, cY);// 找到导数为零的参数值std::vector<double> xSolutions = solveQuadratic(aX, bX, cX);std::vector<double> ySolutions = solveQuadratic(aY, bY, cY);// 存储边界框的点std::vector<Point> bboxPoints;bboxPoints.push_back(Point(x0, y0));bboxPoints.push_back(Point(x3, y3));// 在导数为零的参数值下计算曲线上的点for (double t : xSolutions) {double x = bezierSpline(x0, x1, x2, x3, t);double y = bezierSpline(y0, y1, y2, y3, t);bboxPoints.push_back(Point(x, y));}for (double t : ySolutions) {double x = bezierSpline(x0, x1, x2, x3, t);double y = bezierSpline(y0, y1, y2, y3, t);bboxPoints.push_back(Point(x, y));}// 计算边界框的坐标double minX = bboxPoints[0].x;double minY = bboxPoints[0].y;double maxX = bboxPoints[0].x;double maxY = bboxPoints[0].y;for (const Point &point : bboxPoints) {minX = std::min(minX, point.x);minY = std::min(minY, point.y);maxX = std::max(maxX, point.x);maxY = std::max(maxY, point.y);}// 创建边界框并返回return BoundingBox(minX, minY, maxX - minX, maxY - minY);
}int main()
{// 定义四个控制点的坐标// double x0 = 300.0, y0 = 400.0, x1 = 400.0, y1 = 300.0, x2 = 700.0, y2 = 500.0, x3 = 500.0, y3 = 500.0;double x0 = 300.0, y0 = 400.0, x1 = 400.0, y1 = 600.0, x2 = 700.0, y2 = 500.0, x3 = 500.0, y3 = 500.0;// 计算贝塞尔曲线的边界框BoundingBox bounds = getBoundsOfCurve(x0, y0, x1, y1, x2, y2, x3, y3);std::cout << "Bounding Box: (" << bounds.x << ", " << bounds.y << ") - Width: " << bounds.width << ", Height: " << bounds.height;return 0;
}// Bounding Box: (300, 400) - Width: 267.277, Height: 125