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实现Bezier样条曲线

news 2025/7/18 6:46:46

1.给出n+1 个控制点pk=(xk,yk,zk),这里k可取值0-n,多项式函数公式如下

获取的单个点的代码

void zmBezier::getPoint(float u, double p[3])
{int n = m_count - 1;double x = 0, y = 0, z = 0;for(int k = 0; k <= n; k++){x += m_ctrlPoints[k][0] * BEZ_k_n(n, k, u);y += m_ctrlPoints[k][1] * BEZ_k_n(n, k, u);z += m_ctrlPoints[k][2] * BEZ_k_n(n, k, u);}p[0] = x;p[1] = y;p[2] = z;
}

2.混合函数是如下的多项式

double zmBezier::BEZ_k_n(int n, int k, double u)
{return  C_n_k(n, k) * pow(u, k) * pow(1 - u, n - k);
}

3.二项式系数

double zmBezier::C_n_k(int n, int k)
{n = m_count - 1;return factorial(n) / (factorial(k) * factorial(n - k));
}

4.Bezier样条完整代码，全部用指针表示点


/**
Bezier曲线
给定n+1个控制点 Pk=(Xk,Yk,Zk),k取值0-n
多项式函数
-----------------------------------nP(u)= Σ  Pk × BEZ(u)            0≤u≤1k=0           k,n
-----------------------------------
混合函数
-----------------------------------k       n-kBEZ(u)=C(n,k) × u × (1-u)       0≤u≤1k,n-----------------------------------
二项式系数
-----------------------------------n!C(n,k)=——————————————————k! × (n-k)!-----------------------------------不想使用 点 结构，全部用指针数组表示点集
*/
#ifndef ZMBEZIER_H
#define ZMBEZIER_Hclass zmBezier
{
public:zmBezier();~zmBezier();zmBezier(int n, double (*points)[3]);void getPoint(float u, double p[3]);                //获取参数u时的某一点void getCurve(int n, double (*curve)[3]);           //获取n个插值点，代表曲线void setCtrlPoints(int n, double (*points)[3]);     //设置控制点void getCtrlPoints(int &n, double (*points)[3]);    //获取控制点private:inline double factorial(double n);                  //阶乘inline double C_n_k(int n, int k);                  //二项式系数,参数n为了形式上更接近二项式inline double BEZ_k_n(int n, int k, double u);      //混合函数private:int m_count;                                        //控制点数量double (*m_ctrlPoints)[3];                          //控制点坐标
};#endif // ZMBEZIER_H

#include "zmBezier.h"#include<cmath>
#include<string>zmBezier::zmBezier(){m_count = 0;m_ctrlPoints = nullptr;
}zmBezier::zmBezier(int n, double(*points)[3])
{m_count = n;m_ctrlPoints = new double[n][3];memcpy_s(m_ctrlPoints, sizeof (double)*n * 3, points, sizeof (double)*n * 3);
}zmBezier::~zmBezier()
{m_count = 0;delete [] m_ctrlPoints;
}double zmBezier::C_n_k(int n, int k)
{n = m_count - 1;return factorial(n) / (factorial(k) * factorial(n - k));
}double zmBezier::factorial(double n)
{return tgamma(n + 1);
}void zmBezier::getPoint(float u, double p[3])
{int n = m_count - 1;double x = 0, y = 0, z = 0;for(int k = 0; k <= n; k++){x += m_ctrlPoints[k][0] * BEZ_k_n(n, k, u);y += m_ctrlPoints[k][1] * BEZ_k_n(n, k, u);z += m_ctrlPoints[k][2] * BEZ_k_n(n, k, u);}p[0] = x;p[1] = y;p[2] = z;
}double zmBezier::BEZ_k_n(int n, int k, double u)
{return  C_n_k(n, k) * pow(u, k) * pow(1 - u, n - k);
}void zmBezier::getCurve(int count, double (*curve)[3])
{double point[3] = {0};for(int k = 0; k < count; k++) {getPoint(1.0 * k / (count - 1), point);curve[k][0] = point[0];curve[k][1] = point[1];curve[k][2] = point[2];}
}void zmBezier::setCtrlPoints(int n, double(*points)[3])
{delete [] m_ctrlPoints;m_count = n;m_ctrlPoints = new double[n][3];int size = sizeof (double) * n * 3;memcpy_s(m_ctrlPoints, size, points, size);
}void zmBezier::getCtrlPoints(int &n, double (*points)[3])
{n = m_count;if(m_count){int size = sizeof (double) * n * 3;memcpy_s(points, size, m_ctrlPoints, size);}
}

5. 继承QWidget，定义可显示的控制点

#ifndef MYCTRLPOINT_H
#define MYCTRLPOINT_H#include <QWidget>class myCtrlPoint : public QWidget
{Q_OBJECT
public:myCtrlPoint(QWidget *parent);QPoint getPosition();void setPostion(const QPoint &point);
protected:void paintEvent(QPaintEvent *event) override;void mousePressEvent(QMouseEvent *event) override;void mouseMoveEvent(QMouseEvent *event) override;void mouseReleaseEvent(QMouseEvent *event) override;private:QPoint m_clicked;};#endif // MYCTRLPOINT_H

主要是实现鼠标事件：

5.1 鼠标左键单击，点变成绿色

5.2 鼠标左键拖动，点在父窗口中移动

5.3 鼠标右键，从父类中删除自己


#include"myCanvas.h"
#include"myCtrlPoint.h"#include<QKeyEvent>
#include<QPainter>
#include<QMouseEvent>myCtrlPoint::myCtrlPoint(QWidget *parent): QWidget(parent)
{setFixedSize(20, 20);
}void myCtrlPoint::paintEvent(QPaintEvent *event)
{QWidget::paintEvent(event);QPainter painter(this);if(m_clicked != QPoint())  {painter.setBrush(Qt::green);}else {painter.setBrush(Qt::lightGray);}painter.drawRect(rect());
}void myCtrlPoint::mousePressEvent(QMouseEvent *event)
{if(event->button() == Qt::LeftButton){m_clicked = event->globalPos();update();}else if(event->button() == Qt::RightButton){myCanvas *canvase = (myCanvas *)parent();canvase->m_ctrlWidgets.removeOne(this);this->deleteLater();canvase->update();}}void myCtrlPoint::mouseMoveEvent(QMouseEvent *event)
{if(m_clicked == QPoint()){QWidget::mouseMoveEvent(event);}else{QPoint cur = event->globalPos();QPoint dis = cur - m_clicked;m_clicked = cur;move(mapToParent(QPoint(0, 0)) + dis);((QWidget *)parent())->update();}
}void myCtrlPoint::mouseReleaseEvent(QMouseEvent *event)
{m_clicked = QPoint();update();
}QPoint myCtrlPoint::getPosition()
{return mapToParent(rect().center());
}void myCtrlPoint::setPostion(const QPoint &point)
{QPoint target = point - rect().topLeft();move(target);}

6. 继承QWidget，实现一块画布

#ifndef MYCANVAS_H
#define MYCANVAS_H#include <QWidget>#include"zmBezier.h"class myCtrlPoint;
class myCanvas : public QWidget
{friend class myCtrlPoint;Q_OBJECT
public:explicit myCanvas(QWidget *parent = nullptr);~myCanvas();protected:void paintEvent(QPaintEvent *event) override;void mouseDoubleClickEvent(QMouseEvent *event) override;private:zmBezier m_curve;double m_points[1024][3];                   //不想paintEvent中动态分配内存QVector<myCtrlPoint *>m_ctrlWidgets;
};#endif // MYCANVAS_H

6.1 构造时随机生成4个控制点

6.2 绘制事件中绘制控制点之间的连线、绘制Bezier曲线

6.3 鼠标左键双击空白处会添加一个控制点

6.4 因为不想再绘制事件中动态分配内存，所以用了一个比较大的数组

6.5 控制点是画布的友元类，方便控制点删除自己

#include"myCanvas.h"
#include"myCtrlPoint.h"#include<QTime>
#include<QDebug>
#include<QPainter>
#include<QMouseEvent>
#include<QRandomGenerator>myCanvas::myCanvas(QWidget *parent): QWidget(parent)
{QRandomGenerator random(QTime::currentTime().second());for(int i = 0; i < 4; i++){myCtrlPoint *ctrl = new myCtrlPoint(this);m_ctrlWidgets.append(ctrl);ctrl->setPostion(QPoint(random.generateDouble() * 400, random.generateDouble() * 400));}resize(500, 500);
}myCanvas::~myCanvas()
{}void myCanvas::paintEvent(QPaintEvent *event)
{QWidget::paintEvent(event);QPainter painter(this);painter.drawText(20, 20, "1.左键拖动控制点");painter.drawText(20, 40, "2.右键删除控制点");painter.drawText(20, 60, "3.左键双击空白处添加控制点");int n = m_ctrlWidgets.count();if(n){painter.setPen(QPen(Qt::blue, 1, Qt::DotLine));for(int i = 0; i < n - 1; i++){painter.drawLine(m_ctrlWidgets[i]->getPosition(), m_ctrlWidgets[i + 1]->getPosition());}//        double (*ctrls)[3] = new double[n][3];       尽量别动态分配了,下面限制下点数if(n > 1024) {n = 1024;}for(int i = 0; i < n; i++){
//            ctrls[i][0] = m_ctrlWidgets[i]->getPosition().x();
//            ctrls[i][1] = m_ctrlWidgets[i]->getPosition().y();
//            ctrls[i][2] = 0;m_points[i][0] = m_ctrlWidgets[i]->getPosition().x();m_points[i][1] = m_ctrlWidgets[i]->getPosition().y();m_points[i][2] = 0;}m_curve.setCtrlPoints(n, m_points);
//        m_curve.setCtrlPoints(n, ctrls);
//        delete [] ctrls;int request = 100;
//        double (*points)[3] = new double[request][3];//        m_curve.getCurve(request, points);m_curve.getCurve(request, m_points);painter.setPen(QPen(Qt::green, 1));for(int i = 0; i < request - 1; i++) {painter.drawLine(m_points[i][0], m_points[i][1],m_points[i + 1][0], m_points[i + 1][1]);}//        delete [] points;}
}void myCanvas::mouseDoubleClickEvent(QMouseEvent *event)
{if(event->button() == Qt::LeftButton){QPoint point = event->pos();myCtrlPoint *ctrl = new myCtrlPoint(this);m_ctrlWidgets.append(ctrl);ctrl->setPostion(point);ctrl->show();update();}}

7.直接显示画布


#include<QApplication>#include"myCanvas.h"int main(int argc, char *argv[])
{QApplication a(argc, argv);myCanvas camvas;camvas.show();return a.exec();
}

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