N阶贝塞尔曲线画法

Posted 2022-12-02 Leslie X徐

N阶贝塞尔曲线画法

涉及知识：

• 贝塞尔曲线
• 牛顿二项式
• 杨辉三角
• 组合数

主要逻辑代码：

/**
 * @brief createNBezierCurve 生成N阶贝塞尔曲线点
 * @param src 源贝塞尔控制点
 * @param dest 目的贝塞尔曲线点
 * @param precision 生成精度
 */
static void createNBezierCurve(const QVector<QPointF> &src, QVector<QPointF> &dest, qreal precision)


    int size = src.size();

    //系数数组
    QVector<qreal> coff(size,0);

    //准备二维数组a[n][m]表示n阶杨辉三角,使用new创建动态二维数组
    int** a = new int*[size];
    for(int i=0;i<size;++i)
        a[i] = new int[size];
    
    //求系数，个数对应控制点数。使用杨辉三角+组合数的方法
        for(int i=0;i<size;++i)
        //首尾为 1
            a[i][0]=1;
            a[i][i]=1;
        
        //利用组合数原理 创建杨辉三角 递推 动态规划法
        for(int i=1;i<size;++i)
            for(int j=1;j<i;++j)
                a[i][j] = a[i-1][j-1] + a[i-1][j];

    

    //总循环，时间变化 p=t*p1+(1-t)*p0
    for(qreal t1=0; t1<1; t1+=precision )

        //求每个系数
        qreal t2 = 1 - t1 ;
        int n = size - 1;
        //利用到牛顿二项式展开 (a2+2ab+b2)
        coff[0] = pow(t2,n); coff[n] = pow(t1,n);
        for(int i=1;i<size-1;++i)
            coff[i] = pow(t2,n-i) * pow(t1,i) * a[n][i];
        

        //求曲线上点
        QPointF ret(0,0);
        for(int i=0;i<size;++i)
            ret += src[i] * coff[i];
        

        //把所得的点放入目标数组，最后用QPainterPath连起来
        dest.append(ret);
    

    //释放二维数组
    for(int i = 0; i < size; i ++)
        delete [] a[i];
    delete [] a;

绘制

头文件

#ifndef DRAWBEZIER_H
#define DRAWBEZIER_H

#include <QWidget>
#include <QGraphicsItem>

QT_BEGIN_NAMESPACE
namespace Ui  class DrawBezier; 
QT_END_NAMESPACE

class DrawBezier : public QWidget

    Q_OBJECT

public:
    DrawBezier(QWidget *parent = nullptr);
    ~DrawBezier();
    void draw();

private slots:
    void on_pushButton_clicked();

    void on_pushButton_2_clicked();

    void on_pushButton_3_clicked();

private:
    Ui::DrawBezier *ui;
    QList<QGraphicsItem *>_ctlPts;
    QVector<QPointF> _src;
    QVector<QPointF> _dest;
    QGraphicsItem *_curvenullptr;
    QGraphicsItem *_Linenullptr;
    QGraphicsScene* scene;

    bool isCreatingfalse;
    bool isAlteringfalse;

;
#endif // DRAWBEZIER_H

源文件

#include "DrawBezier.h"
#include "ui_DrawBezier.h"
#include <QDebug>


/**
 * @brief createNBezierCurve 生成N阶贝塞尔曲线点
 * @param src 源贝塞尔控制点
 * @param dest 目的贝塞尔曲线点
 * @param precision 生成精度
 */
static void createNBezierCurve(const QVector<QPointF> &src, QVector<QPointF> &dest, qreal precision)


DrawBezier::DrawBezier(QWidget *parent)
    : QWidget(parent)
    , ui(new Ui::DrawBezier)

    ui->setupUi(this);
    this->resize(600,600);

    //创建场景,必须创建
    scene = new QGraphicsScene(300,300,600,600);
    ui->graphicsView->setScene(scene);
    ui->graphicsView->setDragMode(QGraphicsView::RubberBandDrag);


    //=========================================================

    connect(ui->graphicsView,&QWGraphicsView::mouseClicked,this,[this](QPoint pos)
        if(isCreating==false)return;
        QPointF Scenepos = ui->graphicsView->mapToScene(pos);
        QGraphicsItem* item = scene->addRect(QRectF(Scenepos,QSize(10,10)),QPen(QColor(Qt::blue)),QBrush(QColor(Qt::blue)));
        item->setFlags(QGraphicsItem::ItemIsSelectable|
                       QGraphicsItem::ItemIsFocusable|
                       QGraphicsItem::ItemIsMovable);
        _ctlPts.append(item);
        ui->label->setText(u8"控制点个数："+QString::number(_ctlPts.size()));
ui->graphicsView->viewport()->update();
        draw();
    );

    connect(ui->graphicsView,&QWGraphicsView::mouseClickedFinished,this,[this]()
        isCreating = false;
    );

    connect(ui->graphicsView,&QWGraphicsView::mouseReleased,this,[this]()
        if(isCreating==true)return;
        draw();
    );


DrawBezier::~DrawBezier()

    for(auto i : _ctlPts)delete i;
    if(_curve)delete _curve;
    if(_Line)delete _Line;_Line=nullptr;
    delete scene;
    delete ui;


void DrawBezier::draw()

    if(_ctlPts.size()==0)return;
    _src.clear();
    _dest.clear();

    //利用生成的控制点生成需要的控制点位置，因为在拖动控制点时位置会刷新
    // item在场景中的位置需要使用 scenePos+boundingRect().center()来获取
    for(auto i : _ctlPts)
        _src.append(i->scenePos() + i->boundingRect().center());
    

    createNBezierCurve(_src,_dest,0.01);

    QPainterPath curvepath;
    curvepath.moveTo(_dest[0]);
    for(int i=1 ;i<_dest.size() ;++i)
        curvepath.lineTo(_dest[i]);
    
    if(_curve)delete _curve;_curve=nullptr;
    _curve = scene->addPath(curvepath,QPen(QBrush(Qt::green),3,Qt::SolidLine));

    QPainterPath linepath;
    linepath.moveTo(_src[0]);
    for(int i=1 ;i<_src.size() ;++i)
        linepath.lineTo(_src[i]);
    
    if(_Line)delete _Line;_Line=nullptr;
    _Line = scene->addPath(linepath,QPen(QBrush(Qt::yellow),3,Qt::DotLine));





void DrawBezier::on_pushButton_clicked()

    isCreating = true;



void DrawBezier::on_pushButton_2_clicked()

    for(auto i : _ctlPts)delete i;
    if(_curve)delete _curve;_curve=nullptr;
    if(_Line)delete _Line;_Line=nullptr;
    _ctlPts.clear();
    isCreating = false;



void DrawBezier::on_pushButton_3_clicked()

    for(auto item : scene->selectedItems())
        _ctlPts.removeAll(item);
        delete item;
    
    draw();