QT绘制B样条曲线

² 贝塞尔曲线

贝塞尔曲线是通过一组多边折线的各顶点来定义。在各顶点中，曲线经过第一点和最后一点，其余各点则定义曲线的导数、阶次和形状。第一条和最后一条则表示曲线起点和终点的切线方向。

² B样条曲线

针对贝塞尔曲线存在的一些缺点，数学家们提出了B样条方法，在保留贝塞尔全部优点的同时，克服可贝塞尔方法的弱点。

1) 二次B样条曲线

2) 三次B样条曲线

QT中的QPainter提供了绘制贝塞尔曲线的相关API：

void QPainterPath::quadTo(const QPointF &c, const QPointF &endPoint)

void QPainterPath::cubicTo(const QPointF &c1, const QPointF &c2, const QPointF &endPoint)

void QPainter::drawPath(const QPainterPath &path)

   Widget.h  
 

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/////////////////////////////////////////////////////////////
/// @file   Widget.h
/// @brief  绘制B样条曲线Widget类
///
/// 通过鼠标点击来绘制B样条曲线
/// @author Michael Joessy
/// @date   2019-07-02
/////////////////////////////////////////////////////////////
#ifndef WIDGET_H
#define WIDGET_H

#include <QWidget>
#include <QVector>

class Widget : public QWidget
{
    Q_OBJECT

public :
    Widget(QWidget *parent = nullptr);
    ~Widget();

protected :
     virtual void mousePressEvent(QMouseEvent *event);
     virtual void mouseReleaseEvent(QMouseEvent *event);
     virtual void mouseDoubleClickEvent(QMouseEvent *event);
     virtual void mouseMoveEvent(QMouseEvent *event);
     virtual void paintEvent(QPaintEvent *event);

private :
     void drawSpline();
    qreal N( int k, int i, qreal u);
    qreal N1( int i, qreal u);
    qreal N2( int i, qreal u);
    qreal N3( int i, qreal u);

private :
    QVector<QPointF>    m_ctrlPoints;        // 控制点
    QVector<QPointF>    m_curvePoints;       // 曲线上的点
};

#endif // WIDGET_H

   Widget.pp  
 

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#include "Widget.h"
#include <QMouseEvent>
#include <QPainter>
#include <cmath>

Widget::Widget(QWidget *parent)
    : QWidget(parent)
{
}

Widget::~Widget()
{

}

void Widget::mousePressEvent(QMouseEvent *event)
{
     // 单击鼠标左键获取控制点
     if (event->buttons() == Qt::LeftButton){
        m_ctrlPoints.push_back(event->pos());
    }
     // 单击鼠标右键清空控制点
     else if (event->buttons() == Qt::RightButton) {
        m_ctrlPoints.clear();
    }
    update();
}

void Widget::mouseReleaseEvent(QMouseEvent *event)
{

}

void Widget::mouseDoubleClickEvent(QMouseEvent *event)
{

}

void Widget::mouseMoveEvent(QMouseEvent *event)
{

}

void Widget::paintEvent(QPaintEvent *event)
{
    drawSpline();
}

void Widget::drawSpline()
{
    QPainter painter( this );
     int currentK = 3 ;        // 阶数
    m_curvePoints.clear();
     for (qreal u = currentK; u < m_ctrlPoints.size(); u += 0 . 01 ){
        QPointF pt( 0 . 0 , 0 . 0 );
         for ( int i = 0 ; i < m_ctrlPoints.size(); ++i){
            QPointF pts = m_ctrlPoints[i];
            pts *= N(currentK, i, u);
            pt += pts;
        }
        m_curvePoints.push_back(pt);
    }

     // draw control points
    QPen ctrlPen1(QColor( 0 , 0 , 255 ));
    ctrlPen1.setWidth( 5 );
    painter.setPen(ctrlPen1);
     for ( int i = 0 ; i < m_ctrlPoints.size(); ++i){
        painter.drawPoint(m_ctrlPoints[i]);
    }
     // draw control lines
    QPen ctrlPen2(QColor( 255 , 0 , 0 ));
    ctrlPen2.setWidth( 1 );
    ctrlPen2.setStyle(Qt::DashDotDotLine);
    painter.setPen(ctrlPen2);
     for ( int i = 0 ; i < m_ctrlPoints.size() - 1 ; ++i){
        painter.drawLine(m_ctrlPoints[i], m_ctrlPoints[i + 1 ]);
    }
     // draw spline curve
    QPen curvePen(QColor( 0 , 0 , 0 ));
    curvePen.setWidth( 2 );
    painter.setPen(curvePen);
     for ( int i = 0 ; i < m_curvePoints.size() - 1 ; ++i){
        painter.drawLine(m_curvePoints[i], m_curvePoints[i + 1 ]);
    }
}

qreal Widget::N( int k, int i, qreal u)
{
     switch (k) {
     case 1 :
         return N1(i, u);
     case 2 :
         return   N2(i, u);
     case 3 :
         return   N3(i, u);
     default :
         break ;
    }
}

qreal Widget::N1( int i, qreal u)
{
    qreal t = u - i;
     if ( 0 <= t && t < 1 ){
         return t;
    }
     if ( 1 <= t && t < 2 ){
         return 2 - t;
    }
     return 0 ;
}

qreal Widget::N2( int i, qreal u)
{
    qreal t = u - i;
     if ( 0 <= t && t < 1 ){
         return 0 . 5 * t * t;
    }
     if ( 1 <= t && t < 2 ){
         return 3 * t - t * t - 1 . 5 ;
    }
     if ( 2 <= t && t < 3 ){
         return 0 . 5 * pow( 3 - t, 2 );
    }
     return 0 ;
}

qreal Widget::N3( int i, qreal u)
{
    qreal t = u - i;
    qreal a = 1 . 0 / 6 . 0 ;
     if ( 0 <= t && t < 1 ){
         return a * t * t * t;
    }
     if ( 1 <= t && t < 2 ){
         return a * (- 3 * pow(t - 1 , 3 ) + 3 * pow(t - 1 , 2 ) + 3 * (t - 1 ) + 1 );
    }
     if ( 2 <= t && t < 3 ){
         return a * ( 3 * pow(t - 2 , 3 ) - 6 * pow(t - 2 , 2 ) + 4 );
    }
     if ( 3 <= t && t < 4 ){
         return a * pow( 4 - t, 3 );
    }
     return 0 ;
}

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