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;
}