Bezier 曲线拼接圆

Bezier 曲线拼接

为了描述复杂物体的轮廓曲线，经常需要将多段 Bezier 曲线拼接起来，并在结合处满足一定的连续性条件。

假设两段 三次 Bezier 曲线分别为 $p(t)$ 和 $q(t)$，其控制多边形的顶点分别为 $P_0$，$P_1$ ，$P_2$， $P_3$ 和 $Q_0$，$Q_1$ ，$Q_2$， $Q_3$

两段三次 Bezier 曲线 达到 $G^0$ 连续的条件是， $P_3=Q_0$， 在这一点处有相同的切线方向，$G^1$ 连续的条件是，$P_2$, $p_3或(Q_1)$ 和 $Q_2$ 三点共线。

Bezier 曲线拼接圆

使用一段三次 Bezier 曲线绘制拼接圆

参考 《孔令德 计算几何算法与实现》

#include <QWidget>
#include <QApplication>
#include <QPainter>
#include <QPointF>
#include <vector>
#include <QPainterPath>

#include <iostream>

const double m = 0.5523;  // 魔术常数

std::vector<QPointF> getControlPoints(float radius) {
    std::vector<QPointF> controlPoints = {
        QPointF(radius, 0.0),
        QPointF(radius, m * radius),
        QPointF(m * radius, radius),
        QPointF(0.0, radius),
        QPointF(-m * radius, radius),
        QPointF(-radius, m * radius),
        QPointF(-radius, 0.0),
        QPointF(-radius, -m * radius),
        QPointF(-m * radius, -radius),
        QPointF(0.0, -radius),
        QPointF(m * radius, -radius),
        QPointF(radius, -m * radius)
    };

    return controlPoints;

}


void drawControlPoints(QPainter* painter, const QPointF& center, const std::vector<QPointF>& controlPoints, const QColor& circleColor, int radius = 5)
{
    // 设置空心圆圈的颜色
    painter->setPen(circleColor);
    painter->setBrush(Qt::NoBrush);  // 空心圆圈
    // 计算控制点的绝对坐标

    for (int i=0; i<controlPoints.size(); i++)
    {
        QPointF point = center + controlPoints[i];
        painter->drawEllipse(point, radius, radius);

    }
}


void drawControlPolygon(QPainter* painter, const QPointF& center, const std::vector<QPointF>& controlPoints, const QColor& lineColor) {
    // 设置线条颜色
    QPen pen(lineColor);
    painter->setPen(pen);

    // 绘制控制点之间的直线
    for (int i = 0; i < controlPoints.size(); ++i) {
        QPointF startPoint = center + controlPoints[i];
        QPointF endPoint = center + controlPoints[(i + 1) % controlPoints.size()];
        painter->drawLine(startPoint, endPoint);
    }
}

QPointF calculateCubicBezierPoint(float t, const QPointF& P0, const QPointF& P1, const QPointF& P2, const QPointF& P3)
{
    float u = 1 - t;
    float b0 = u * u * u;
    float b1 = 3 * u * u * t;
    float b2 = 3 * u * t * t;
    float b3 = t * t * t;
    return b0 * P0 + b1 * P1 + b2 * P2 + b3 * P3;
}

void CubicBezierCurve(QPainter* painter, const QPointF& P0, const QPointF& P1, const QPointF& P2, const QPointF& P3)
{
    float tStep = 0.01f;
    QPainterPath bezierPath;
    QColor curveColor(255, 0, 0);  // 红色

    painter->setPen(curveColor);
    bezierPath.moveTo(P0);

    for (float t = tStep; t <= 1.0f; t += tStep)
    {
        QPointF P = calculateCubicBezierPoint(t, P0, P1, P2, P3);
        bezierPath.lineTo(P);
    }

    painter->drawPath(bezierPath);
}




class MyWidget : public QWidget
{
public:
    MyWidget(QWidget* parent = nullptr) : QWidget(parent) {
        setFixedSize(1000, 800);
    }

protected:
    void paintEvent(QPaintEvent* event) override {
        Q_UNUSED(event);

        QPainter painter(this);
        painter.setRenderHint(QPainter::Antialiasing, true);

        float radius = 200.0;
        QPointF center(500, 400);  // 圆心坐标

        std::vector<QPointF> controlPoints = getControlPoints(radius);

        for (int i = 0; i < controlPoints.size(); i += 3) {
            CubicBezierCurve(&painter, center + controlPoints[i], center + controlPoints[i + 1], center + controlPoints[i + 2], center + controlPoints[(i + 3) % controlPoints.size()]);
        }
        QColor lineColor(0, 0, 255);

        // 调用绘制控制多边形的函数，并传入线条颜色
        drawControlPolygon(&painter, center, controlPoints, lineColor);

        QColor circleColor(0, 255, 0);
        drawControlPoints(&painter, center, controlPoints, circleColor,6);
    }

};

int main(int argc, char* argv[]) {
    QApplication app(argc, argv);

    MyWidget widget;
    widget.show();

    return app.exec();
}