圆中点算法

中心在原点，半径为 R 的圆的隐式函数方程为

$$F(x,y)=x^2+y^2?R^2=0$$

把像素上的点的坐标代入上述隐式方程

八分法画圆算法

利用坐标轴和与坐标轴夹角 45 度的直线

原理

假设圆弧起点 $x=0$， $y=R$ 精确地落在像素点上，中电算法要从 $x=0$ 顺时针绘制到 $x=y$

$$d_i=F(x_i+1,y_i?0.5)=(x_i+1{)}^2+(y_i?0.5{)}^2?R^2$$

$$y_{i+1}=\{\begin{array}{ll}{y}_i& d<0\\ y_i?1& d\ge 0\end{array}$$
进一步推导

当 $d_i<0$ 时， 下一个点的坐标是 $(x_i+2,y_i?0.5)$, 代入隐函数 $F(x_i+2,y_i?0.5)$
$$\begin{gathered} F(x_i+2,y_i?0.5)=F(x_I=1，y_i?0.5)+2x_i+3 \\ {d}_{i+1}=d_i+2x_i+3 \end{gathered}$$

同理 $d_i\ge 0$ 有
$$d_{i+1}=d_i+2(x_i+y_i)+5$$

$$d_{i+1}=\{\begin{array}{ll}{d}_i+2x_i+3& d_i<0\\ d_i+2(x_i?y_i)+5& d_i\ge 0\end{array}$$
也就是 $d<0$ 时，$d$的增量 $2x_i+3$；$d\ge 0$ 时，$d$ 的增量 $2(x_i?y_i)+5$
初始值

$$d_i=F(1,R?0.5)=1+(R?0.5{)}^2?R^2=1.25?R$$

算法

1. 圆的半径为 $R$
2. 定义圆当前点坐标 $x,y$ 定义中点误差项 $d$、 定义像素点颜色 $color$
3. 计算 $d=1.25?R$, $x=0,y=R$
4. 绘制点 $(x,y)$ 及其在八分圆中的另外 7 个对称点
5. 判断 $d$ 的符号。若 $d<0$ 则 (x, y) 更新为 $(x+1,y)$ $d$ 更新为 $d+2x+3$, 否则, (x, y) 更新为 $(x+1,y?1)$ $d$ 更新为 $d+2(x?y)+5$
6. 当 $x\le y$, 重复 步骤 4 和步骤 5

void MidPointCircle(CDC* pDC, int R) {
	int x, y;
	double d = 1.25 - R;
	x = 0; y = R;	// y 轴上方开始
	for (x = 0; x<y;x++) {
		if (d < 0) {
			d += 2.0 * x + 3;
		}
		else {
			d += 2.0 * (x-y)+ 5;
			y--;
		}
	}
}

void CirclePoint(CDC * pDC, ine x, int y) {
	COLORREF crColor = RGB(0, 0, 0);
	pDC->SetPixelV(x, y, crColor);
	pDC->SetPixelV(y, x, crColor);
	pDC->SetPixelV(y, -x, crColor);
	pDC->SetPixelV(x, -y, crColor);
	pDC->SetPixelV(-x, -y, crColor);
	pDC->SetPixelV(-y, -x, crColor);
	pDC->SetPixelV(-y, x, crColor);
	pDC->SetPixelV(-x, y, crColor);
}

void CirclePoint(QPainter* painter, int x, int y) {
    QColor color(0, 0, 0);
    painter->setPen(color);

    painter->drawPoint(x, y);
    painter->drawPoint(y, x);
    painter->drawPoint(y, -x);
    painter->drawPoint(x, -y);
    painter->drawPoint(-x, -y);
    painter->drawPoint(-y, -x);
    painter->drawPoint(-y, x);
    painter->drawPoint(-x, y);
}

void MidPointCircle(QPainter* painter, int R) {
    int x, y;
    double d = 1.25 - R;
    x = 0; y = R;    // 在y轴上方开始
    while (x < y) {
        if (d < 0) {
            d += 2.0 * x + 3;
        }
        else {
            d += 2.0 * (x - y) + 5;
            y--;
        }

        CirclePoint(painter, x, y);
    }
}

#include <QApplication>
#include <QPainter>
#include <QWidget>



void CirclePoint(QPainter* painter, int x, int y) {
    QColor color(255, 0, 0);  // 设置为红色
    painter->setPen(color);

    // 平移到正确的位置
    int centerX = 400;  // 窗口宽度的一半
    int centerY = 300;  // 窗口高度的一半

    painter->drawPoint(centerX + x, centerY + y);
    painter->drawPoint(centerX + y, centerY + x);
    painter->drawPoint(centerX + y, centerY - x);
    painter->drawPoint(centerX + x, centerY - y);
    painter->drawPoint(centerX - x, centerY - y);
    painter->drawPoint(centerX - y, centerY - x);
    painter->drawPoint(centerX - y, centerY + x);
    painter->drawPoint(centerX - x, centerY + y);
}

void MidPointCircle(QPainter* painter, int R) {
    int x = 0, y = R;
    double d = 1.25 - R;
    while (x < y) {
        if (d < 0) {
            d += 2.0 * x + 3;
        }
        else {
            d += 2.0 * (x - y) + 5;
            y--;
        }

        CirclePoint(painter, x, y);
        x++;
    }
}

class MyWidget : public QWidget {
public:
    MyWidget(QWidget* parent = nullptr) : QWidget(parent) {
        setFixedSize(800, 600);
    }

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

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

        int radius = 50;  // 圆的半径

        MidPointCircle(&painter, radius);
    }
};

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

    MyWidget widget;
    widget.show();

    return app.exec();
}