圆中点算法

中心在原点，半径为 R 的圆的隐式函数方程为

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

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把像素上的点的坐标代入上述隐式方程

八分法画圆算法

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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 $x = y$

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$d_i = F(x_i + 1, y_i - 0.5) = (x_i + 1)^2 + (y_i -0.5)^2 - R^2$

$y_{i+1} = \begin{cases} y_i & d < 0\\ y_i - 1 & d \geq 0 \end{cases}$
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 $F(x_i+2 , y_i -0.5)$
$F(x_i+2 , y_i -0.5) = F(x_I =1， y_i-0.5) + 2 x_i +3\\ d_{i+1} = d_{i} + 2 x_i + 3$

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

$d_{i+1} = \begin{cases} d_i + 2 x_i + 3 & d_ i < 0\\ d_i + 2 (x_i - y_i) + 5 & d_i \geq 0 \end{cases}$
也就是 $d < 0$ 时， $d$ 的增量 $2x_i + 3$ ； $d\geq 0$ 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

圆的半径为 $R$
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 $co l or$
计算 $d = 1.25 - R$ , $x = 0, y = R$
绘制点 $(x, y)$ 及其在八分圆中的另外 7 个对称点
判断 $d$ 的符号。若 $d < 0$ 则 (x, y) 更新为 $(x + 1, y)$ $d$ 更新为 $d + 2 x + 3$ , 否则, (x, y) 更新为 $(x + 1, y - 1)$ $d$ 更新为 $d + 2 (x - y) + 5$
当 $x\leq y$ , 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();
}