圆中点算法

2023-12-25 11:30:27

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 F(x, y) = x^2 + y^2 - R^2 = 0 F(x,y)=x2+y2?R2=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 x=0 x=0 y = R y= R y=R 精确地落在像素点上,中电算法要从 x = 0 x=0 x=0 顺时针绘制到 x = y x=y 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 d_i = F(x_i + 1, y_i - 0.5) = (x_i + 1)^2 + (y_i -0.5)^2 - R^2 di?=F(xi?+1,yi??0.5)=(xi?+1)2+(yi??0.5)2?R2
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y i + 1 = { y i d < 0 y i ? 1 d ≥ 0 y_{i+1} = \begin{cases} y_i & d < 0\\ y_i - 1 & d \geq 0 \end{cases} yi+1?={yi?yi??1?d<0d0?
进一步推导

d i < 0 d _i < 0 di?<0 时, 下一个点的坐标是 ( x i + 2 , y i ? 0.5 ) (x_i +2, y_i -0.5) (xi?+2,yi??0.5), 代入隐函数 F ( x i + 2 , y i ? 0.5 ) F(x_i+2 , y_i -0.5) F(xi?+2,yi??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 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 F(xi?+2,yi??0.5)=F(xI?=1yi??0.5)+2xi?+3di+1?=di?+2xi?+3

同理 d i ≥ 0 d_i \geq 0 di?0
d i + 1 = d i + 2 ( x i + y i ) + 5 d_{i+1} = d_{i} + 2 (x_i + y_i) + 5 di+1?=di?+2(xi?+yi?)+5

d i + 1 = { d i + 2 x i + 3 d i < 0 d i + 2 ( x i ? y i ) + 5 d i ≥ 0 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} di+1?={di?+2xi?+3di?+2(xi??yi?)+5?di?<0di?0?
也就是 d < 0 d< 0 d<0 时, d d d的增量 2 x i + 3 2x_i + 3 2xi?+3 d ≥ 0 d\geq 0 d0 时, d d d 的增量 2 ( x i ? y i ) + 5 2(x_i -y_i) + 5 2(xi??yi?)+5
初始值

d i = F ( 1 , R ? 0.5 ) = 1 + ( R ? 0.5 ) 2 ? R 2 = 1.25 ? R d_i = F(1, R-0.5) = 1+ (R-0.5)^2 - R^2 = 1.25 -R di?=F(1,R?0.5)=1+(R?0.5)2?R2=1.25?R

算法

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

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文章来源:https://blog.csdn.net/weixin_43862398/article/details/135191824
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