三点确定一个圆的计算方法

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三点确定一个圆的计算方法

最近在写的一个软件需要根据三个坐标点来计算一个圆。因此花了点时间推导了相关的公式。这个推导不算太难，放在这里主要是做个备忘。

我们设一个圆的圆心坐标为 (x 0 ,y) (x0,y)(x_0, y)，半径为 r rr。那么这个圆的方程可以写为：
(x−x 0 ) 2 +(y−y 0 ) 2 =r 2  (x−x0)2+(y−y0)2=r2
(x – x_0)^2 + (y – y_0)^2 = r^2

⎧ ⎩ ⎨ ⎪ ⎪ (x 1 −x 0 ) 2 +(y 1 −y 0 ) 2 =r 2 (x 2 −x 0 ) 2 +(y 2 −y 0 ) 2 =r 2 (x 3 −x 0 ) 2 +(y 3 −y 0 ) 2 =r 2  (1)(2)(3)  {(x1−x0)2+(y1−y0)2=r2(1)(x2−x0)2+(y2−y0)2=r2(2)(x3−x0)2+(y3−y0)2=r2(3)
\begin{cases}
(x_1 – x_0)^2 + (y_1-y_0)^2 = r^2 & (1)\\
(x_2 – x_0)^2 + (y_2-y_0)^2 = r^2 & (2)\\
(x_3 – x_0)^2 + (y_3-y_0)^2 = r^2 & (3)
\end{cases}

公式（1）（2）相减，（1）（3）相减之后经过化简可以得到：
(x 1 −x 2 )x 0 +(y 1 −y 2 )y 0 =(x 2 1 −x 2 2 )−(y 2 2 −y 2 1 )2 (x 1 −x 3 )x 0 +(y 1 −y 3 )y 0 =(x 2 1 −x 2 3 )−(y 2 3 −y 2 1 )2  (x1−x2)x0+(y1−y2)y0=(x12−x22)−(y22−y12)2(x1−x3)x0+(y1−y3)y0=(x12−x32)−(y32−y12)2
(x_1 – x_2) x_0 + (y_1- y_2) y_0 = \frac{(x_1^2-x_2^2)-(y_2^2-y_1^2)}{2}\\
(x_1 – x_3) x_0 + (y_1- y_3) y_0 = \frac{(x_1^2-x_3^2)-(y_3^2-y_1^2)}{2}\\

x 0 ,y 0  x0,y0x_0,y_0有唯一解的条件是系数行列式不为 0 00：
∣ ∣ ∣ (x 1 −x 2 )(x 1 −x 3 ) (y 1 −y 2 )(y 1 −y 3 ) ∣ ∣ ∣ ≠0 |(x1−x2)(y1−y2)(x1−x3)(y1−y3)|≠0
\begin{vmatrix}
(x_1 – x_2) & (y_1- y_2) \\
(x_1 – x_3) & (y_1- y_3)
\end{vmatrix} \neq 0

简单变变型也就是：
x 1 −x 2 y 1 −y 2  ≠x 1 −x 3 y 1 −y 3   x1−x2y1−y2≠x1−x3y1−y3
\frac{x_1 – x_2}{y_1-y_2} \neq \frac{x_1 – x_3}{y_1-y_3}

设:
a=x 1 −x 2 b=y 1 −y 2 c=x 1 −x 3 d=y 1 −y 3 e=(x 2 1 −x 2 2 )−(y 2 2 −y 2 1 )2 f=(x 2 1 −x 2 3 )−(y 2 3 −y 2 1 )2  a=x1−x2b=y1−y2c=x1−x3d=y1−y3e=(x12−x22)−(y22−y12)2f=(x12−x32)−(y32−y12)2a = x_1-x_2\\
b = y_1-y_2\\
c = x_1-x_3\\
d = y_1-y_3\\
e = \frac{(x_1^2-x_2^2)-(y_2^2-y_1^2)}{2}\\
f = \frac{(x_1^2-x_3^2)-(y_3^2-y_1^2)}{2}

那么 :
x 0 =−de−bfbc−ad y 0 =−af−cebc−ad  x0=−de−bfbc−ady0=−af−cebc−ad
x_0 = -\frac{d e-b f}{b c-a d}\\
y_0 = -\frac{a f-c e}{b c-a d}

下面是个 C++ 代码（用到了Qt 的 QPointF 类型）：

    #include <math.h>
    #include <limits>
    #include <QPoint>
    #include <QDebug>
    QPointF tcircle(QPointF pt1, QPointF pt2, QPointF pt3, double &radius)
    {

        double x1 = pt1.x(), x2 = pt2.x(), x3 = pt3.x();
        double y1 = pt1.y(), y2 = pt2.y(), y3 = pt3.y();
        double a = x1 – x2;
        double b = y1 – y2;
        double c = x1 – x3;
        double d = y1 – y3;
        double e = ((x1 * x1 – x2 * x2) + (y1 * y1 – y2 * y2)) / 2.0;
        double f = ((x1 * x1 – x3 * x3) + (y1 * y1 – y3 * y3)) / 2.0;
        double det = b * c – a * d;
        if( fabs(det) < 1e-5)
        {

            radius = -1;
            return QPointF(0,0);
        }

        double x0 = -(d * e – b * f) / det;
        double y0 = -(a * f – c * e) / det;
        radius = hypot(x1 – x0, y1 – y0);
        return QPointF(x0, y0);
    }