MATLAB实现利用三个不共线的点绘制圆（包括圆心和半径的求解）

通过三个不共线的平面点绘制圆形/MATLAB实现

1: 三个不共线的点求解圆心和半径

假设三个不共线的点的坐标分别为P1(x1,y1)、P2(x2,y2)P1(x1,y1)、P2(x2,y2)P1(x1,y1)、P2(x2,y2)和P3(x3,y3)P3(x3,y3)P3(x3,y3)，设过三个不共线点的圆的圆心和半径分别为(x,y)和R(x,y)和R(x,y)和R，根据圆的公式可以得到：{(x−x1)2+(y−y1)2=R2(x−x2)2+(y−y2)2=R2(x−x3)2+(y−y3)2=R2\begin{cases}(x-x1)^2+(y-y1)^2=R^2\\(x-x2)^2+(y-y2)^2=R^2\\(x-x3)^2+(y-y3)^2=R^2\end{cases} ⎩⎪⎨⎪⎧(x−x1)2+(y−y1)2=R2(x−x2)2+(y−y2)2=R2(x−x3)2+(y−y3)2=R2将上述公式展开移项后得到：{x2+y2+x12+y12=2x1x+2y1y+R2x2+y2+x22+y22=2x2x+2y2y+R2x2+y2+x32+y32=2x3x+2y3y+R2\begin{cases}x^2+y^2+x1^2+y1^2=2x1x+2y1y+R^2\\x^2+y^2+x2^2+y2^2=2x2x+2y2y+R^2\\x^2+y^2+x3^2+y3^2=2x3x+2y3y+R^2\end{cases} ⎩⎪⎨⎪⎧x2+y2+x12+y12=2x1x+2y1y+R2x2+y2+x22+y22=2x2x+2y2y+R2x2+y2+x32+y32=2x3x+2y3y+R2通过对上述公式合并整理可以得到：
{(x2−x1)x+(y2−y1)y=0.5(x22+y22−x12−y12)(x3−x1)x+(y3−y1)y=0.5(x32+y32−x12−y12)(x3−x2)x+(y3−y2)y=0.5(x32+y32−x22−y22)\begin{cases}(x2-x1)x+(y2-y1)y=0.5(x2^2+y2^2-x1^2-y1^2)\\(x3-x1)x+(y3-y1)y=0.5(x3^2+y3^2-x1^2-y1^2)\\(x3-x2)x+(y3-y2)y=0.5(x3^2+y3^2-x2^2-y2^2)\\\end{cases} ⎩⎪⎨⎪⎧(x2−x1)x+(y2−y1)y=0.5(x22+y22−x12−y12)(x3−x1)x+(y3−y1)y=0.5(x32+y32−x12−y12)(x3−x2)x+(y3−y2)y=0.5(x32+y32−x22−y22)将上述方程构成矩阵形式，既有：Aθ=BA \theta = BAθ=B其中A=[x2−x1y2−y1x3−x1y3−y1x3−x2y3−y2],B=0.5[x22+y22−x12−y12x32+y32−x12−y12x32+y32−x22−y22]A=\begin{bmatrix} x2-x1 & y2-y1 \\ x3-x1 & y3-y1 \\ x3-x2 & y3-y2 \end{bmatrix},B=0.5\begin{bmatrix} x2^2+y2^2-x1^2-y1^2 \\ x3^2+y3^2-x1^2-y1^2 \\ x3^2+y3^2-x2^2-y2^2 \end{bmatrix}A=⎣⎡x2−x1x3−x1x3−x2y2−y1y3−y1y3−y2⎦⎤,B=0.5⎣⎡x22+y22−x12−y12x32+y32−x12−y12x32+y32−x22−y22⎦⎤
由于上述方程为线性方程，也不存在误差，利用最小二乘求得圆心的坐标为：θ=(ATA)−1ATB\theta =(A^{T}A) ^{-1}A^{T}Bθ=(ATA)−1ATB
那么圆心的坐标有：x=θ(1),y=θ(2)x=\theta(1),y=\theta(2)x=θ(1),y=θ(2)
利用得到的圆心可以直接求得圆的半径为：R=(x−x1)2+(y−y1)2R=\sqrt{(x-x1)^2+(y-y1)^2}R=(x−x1)2+(y−y1)2

2: 得到圆的边的坐标

当得到过三点圆的圆心和半径后，可以利用圆的参数方程得到圆的边的坐标，既有：{x=x0+Rcosαy=y0+Rsinα，其中α∈[0,2π]\begin{cases} x=x0+Rcos{\alpha}\\ y=y0+Rsin{\alpha}\end{cases} ，其中 \alpha \in{[0,2\pi]}{x=x0+Rcosαy=y0+Rsinα，其中α∈[0,2π]其中，xxx为圆边上某一点的横坐标，yyy为圆边上某一点的纵坐标，α\alphaα为圆边上某一点对应的角度，x0x0x0和y0y0y0为该圆的圆心，RRR为该圆的半径。

3: 代码实现-MATLAB

function Result = ThreePoint2Circle(P1, P2, P3)
%% 求圆心和半径
x1 = P1(1);    x2 = P2(1);    x3 = P3(1);
y1 = P1(2);    y2 = P2(2);    y3 = P3(2);
z1 = x2^2 + y2^2 - x1^2 - y1^2;
z2 = x3^2 + y3^2 - x1^2 - y1^2;
z3 = x3^2 + y3^2 - x2^2 - y2^2;
A = [(x2-x1), (y2-y1); (x3-x1), (y3-y1); (x3-x2), (y3-y2)];
B = 0.5*[z1;  z2;  z3];
P0 = (A'*A)\A'*B;
R1 = sqrt( (P0(1) - P1(1))^2 + (P0(2) - P1(2))^2 );
R2 = sqrt( (P0(1) - P2(1))^2 + (P0(2) - P2(2))^2 );
R3 = sqrt( (P0(1) - P3(1))^2 + (P0(2) - P3(2))^2 );
R = (R1 + R2 + R3)/3;
%% 绘制圆
theta = (0:pi/360:2*pi)';
Result = zeros(size(theta,1),4);
for i = 1: size(theta,1)Result(i,1) = i;Result(i,2) = theta(i);Result(i,3) = P0(1) + R*cos(theta(i));Result(i,4) = P0(2) + R*sin(theta(i));
end
figure();plot(Result(:,3),Result(:,4),'b-');hold on;
grid on; xlabel('x');ylabel('y'); axis equal;
end

输出的结果为0.5°间隔，其中第1列为点号，第2列为角度，第3列为横坐标，第4列为纵坐标。

4: 代码测试

(1)过(1,1)、(3,2)、(5,1)(1,1)、(3,2)、(5,1)(1,1)、(3,2)、(5,1)画圆

(2)过(1.2,1.2)、(8,0.5)、(13.8,1.2)(1.2,1.2)、(8,0.5)、(13.8,1.2)(1.2,1.2)、(8,0.5)、(13.8,1.2)画圆

结束！