一、2D圆弧拟合

1、不经过给定起点与终点

??平面圆的一般方程为:
$x^2 + y^2 + ax + by +c = 0\tag 1$
??其中， $a,b,c\in R$ 。
??式(1)配方，可以得到:
$a/2)^2 + (x + b/2)^2 = a^2/4 + b^2/4 - c \tag 2$

??对于给定的一系列二维数据 $x_i,y_i),i=0,...,n$ ，根据式(1)可以列出 $n + 1$ 个线性方程，然后可以采用最小二乘法求解，非常简单有效。

2、精确经过给定起点与终点

??有时候我们需要约束圆弧精确经过给定起点与终点，设起点坐标为 $x_0,y_0)$ ，终点坐标为 $x_n,y_n)$ ，则有约束等式:
$\begin{cases} x_0^2 + y_0^2 + ax_0 + by_0 +c = 0 \\ x_n^2 + y_n^2 + ax_n + by_n +c = 0 \\ \tag 3 \end{cases}$
??(1)若起点与终点坐标重合，则式(3)退化为一个约束等式，将参数 $c$ 的表达式代入到式(1)，利用最小二乘法求解参数 $a, b$ ，再计算参数 $c$ 。
??(2)若 $x_0\ne x_n$ ，根据式(3)解出参数 $a, c$ 的表达式，然后代入到式(1)，利用最小二乘法求解参数 $b$ ，再计算参数 $a, c$ 。
??(3)若 $y_0\ne y_n$ ，根据式(3)解出参数 $b, c$ 的表达式，然后代入到式(1)，利用最小二乘法求解参数 $a$ ，再计算参数 $b, c$ 。

??circle_fitting_2D.m

function [ center, R, fittingError ] = circle_fitting_2D( points, pointCount, crossStartAndEndPointFlag )
%{
Function: circle_fitting_2D
Description: 2D圆弧拟合
Input: 二维点points, 点个数pointCount, 是否经过起点/终点标志位crossStartAndEndPointFlag
Output: 圆心center, 半径R, 拟合误差fittingError
Author: Marc Pony(marc_pony@163.com)
圆的方程:x^2 + y^2 + a*x + b*y +c = 0  -> (x + a/2)^2 + (x + b/2)^2 = a^2/4 + b^2/4 - c
%}
if crossStartAndEndPointFlag == 0
    x = points(:, 1);
    y = points(:, 2);
    A = [x, y, ones(size(x))];
    B = -x.^2 - y.^2;
    temp = A \ B;
    
    a = temp(1);
    b = temp(2);
    c = temp(3);
else
    x0 = points(1, 1);
    y0 = points(1, 2);
    xn = points(pointCount, 1);
    yn = points(pointCount, 2);
    x = points(2 : pointCount - 1, 1);
    y = points(2 : pointCount - 1, 2);
    EPS = 1.0e-4;
    
    if abs(x0 - xn) < EPS && abs(y0 - yn) < EPS %起点与终点重合
        A = [x - x0, y - y0];
        B = x0^2 + y0^2 - x.^2 - y.^2;
        temp = A \ B;
        a = temp(1);
        b = temp(2);
        c = -x0^2 - y0^2 - a * x0 - b * y0;
    else
        if abs(x0 - xn) > abs(y0 - yn)
            A = x * (yn - y0) / (x0 - xn) + y - (x0 * yn - xn * y0) / (x0 - xn);
            B = -x.^2 - y.^2 - x * (xn^2 + yn^2 - x0^2 - y0^2) / (x0 - xn) + ((xn^2 + yn^2) * x0 - (x0^2 + y0^2) * xn) / (x0 - xn);
            b = A \ B;
            P = -x0^2 - y0^2 - b * y0;
            Q = -xn^2 - yn^2 - b * yn;
            a = (P - Q) / (x0 - xn);
            c = -(P * xn - Q * x0) / (x0 - xn);
        else
            A = x * (xn - x0) / (y0 - yn) + y - (y0 * xn - yn * x0) / (y0 - yn);
            B = -x.^2 - y.^2 - x * (yn^2 + xn^2 - y0^2 - x0^2) / (y0 - yn) + ((yn^2 + xn^2) * y0 - (y0^2 + x0^2) * yn) / (y0 - yn);
            a = A \ B;
            P = -y0^2 - x0^2 - a * x0;
            Q = -yn^2 - xn^2 - a * xn;
            b = (P - Q) / (y0 - yn);
            c = -(P * yn - Q * y0) / (y0 - yn);
        end
    end
end

R = sqrt(a^2 / 4 + b^2 / 4 - c);
center(1) = -a / 2;
center(2) = -b / 2;
fittingError = sqrt((points(:, 1) - center(1)).^2 + (points(:, 2) - center(2)).^2) - R;

end

??test_circle_fitting_2D.m

clc
clear
close all

%% 绘参考圆
figure
axis([0 100 0 100])
theta = linspace(0, 2*pi, 1000);
r = 30;
x = 50 + r*cos(theta);
y = 50 + r*sin(theta);
plot(x,y,'g--')
hold on
axis equal

%% 左键点击取点，按回车键退出
pos = ginput();
%pos = [pos;pos(1,1), pos(1,2)];  %起点与终点重合
pointCount = size(pos, 1);
plot(pos(1, 1), pos(1, 2), 'k+')
plot(pos(pointCount, 1), pos(pointCount, 2), 'k+')
plot(pos(2:pointCount-1, 1), pos(2:pointCount-1, 2), 'o')

%% 圆最小二乘拟合
crossStartAndEndPointFlag = 1; %0:不经过给定起点与终点;  1:精确经过给定起点与终点
[ center, R, fittingError ] = circle_fitting_2D( pos, pointCount, crossStartAndEndPointFlag )

x = center(1) + R * cos(theta);
y = center(2) + R * sin(theta);
plot(x, y, 'b')                   
plot(center(1), center(2), 'b+')
plot(center(1), center(2), 'bo')