function?[z?r?residual]?=?fitcircle（x?varargin）
%FITCIRCLE????least?squares?circle?fit
%???
%???[Z?R]?=?FITCIRCLE（X）?fits?a?circle?to?the?N?points?in?X?minimising
%???geometric?error?（sum?of?squared?distances?from?the?points?to?the?fitted
%???circle）?using?nonlinear?least?squares?（Gauss?Newton）
%???????Input
%???????????X?:?2xN?array?of?N?2D?points?with?N?>=?3
%???????Output
%???????????Z?:?center?of?the?fitted?circle
%???????????R?:?radius?of?the?fitted?circle
%
%???[Z?R]?=?FITCIRCLE（X?‘linear‘）?fits?a?circle?using?linear?least
%???squares?minimising?the?algebraic?error?（residual?from?fitting?system
%???of?the?form?ax‘x?+?b‘x?+?c?=?0）
%
%???[Z?R]?=?FITCIRCLE（X?Property?Value?...）?allows?parameters?to?be
%???passed?to?the?internal?Gauss?Newton?method.?Property?names?can?be
%???supplied?as?any?unambiguous?contraction?of?the?property?name?and?are?
%???case?insensitive?e.g.?FITCIRCLE（X?‘t‘?1e-4）?is?equivalent?to
%???FITCIRCLE（X?‘tol‘?1e-4）.?Valid?properties?are:
%
%???????Property:?????????????????Value:
%???????--------------------------------
%???????maxits????????????????????positive?integer?default?100
%???????????Sets?the?maximum?number?of?iterations?of?the?Gauss?Newton
%???????????method
%
%???????tol???????????????????????positive?constant?default?1e-5
%???????????Gauss?Newton?converges?when?the?relative?change?in?the?solution
%???????????is?less?than?tol
%
%???[X?R?RES]?=?fitcircle（...）?returns?the?2?norm?of?the?residual?from?
%???the?least?squares?fit
%
%???Example:
%???????x?=?[1?2?5?7?9?3;?7?6?8?7?5?7];
%???????%?Get?linear?least?squares?fit
%???????[zl?rl]?=?fitcircle（x?‘linear‘）
%???????%?Get?true?best?fit
%???????[z?r]?=?fitcircle（x）
%
%???Reference:?“Least-squares?fitting?of?circles?and?ellipses“?W.?Gander
%???G.?Golub?R.?Strebel?-?BIT?Numerical?Mathematics?1994?Springer

%?This?implementation?copyright?Richard?Brown?2007?but?is?freely
%?available?to?copy?use?or?modify?as?long?as?this?line?is?maintained

error（nargchk（1?5?nargin?‘struct‘））

%?Default?parameters?for?Gauss?Newton?minimisation
params.maxits?=?100;
params.tol????=?1e-5;

%?Check?x?and?get?user?supplied?parameters
[x?fNonlinear?params]?=?parseinputs（x?params?varargin{:}）;

%?Convenience?variables
m??=?size（x?2）;
x1?=?x（1?:）‘;
x2?=?x（2?:）‘;


%?1）?Compute?best?fit?w.r.t.?algebraic?error?using?linear?least?squares
%?
%?Circle?is?represented?as?a?matrix?quadratic?form
%?????ax‘x?+?b‘x?+?c?=?0
%?Linear?least?squares?estimate?found?by?minimising?Bu?=?0?s.t.?norm（u）?=?1
%?????where?u?=?[a;?b;?c]

%?Form?the?coefficient?matrix
B?=?[x1.^2?+?x2.^2?x1?x2?ones（m?1）];

%?Least?squares?estimate?is?right?singular?vector?corresp.?to?smallest
%?singular?value?of?B
[U?S?V]?=?svd（B）;
u?=?V（:?4）;

%?For?clarity?set?the?quadratic?form?variables
a?=?u（1）;
b?=?u（2:3）;
c?=?u（4）;

%?Convert?to?centre/radius
z?=?-b?/?（2*a）;
r?=?sqrt（（norm（b）/（2*a））^2?-?c/a）;

%?2