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## Copyright (C) 2024 David Legland
## All rights reserved.
##
## Redistribution and use in source and binary forms, with or without
## modification, are permitted provided that the following conditions are met:
##
## 1 Redistributions of source code must retain the above copyright notice,
## this list of conditions and the following disclaimer.
## 2 Redistributions in binary form must reproduce the above copyright
## notice, this list of conditions and the following disclaimer in the
## documentation and/or other materials provided with the distribution.
##
## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ''AS IS''
## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
## IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
## ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR
## ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
## DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
## SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
## CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
## OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
##
## The views and conclusions contained in the software and documentation are
## those of the authors and should not be interpreted as representing official
## policies, either expressed or implied, of the copyright holders.
function [sphere, residuals] = fitSphere(x,y,z)
%FITSPHERE Fit a sphere to 3D points using the least squares approach.
%
% SPHERE = fitSphere(PTS)
% Fits the equation of a sphere in Cartesian coordinates to the N-by-3
% array PTS using the least squares approach. The sphere is represented
% by its center [xc yc zc] and its radius r: SPHERE = [xc yc zc r].
%
% SPHERE = fitSphere(X, Y, Z)
% Use three vectors X, Y and Z with the length N instead of a the
% N-by-3 array PTS.
%
% [SPHERE, RESIDUALS] = fitSphere(...)
% Additionally outputs the residuals in the radial direction.
%
% Example:
% center=-100 + 200*rand(1,3);
% radius = randi([10 100]);
% [x,y,z]=drawSphere(center, radius);
% x=x+rand(size(x)); y=y+rand(size(y)); z=z+rand(size(z));
% sampleIdx = randi(numel(x),[1,randi([4, numel(x)])]);
% x=x(sampleIdx); y=y(sampleIdx); z=z(sampleIdx);
% sphere = fitSphere(x,y,z);
% figure('color','w'); hold on; axis equal tight; view(3)
% drawPoint3d(x,y,z)
% drawSphere(sphere,'FaceAlpha',0.5)
%
% See also
% createSphere, drawSphere, intersectLineSphere, intersectPlaneSphere
%
% Source:
% Levente Hunyadi - Fitting quadratic curves and surfaces:
% https://de.mathworks.com/matlabcentral/fileexchange/45356
% ------
% Author: Levente Hunyadi, oqilipo (minor adaptions for matGeom)
% E-mail: N/A
% Created: 2010
% Copyright 2010-2023 Levente Hunyadi
narginchk(1,3);
switch nargin % n x 3 matrix
case 1
n = size(x,1);
validateattributes(x, {'numeric'}, {'2d','real','size',[n,3]});
z = x(:,3);
y = x(:,2);
x = x(:,1);
otherwise % three x,y,z vectors
n = length(x(:));
x = x(:); % force into columns
y = y(:);
z = z(:);
validateattributes(x, {'numeric'}, {'real','size',[n,1]});
validateattributes(y, {'numeric'}, {'real','size',[n,1]});
validateattributes(z, {'numeric'}, {'real','size',[n,1]});
end
% need four or more data points
if n < 4
error('spherefit:InsufficientData', ...
'At least four points are required to fit a unique sphere.');
end
% solve linear system of normal equations
A = [x, y, z, ones(size(x))];
b = -(x.^2 + y.^2 + z.^2);
a = A \ b;
% return center coordinates and sphere radius
center = -a(1:3)./2;
radius = realsqrt(sum(center.^2)-a(4));
sphere = [center' radius];
% calculate residuals
residuals = radius - sqrt(sum(bsxfun(@minus,[x y z],center.').^2,2));
end
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