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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 elli = polygonEquivalentEllipse(poly)
%POLYGONEQUIVALENTELLIPSE Compute equivalent ellipse with same second order moments as polygon.
%
% ELLI = polygonEquivalentEllipse(POLY)
%
% Example
% % convert an ellipse to polygon, and check that it equivalent ellipse
% is close to original ellipse
% elli = [50 50 50 30 20];
% poly = ellipseToPolygon(elli, 1000);
% polygonEquivalentEllipse(poly)
% ans =
% 50.0000 50.0000 49.9998 29.9999 20.0000
%
% % compute equivalent ellipse of more complex figure
% img = imread('circles.png');
% img = imfill(img, 'holes');
% figure; imshow(img); hold on;
% B = bwboundaries(img);
% poly = B{1}(:,[2 1]);
% drawPolygon(poly, 'r');
% elli = polygonEquivalentEllipse(poly);
% drawEllipse(elli, 'color', 'g', 'linewidth', 2);
%
%
% See also
% polygons2d, polygonSecondAreaMoments, polygonCentroid,
% equivalentEllipse, ellipseToPolygon
%
% ------
% Author: David Legland
% E-mail: david.legland@inrae.fr
% Created: 2017-09-08, using Matlab 9.1.0.441655 (R2016b)
% Copyright 2017-2023 INRA - Cepia Software Platform
% first re-center the polygon
centroid = polygonCentroid(poly);
poly = bsxfun(@minus, poly, centroid);
% compute non-normalized inertia moments
[Ix, Iy, Ixy] = polygonSecondAreaMoments(poly);
% normalize with polygon area
area = polygonArea(poly);
Ix = Ix / area;
Iy = Iy / area;
Ixy = Ixy / area;
% compute ellipse semi-axis lengths
common = sqrt((Ix - Iy)^2 + 4 * Ixy^2);
ra = sqrt(2) * sqrt(Ix + Iy + common);
rb = sqrt(2) * sqrt(Ix + Iy - common);
% compute ellipse angle and convert into degrees
% (different formula from the equivalentEllipse function, as the definition
% for Ix and Iy do not refer to same axes)
theta = atan2(2 * Ixy, Iy - Ix) / 2;
theta = theta * 180 / pi;
% compute centroid and concatenate results into ellipse format
elli = [centroid ra rb theta];
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