polygonSecondAreaMoments

POLYGONSECONDAREAMOMENTS Compute second-order area moments of a polygon.

   [IXX, IYY, IXY] = polygonSecondAreaMoments(POLY)
   Compute the second-order inertia moments of a polygon. The polygon is
   specified by the N-by-2 list of vertex coordinates.

   Example
   polygonSecondAreaMoments

   References
   * http://paulbourke.net/geometry/polygonmesh/
   * https://en.wikipedia.org/wiki/Second_moment_of_area

   See also
     polygons2d, polygonEquivalentEllipse, polygonArea, polygonCentroid

0001 function [Ixx, Iyy, Ixy] = polygonSecondAreaMoments(poly)
0002 %POLYGONSECONDAREAMOMENTS Compute second-order area moments of a polygon.
0003 %
0004 %   [IXX, IYY, IXY] = polygonSecondAreaMoments(POLY)
0005 %   Compute the second-order inertia moments of a polygon. The polygon is
0006 %   specified by the N-by-2 list of vertex coordinates.
0007 %
0008 %   Example
0009 %   polygonSecondAreaMoments
0010 %
0011 %   References
0012 %   * http://paulbourke.net/geometry/polygonmesh/
0013 %   * https://en.wikipedia.org/wiki/Second_moment_of_area
0014 %
0015 %   See also
0016 %     polygons2d, polygonEquivalentEllipse, polygonArea, polygonCentroid
0017  
0018 % ------
0019 % Author: David Legland
0020 % e-mail: david.legland@inra.fr
0021 % Created: 2017-09-08,    using Matlab 9.1.0.441655 (R2016b)
0022 % Copyright 2017 INRA - Cepia Software Platform.
0023 
0024 % get vertex coordinates, and recenter polygon
0025 centroid = polygonCentroid(poly);
0026 px = poly(:,1) - centroid(1);
0027 py = poly(:,2) - centroid(2);
0028 
0029 % vertex indices
0030 N = length(px);
0031 iNext = [2:N 1];
0032 
0033 % compute twice signed area of each triangle
0034 common = px .* py(iNext) - px(iNext) .* py;
0035 
0036 % compute each term
0037 Ixx = sum( (py.^2 + py .* py(iNext) + py(iNext).^2) .* common) / 12;
0038 Iyy = sum( (px.^2 + px .* px(iNext) + px(iNext).^2) .* common) / 12;
0039 Ixy = sum( ...
0040     (px .* py(iNext) + 2 * px .* py + 2 * px(iNext) .* py(iNext) ...
0041     + px(iNext) .* py ) .* common) / 24;