v_rotro2pl

V_ROTRO2PL find the plane and rotation angle of a rotation matrix [u,v,t]=r
 Inputs:

     R(n,n)   Rotation matrix

 Outputs:

     U(n,1) and V(n,1) are orthonormal vectors defining a plane in n-dimensional space
     T is the rotation angle in radians from U towards V with 0<=T<=pi. If T
       is omitted it U and V will be separated by T instead of being orthogonal

0001 function [u,v,t]=v_rotro2pl(r)
0002 %V_ROTRO2PL find the plane and rotation angle of a rotation matrix [u,v,t]=r
0003 % Inputs:
0004 %
0005 %     R(n,n)   Rotation matrix
0006 %
0007 % Outputs:
0008 %
0009 %     U(n,1) and V(n,1) are orthonormal vectors defining a plane in n-dimensional space
0010 %     T is the rotation angle in radians from U towards V with 0<=T<=pi. If T
0011 %       is omitted it U and V will be separated by T instead of being orthogonal
0012 
0013 %
0014 %      Copyright (C) Mike Brookes 2007-2018
0015 %      Version: $Id: v_rotro2pl.m 10865 2018-09-21 17:22:45Z dmb $
0016 %
0017 %   VOICEBOX is a MATLAB toolbox for speech processing.
0018 %   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html
0019 %
0020 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0021 %   This program is free software; you can redistribute it and/or modify
0022 %   it under the terms of the GNU General Public License as published by
0023 %   the Free Software Foundation; either version 2 of the License, or
0024 %   (at your option) any later version.
0025 %
0026 %   This program is distributed in the hope that it will be useful,
0027 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
0028 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
0029 %   GNU General Public License for more details.
0030 %
0031 %   You can obtain a copy of the GNU General Public License from
0032 %   http://www.gnu.org/copyleft/gpl.html or by writing to
0033 %   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.
0034 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0035 
0036 n=size(r,1);
0037 [q,e]=schur(r);
0038 [m,i]=max(abs(e(2:n+1:n^2)));
0039 z=e(i+1,i)<0; % =1 if negative
0040 uv=q(:,i+z:1-2*z:i+1-z);
0041 u=uv(:,1);
0042 % the following code selects unique values of u and v
0043 % v=uv(:,2);
0044 % f=u.*v;         % maximize inner product of u.^2 and v.^2
0045 % g=(v+u).*(v-u);
0046 % t=atan2(sum(f.*g),sum(g.^2/4-f.^2))/4;
0047 % c=cos(t);
0048 % s=sin(t);
0049 % uv=uv*[c s; -s c];
0050 % a=sum(uv)<0;
0051 % uv=uv*[1-a(1)-a(2) a(2)-a(1); a(1)-a(2) 1-a(1)-a(2)];
0052 % u=uv(:,1);
0053 if nargout>2
0054     v=uv(:,2);
0055     t=atan2(abs(e(i+1,i)),e(i,i));
0056 elseif nargout
0057     [s,c]=v_atan2sc(abs(e(i+1,i)),e(i,i));
0058     v=uv*[c;s];
0059 elseif n==3 % if a 3D rotation
0060     v_rotqr2ro(v_rotro2qr(r)); % plot a rotated cube
0061 end