# v_rotro2pl

## PURPOSE

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

## SYNOPSIS

function [u,v,t]=v_rotro2pl(r)

## DESCRIPTION

    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

## CROSS-REFERENCE INFORMATION

This function calls:

- v_atan2sc V_ATAN2SC sin and cosine of atan(y/x) [S,C,R,T]=(Y,X)
- v_rotqr2ro ROTQR2RO converts a real quaternion to a 3x3 rotation matrix
- v_rotro2qr V_ROTRO2QR converts a 3x3 rotation matrix to a real quaternion

This function is called by:

## SOURCE CODE

    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 Lesser General Public License as published by
    0023 %   the Free Software Foundation; either version 3 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 Lesser General Public License for more details.
    0030 %
    0031 %   You can obtain a copy of the GNU Lesser General Public License from
    0032 %   https://www.gnu.org/licenses/ .
    0033 %
    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