# v_rotax2qr

## PURPOSE

V_ROTQR2AX converts a rotation axis and angle to the corresponding real quaternion

## SYNOPSIS

function q=v_rotax2qr(a,t)

## DESCRIPTION

    V_ROTQR2AX converts a rotation axis and angle to the corresponding real quaternion
     Inputs:

         A(3,1)   Vector in the direction of the rotation axis.
         T        Rotation angle in radians

     Output:

         Q(4,1)   normalized real-valued quaternion

     In the quaternion representation of a rotation, and q(1) = cos(t/2)
     where t is the angle of rotation and q(2:4)/sin(t/2) is a unit vector
     lying along the axis of rotation.
     A positive rotation about [0 0 1] takes the X axis towards the Y axis.

          Copyright (C) Mike Brookes 2007-2018
          Version: $Id: v_rotax2qr.m 10865 2018-09-21 17:22:45Z dmb $

       VOICEBOX is a MATLAB toolbox for speech processing.
       Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
       This program is free software; you can redistribute it and/or modify
       it under the terms of the GNU Lesser General Public License as published by
       the Free Software Foundation; either version 3 of the License, or
       (at your option) any later version.

       This program is distributed in the hope that it will be useful,
       but WITHOUT ANY WARRANTY; without even the implied warranty of
       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
       GNU Lesser General Public License for more details.

       You can obtain a copy of the GNU Lesser General Public License from
       https://www.gnu.org/licenses/ .

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

## CROSS-REFERENCE INFORMATION

This function calls:

- v_rotqr2ro ROTQR2RO converts a real quaternion to a 3x3 rotation matrix

This function is called by:

## SOURCE CODE

    0001 function q=v_rotax2qr(a,t)
    0002 %V_ROTQR2AX converts a rotation axis and angle to the corresponding real quaternion
    0003 % Inputs:
    0004 %
    0005 %     A(3,1)   Vector in the direction of the rotation axis.
    0006 %     T        Rotation angle in radians
    0007 %
    0008 % Output:
    0009 %
    0010 %     Q(4,1)   normalized real-valued quaternion
    0011 %
    0012 % In the quaternion representation of a rotation, and q(1) = cos(t/2)
    0013 % where t is the angle of rotation and q(2:4)/sin(t/2) is a unit vector
    0014 % lying along the axis of rotation.
    0015 % A positive rotation about [0 0 1] takes the X axis towards the Y axis.
    0016 %
    0017 %      Copyright (C) Mike Brookes 2007-2018
    0018 %      Version: $Id: v_rotax2qr.m 10865 2018-09-21 17:22:45Z dmb $
    0019 %
    0020 %   VOICEBOX is a MATLAB toolbox for speech processing.
    0021 %   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html
    0022 %
    0023 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    0024 %   This program is free software; you can redistribute it and/or modify
    0025 %   it under the terms of the GNU Lesser General Public License as published by
    0026 %   the Free Software Foundation; either version 3 of the License, or
    0027 %   (at your option) any later version.
    0028 %
    0029 %   This program is distributed in the hope that it will be useful,
    0030 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
    0031 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    0032 %   GNU Lesser General Public License for more details.
    0033 %
    0034 %   You can obtain a copy of the GNU Lesser General Public License from
    0035 %   https://www.gnu.org/licenses/ .
    0036 %
    0037 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    0038 if all(a==0)
    0039     a=[1;0;0];
    0040 end
    0041 m=sqrt(a(:)'*a(:));
    0042 q=[cos(0.5*t); sin(0.5*t)*a(:)/m];
    0043 if ~nargout
    0044     v_rotqr2ro(q); % plot a rotated cube
    0045 end
    0046