MT_ExpMap.cpp

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/*
 * ***** BEGIN GPL LICENSE BLOCK *****
 *
 * This program is free software; you can redistribute it and/or
 * modify it under the terms of the GNU General Public License
 * as published by the Free Software Foundation; either version 2
 * 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 General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software Foundation,
 * Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
 *
 * The Original Code is Copyright (C) 2001-2002 by NaN Holding BV.
 * All rights reserved.
 *
 * The Original Code is: all of this file.
 *
 * Original Author: Laurence
 * Contributor(s): Brecht
 *
 * ***** END GPL LICENSE BLOCK *****
 */

#include "MT_ExpMap.h"

    void
MT_ExpMap::
setRotation(
    const MT_Quaternion &q
) {
    // ok first normalize the quaternion
    // then compute theta the axis-angle and the normalized axis v
    // scale v by theta and that's it hopefully!

    m_q = q.normalized();
    m_v = MT_Vector3(m_q.x(), m_q.y(), m_q.z());

    MT_Scalar cosp = m_q.w();
    m_sinp = m_v.length();
    m_v /= m_sinp;

    m_theta = atan2(double(m_sinp),double(cosp));
    m_v *= m_theta;
}
    
    const MT_Quaternion&
MT_ExpMap::
getRotation(
) const {
    return m_q;
}
    
    MT_Matrix3x3
MT_ExpMap::
getMatrix(
) const {
    return MT_Matrix3x3(m_q);
}

    void
MT_ExpMap::
update(
    const MT_Vector3& dv
){
    m_v += dv;

    angleUpdated();
}

    void
MT_ExpMap::
partialDerivatives(
    MT_Matrix3x3& dRdx,
    MT_Matrix3x3& dRdy,
    MT_Matrix3x3& dRdz
) const {
    
    MT_Quaternion dQdx[3];

    compute_dQdVi(dQdx);

    compute_dRdVi(dQdx[0], dRdx);
    compute_dRdVi(dQdx[1], dRdy);
    compute_dRdVi(dQdx[2], dRdz);
}

    void
MT_ExpMap::
compute_dRdVi(
    const MT_Quaternion &dQdvi,
    MT_Matrix3x3 & dRdvi
) const {

    MT_Scalar  prod[9];
    
    /* This efficient formulation is arrived at by writing out the
     * entire chain rule product dRdq * dqdv in terms of 'q' and 
     * noticing that all the entries are formed from sums of just
     * nine products of 'q' and 'dqdv' */

    prod[0] = -MT_Scalar(4)*m_q.x()*dQdvi.x();
    prod[1] = -MT_Scalar(4)*m_q.y()*dQdvi.y();
    prod[2] = -MT_Scalar(4)*m_q.z()*dQdvi.z();
    prod[3] = MT_Scalar(2)*(m_q.y()*dQdvi.x() + m_q.x()*dQdvi.y());
    prod[4] = MT_Scalar(2)*(m_q.w()*dQdvi.z() + m_q.z()*dQdvi.w());
    prod[5] = MT_Scalar(2)*(m_q.z()*dQdvi.x() + m_q.x()*dQdvi.z());
    prod[6] = MT_Scalar(2)*(m_q.w()*dQdvi.y() + m_q.y()*dQdvi.w());
    prod[7] = MT_Scalar(2)*(m_q.z()*dQdvi.y() + m_q.y()*dQdvi.z());
    prod[8] = MT_Scalar(2)*(m_q.w()*dQdvi.x() + m_q.x()*dQdvi.w());

    /* first row, followed by second and third */
    dRdvi[0][0] = prod[1] + prod[2];
    dRdvi[0][1] = prod[3] - prod[4];
    dRdvi[0][2] = prod[5] + prod[6];

    dRdvi[1][0] = prod[3] + prod[4];
    dRdvi[1][1] = prod[0] + prod[2];
    dRdvi[1][2] = prod[7] - prod[8];

    dRdvi[2][0] = prod[5] - prod[6];
    dRdvi[2][1] = prod[7] + prod[8];
    dRdvi[2][2] = prod[0] + prod[1];
}

// compute partial derivatives dQ/dVi

    void
MT_ExpMap::
compute_dQdVi(
    MT_Quaternion *dQdX
) const {

    /* This is an efficient implementation of the derivatives given
     * in Appendix A of the paper with common subexpressions factored out */

    MT_Scalar sinc, termCoeff;

    if (m_theta < MT_EXPMAP_MINANGLE) {
        sinc = 0.5 - m_theta*m_theta/48.0;
        termCoeff = (m_theta*m_theta/40.0 - 1.0)/24.0;
    }
    else {
        MT_Scalar cosp = m_q.w();
        MT_Scalar ang = 1.0/m_theta;

        sinc = m_sinp*ang;
        termCoeff = ang*ang*(0.5*cosp - sinc);
    }

    for (int i = 0; i < 3; i++) {
        MT_Quaternion& dQdx = dQdX[i];
        int i2 = (i+1)%3;
        int i3 = (i+2)%3;

        MT_Scalar term = m_v[i]*termCoeff;
        
        dQdx[i] = term*m_v[i] + sinc;
        dQdx[i2] = term*m_v[i2];
        dQdx[i3] = term*m_v[i3];
        dQdx.w() = -0.5*m_v[i]*sinc;
    }
}

// reParametize away from singularity, updating
// m_v and m_theta

    void
MT_ExpMap::
reParametrize(
){
    if (m_theta > MT_PI) {
        MT_Scalar scl = m_theta;
        if (m_theta > MT_2_PI){ /* first get theta into range 0..2PI */
            m_theta = MT_Scalar(fmod(m_theta, MT_2_PI));
            scl = m_theta/scl;
            m_v *= scl;
        }
        if (m_theta > MT_PI){
            scl = m_theta;
            m_theta = MT_2_PI - m_theta;
            scl = MT_Scalar(1.0) - MT_2_PI/scl;
            m_v *= scl;
        }
    }
}

// compute cached variables

    void
MT_ExpMap::
angleUpdated(
){
    m_theta = m_v.length();

    reParametrize();

    // compute quaternion, sinp and cosp

    if (m_theta < MT_EXPMAP_MINANGLE) {
        m_sinp = MT_Scalar(0.0);

        /* Taylor Series for sinc */
        MT_Vector3 temp = m_v * MT_Scalar(MT_Scalar(.5) - m_theta*m_theta/MT_Scalar(48.0));
        m_q.x() = temp.x();
        m_q.y() = temp.y();
        m_q.z() = temp.z();
        m_q.w() = MT_Scalar(1.0);
    } else {
        m_sinp = MT_Scalar(sin(.5*m_theta));

        /* Taylor Series for sinc */
        MT_Vector3 temp = m_v * (m_sinp/m_theta);
        m_q.x() = temp.x();
        m_q.y() = temp.y();
        m_q.z() = temp.z();
        m_q.w() = MT_Scalar(cos(.5*m_theta));
    }
}