dust3d/thirdparty/cgal/CGAL-4.13/include/CGAL/Qt/quaternion_impl.h

/****************************************************************************

 Copyright (c) 2018  GeometryFactory Sarl (France).
 Copyright (C) 2002-2014 Gilles Debunne. All rights reserved.

 This file is part of a fork of the QGLViewer library version 2.7.0.
 http://www.libqglviewer.com - contact@libqglviewer.com

 This file may be used under the terms of the GNU General Public License 
 version 3.0 as published by the Free Software Foundation and
 appearing in the LICENSE file included in the packaging of this file.

 This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
 WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.

*****************************************************************************/
// $URL$
// $Id$
// SPDX-License-Identifier: GPL-3.0

#ifdef CGAL_HEADER_ONLY
#define CGAL_INLINE_FUNCTION inline

#include <CGAL/license/GraphicsView.h>

#else
#define CGAL_INLINE_FUNCTION
#endif
#include <CGAL/number_type_config.h>
#include <CGAL/Qt/quaternion.h>
#include <CGAL/Qt/domUtils.h>
#include <stdlib.h> // RAND_MAX

// All the methods are declared inline in Quaternion.h
namespace CGAL{
namespace qglviewer{


/*! Constructs a Quaternion that will rotate from the \p from direction to the
\p to direction.

Note that this rotation is not uniquely defined. The selected axis is usually
orthogonal to \p from and \p to, minimizing the rotation angle. This method is
robust and can handle small or almost identical vectors. */
CGAL_INLINE_FUNCTION
Quaternion::Quaternion(const Vec &from, const Vec &to) {
  const qreal epsilon = 1E-10;

  const qreal fromSqNorm = from.squaredNorm();
  const qreal toSqNorm = to.squaredNorm();
  // Identity Quaternion when one vector is null
  if ((fromSqNorm < epsilon) || (toSqNorm < epsilon)) {
    q[0] = q[1] = q[2] = 0.0;
    q[3] = 1.0;
  } else {
    Vec axis = cross(from, to);
    const qreal axisSqNorm = axis.squaredNorm();

    // Aligned vectors, pick any axis, not aligned with from or to
    if (axisSqNorm < epsilon)
      axis = from.orthogonalVec();

    qreal angle = asin(sqrt(axisSqNorm / (fromSqNorm * toSqNorm)));

    if (from * to < 0.0)
      angle = CGAL_PI - angle;

    setAxisAngle(axis, angle);
  }
}

/*! Returns the image of \p v by the Quaternion inverse() rotation.

rotate() performs an inverse transformation. Same as inverse().rotate(v). */
CGAL_INLINE_FUNCTION
Vec Quaternion::inverseRotate(const Vec &v) const {
  return inverse().rotate(v);
}

/*! Returns the image of \p v by the Quaternion rotation.

See also inverseRotate() and operator*(const Quaternion&, const Vec&). */
CGAL_INLINE_FUNCTION
Vec Quaternion::rotate(const Vec &v) const {
  const qreal q00 = 2.0 * q[0] * q[0];
  const qreal q11 = 2.0 * q[1] * q[1];
  const qreal q22 = 2.0 * q[2] * q[2];

  const qreal q01 = 2.0 * q[0] * q[1];
  const qreal q02 = 2.0 * q[0] * q[2];
  const qreal q03 = 2.0 * q[0] * q[3];

  const qreal q12 = 2.0 * q[1] * q[2];
  const qreal q13 = 2.0 * q[1] * q[3];

  const qreal q23 = 2.0 * q[2] * q[3];

  return Vec((1.0 - q11 - q22) * v[0] + (q01 - q23) * v[1] + (q02 + q13) * v[2],
             (q01 + q23) * v[0] + (1.0 - q22 - q00) * v[1] + (q12 - q03) * v[2],
             (q02 - q13) * v[0] + (q12 + q03) * v[1] +
                 (1.0 - q11 - q00) * v[2]);
}

/*! Set the Quaternion from a (supposedly correct) 3x3 rotation matrix.

  The matrix is expressed in European format: its three \e columns are the
  images by the rotation of the three vectors of an orthogonal basis. Note that
  OpenGL uses a symmetric representation for its matrices.

  setFromRotatedBasis() sets a Quaternion from the three axis of a rotated
  frame. It actually fills the three columns of a matrix with these rotated
  basis vectors and calls this method. */
CGAL_INLINE_FUNCTION
void Quaternion::setFromRotationMatrix(const qreal m[3][3]) {
  // Compute one plus the trace of the matrix
  const qreal onePlusTrace = 1.0 + m[0][0] + m[1][1] + m[2][2];

  if (onePlusTrace > 1E-5) {
    // Direct computation
    const qreal s = sqrt(onePlusTrace) * 2.0;
    q[0] = (m[2][1] - m[1][2]) / s;
    q[1] = (m[0][2] - m[2][0]) / s;
    q[2] = (m[1][0] - m[0][1]) / s;
    q[3] = 0.25 * s;
  } else {
    // Computation depends on major diagonal term
    if ((m[0][0] > m[1][1]) & (m[0][0] > m[2][2])) {
      const qreal s = sqrt(1.0 + m[0][0] - m[1][1] - m[2][2]) * 2.0;
      q[0] = 0.25 * s;
      q[1] = (m[0][1] + m[1][0]) / s;
      q[2] = (m[0][2] + m[2][0]) / s;
      q[3] = (m[1][2] - m[2][1]) / s;
    } else if (m[1][1] > m[2][2]) {
      const qreal s = sqrt(1.0 + m[1][1] - m[0][0] - m[2][2]) * 2.0;
      q[0] = (m[0][1] + m[1][0]) / s;
      q[1] = 0.25 * s;
      q[2] = (m[1][2] + m[2][1]) / s;
      q[3] = (m[0][2] - m[2][0]) / s;
    } else {
      const qreal s = sqrt(1.0 + m[2][2] - m[0][0] - m[1][1]) * 2.0;
      q[0] = (m[0][2] + m[2][0]) / s;
      q[1] = (m[1][2] + m[2][1]) / s;
      q[2] = 0.25 * s;
      q[3] = (m[0][1] - m[1][0]) / s;
    }
  }
  normalize();
}


/*! Sets the Quaternion from the three rotated vectors of an orthogonal basis.

  The three vectors do not have to be normalized but must be orthogonal and
  direct (X^Y=k*Z, with k>0).

  \code
  Quaternion q;
  q.setFromRotatedBasis(X, Y, Z);
  // Now q.rotate(Vec(1,0,0)) == X and q.inverseRotate(X) == Vec(1,0,0)
  // Same goes for Y and Z with Vec(0,1,0) and Vec(0,0,1).
  \endcode

  See also setFromRotationMatrix() and Quaternion(const Vec&, const Vec&). */
CGAL_INLINE_FUNCTION
void Quaternion::setFromRotatedBasis(const Vec &X, const Vec &Y, const Vec &Z) {
  qreal m[3][3];
  qreal normX = X.norm();
  qreal normY = Y.norm();
  qreal normZ = Z.norm();

  for (int i = 0; i < 3; ++i) {
    m[i][0] = X[i] / normX;
    m[i][1] = Y[i] / normY;
    m[i][2] = Z[i] / normZ;
  }

  setFromRotationMatrix(m);
}

/*! Returns the axis vector and the angle (in radians) of the rotation
 represented by the Quaternion. See the axis() and angle() documentations. */
CGAL_INLINE_FUNCTION
void Quaternion::getAxisAngle(Vec &axis, qreal &angle) const {
  angle = 2.0 * acos(q[3]);
  axis = Vec(q[0], q[1], q[2]);
  const qreal sinus = axis.norm();
  if (sinus > 1E-8)
    axis /= sinus;

  if (angle > CGAL_PI) {
    angle = 2.0 * qreal(CGAL_PI) - angle;
    axis = -axis;
  }
}

/*! Returns the normalized axis direction of the rotation represented by the
Quaternion.

It is null for an identity Quaternion. See also angle() and getAxisAngle(). */
CGAL_INLINE_FUNCTION
Vec Quaternion::axis() const {
  Vec res = Vec(q[0], q[1], q[2]);
  const qreal sinus = res.norm();
  if (sinus > 1E-8)
    res /= sinus;
  return (acos(q[3]) <= CGAL_PI / 2.0) ? res : -res;
}

/*! Returns the angle (in radians) of the rotation represented by the
 Quaternion.

 This value is always in the range [0-pi]. Larger rotational angles are obtained
 by inverting the axis() direction.

 See also axis() and getAxisAngle(). */
CGAL_INLINE_FUNCTION
qreal Quaternion::angle() const {
  const qreal angle = 2.0 * acos(q[3]);
  return (angle <= CGAL_PI) ? angle : 2.0 * CGAL_PI - angle;
}

/*! Returns an XML \c QDomElement that represents the Quaternion.

 \p name is the name of the QDomElement tag. \p doc is the \c QDomDocument
 factory used to create QDomElement.

 When output to a file, the resulting QDomElement will look like:
 \code
 <name q0=".." q1=".." q2=".." q3=".." />
 \endcode

 Use initFromDOMElement() to restore the Quaternion state from the resulting \c
 QDomElement. See also the Quaternion(const QDomElement&) constructor.

CGAL_INLINE_FUNCTION
 See the Vec::domElement() documentation for a complete QDomDocument creation
 and saving example.

CGAL_INLINE_FUNCTION
 See also Frame::domElement(), Camera::domElement(),
CGAL_INLINE_FUNCTION
 KeyFrameInterpolator::domElement()... */
CGAL_INLINE_FUNCTION
QDomElement Quaternion::domElement(const QString &name,
                                   QDomDocument &document) const {
  QDomElement de = document.createElement(name);
  de.setAttribute("q0", QString::number(q[0]));
  de.setAttribute("q1", QString::number(q[1]));
  de.setAttribute("q2", QString::number(q[2]));
  de.setAttribute("q3", QString::number(q[3]));
  return de;
}

/*! Restores the Quaternion state from a \c QDomElement created by domElement().

 The \c QDomElement should contain the \c q0, \c q1 , \c q2 and \c q3
 attributes. If one of these attributes is missing or is not a number, a warning
 is displayed and these fields are respectively set to 0.0, 0.0, 0.0 and 1.0
 (identity Quaternion).

 See also the Quaternion(const QDomElement&) constructor. */
CGAL_INLINE_FUNCTION
void Quaternion::initFromDOMElement(const QDomElement &element) {
  Quaternion q(element);
  *this = q;
}

/*! Constructs a Quaternion from a \c QDomElement representing an XML code of
  the form \code< anyTagName q0=".." q1=".." q2=".." q3=".." />\endcode

  If one of these attributes is missing or is not a number, a warning is
  displayed and the associated value is respectively set to 0, 0, 0 and 1
  (identity Quaternion).

  See also domElement() and initFromDOMElement(). */
CGAL_INLINE_FUNCTION
Quaternion::Quaternion(const QDomElement &element) {
  QStringList attribute;
  attribute << "q0"
            << "q1"
            << "q2"
            << "q3";
  for (int i = 0; i < attribute.size(); ++i)
    q[i] = DomUtils::qrealFromDom(element, attribute[i], ((i < 3) ? 0.0 : 1.0));
}

/*! Returns the Quaternion associated 4x4 OpenGL rotation matrix.

 Use \c glMultMatrixd(q.matrix()) to apply the rotation represented by
 Quaternion \c q to the current OpenGL matrix.

 See also getMatrix(), getRotationMatrix() and inverseMatrix().

 \attention The result is only valid until the next call to matrix(). Use it
 immediately (as shown above) or consider using getMatrix() instead.

 \attention The matrix is given in OpenGL format (row-major order) and is the
 transpose of the actual mathematical European representation. Consider using
 getRotationMatrix() instead. */
CGAL_INLINE_FUNCTION
const GLdouble *Quaternion::matrix() const {
  static GLdouble m[4][4];
  getMatrix(m);
  return (const GLdouble *)(m);
}

/*! Fills \p m with the OpenGL representation of the Quaternion rotation.

Use matrix() if you do not need to store this matrix and simply want to alter
the current OpenGL matrix. See also getInverseMatrix() and Frame::getMatrix().
*/
CGAL_INLINE_FUNCTION
void Quaternion::getMatrix(GLdouble m[4][4]) const {
  const qreal q00 = 2.0 * q[0] * q[0];
  const qreal q11 = 2.0 * q[1] * q[1];
  const qreal q22 = 2.0 * q[2] * q[2];

  const qreal q01 = 2.0 * q[0] * q[1];
  const qreal q02 = 2.0 * q[0] * q[2];
  const qreal q03 = 2.0 * q[0] * q[3];

  const qreal q12 = 2.0 * q[1] * q[2];
  const qreal q13 = 2.0 * q[1] * q[3];

  const qreal q23 = 2.0 * q[2] * q[3];

  m[0][0] = 1.0 - q11 - q22;
  m[1][0] = q01 - q23;
  m[2][0] = q02 + q13;

  m[0][1] = q01 + q23;
  m[1][1] = 1.0 - q22 - q00;
  m[2][1] = q12 - q03;

  m[0][2] = q02 - q13;
  m[1][2] = q12 + q03;
  m[2][2] = 1.0 - q11 - q00;

  m[0][3] = 0.0;
  m[1][3] = 0.0;
  m[2][3] = 0.0;

  m[3][0] = 0.0;
  m[3][1] = 0.0;
  m[3][2] = 0.0;
  m[3][3] = 1.0;
}

/*! Same as getMatrix(), but with a \c GLdouble[16] parameter. See also
 * getInverseMatrix() and Frame::getMatrix(). */
CGAL_INLINE_FUNCTION
void Quaternion::getMatrix(GLdouble m[16]) const {
  static GLdouble mat[4][4];
  getMatrix(mat);
  int count = 0;
  for (int i = 0; i < 4; ++i)
    for (int j = 0; j < 4; ++j)
      m[count++] = mat[i][j];
}

/*! Fills \p m with the 3x3 rotation matrix associated with the Quaternion.

  See also getInverseRotationMatrix().

  \attention \p m uses the European mathematical representation of the rotation
  matrix. Use matrix() and getMatrix() to retrieve the OpenGL transposed
  version. */
CGAL_INLINE_FUNCTION
void Quaternion::getRotationMatrix(qreal m[3][3]) const {
  static GLdouble mat[4][4];
  getMatrix(mat);
  for (int i = 0; i < 3; ++i)
    for (int j = 0; j < 3; ++j)
      // Beware of transposition
      m[i][j] = qreal(mat[j][i]);
}

/*! Returns the associated 4x4 OpenGL \e inverse rotation matrix. This is simply
  the matrix() of the inverse().

  \attention The result is only valid until the next call to inverseMatrix().
  Use it immediately (as in \c glMultMatrixd(q.inverseMatrix())) or use
  getInverseMatrix() instead.

  \attention The matrix is given in OpenGL format (row-major order) and is the
  transpose of the actual mathematical European representation. Consider using
  getInverseRotationMatrix() instead. */
CGAL_INLINE_FUNCTION
const GLdouble *Quaternion::inverseMatrix() const {
  static GLdouble m[4][4];
  getInverseMatrix(m);
  return (const GLdouble *)(m);
}

/*! Fills \p m with the OpenGL matrix corresponding to the inverse() rotation.

Use inverseMatrix() if you do not need to store this matrix and simply want to
alter the current OpenGL matrix. See also getMatrix(). */
CGAL_INLINE_FUNCTION
void Quaternion::getInverseMatrix(GLdouble m[4][4]) const {
  inverse().getMatrix(m);
}

/*! Same as getInverseMatrix(), but with a \c GLdouble[16] parameter. See also
 * getMatrix(). */
CGAL_INLINE_FUNCTION
void Quaternion::getInverseMatrix(GLdouble m[16]) const {
  inverse().getMatrix(m);
}

/*! \p m is set to the 3x3 \e inverse rotation matrix associated with the
 Quaternion.

 \attention This is the classical mathematical rotation matrix. The OpenGL
 format uses its transposed version. See inverseMatrix() and getInverseMatrix().
 */
CGAL_INLINE_FUNCTION
void Quaternion::getInverseRotationMatrix(qreal m[3][3]) const {
  static GLdouble mat[4][4];
  getInverseMatrix(mat);
  for (int i = 0; i < 3; ++i)
    for (int j = 0; j < 3; ++j)
      // Beware of transposition
      m[i][j] = qreal(mat[j][i]);
}

/*! Returns the slerp interpolation of Quaternions \p a and \p b, at time \p t.

 \p t should range in [0,1]. Result is \p a when \p t=0 and \p b when \p t=1.

 When \p allowFlip is \c true (default) the slerp interpolation will always use
 the "shortest path" between the Quaternions' orientations, by "flipping" the
 source Quaternion if needed (see negate()). */
CGAL_INLINE_FUNCTION
Quaternion Quaternion::slerp(const Quaternion &a, const Quaternion &b, qreal t,
                             bool allowFlip) {
  qreal cosAngle = Quaternion::dot(a, b);

  qreal c1, c2;
  // Linear interpolation for close orientations
  if ((1.0 - fabs(cosAngle)) < 0.01) {
    c1 = 1.0 - t;
    c2 = t;
  } else {
    // Spherical interpolation
    qreal angle = acos(fabs(cosAngle));
    qreal sinAngle = sin(angle);
    c1 = sin(angle * (1.0 - t)) / sinAngle;
    c2 = sin(angle * t) / sinAngle;
  }

  // Use the shortest path
  if (allowFlip && (cosAngle < 0.0))
    c1 = -c1;

  return Quaternion(c1 * a[0] + c2 * b[0], c1 * a[1] + c2 * b[1],
                    c1 * a[2] + c2 * b[2], c1 * a[3] + c2 * b[3]);
}

/*! Returns the slerp interpolation of the two Quaternions \p a and \p b, at
  time \p t, using tangents \p tgA and \p tgB.

  The resulting Quaternion is "between" \p a and \p b (result is \p a when \p
  t=0 and \p b for \p t=1).

  Use squadTangent() to define the Quaternion tangents \p tgA and \p tgB. */
CGAL_INLINE_FUNCTION
Quaternion Quaternion::squad(const Quaternion &a, const Quaternion &tgA,
                             const Quaternion &tgB, const Quaternion &b,
                             qreal t) {
  Quaternion ab = Quaternion::slerp(a, b, t);
  Quaternion tg = Quaternion::slerp(tgA, tgB, t, false);
  return Quaternion::slerp(ab, tg, 2.0 * t * (1.0 - t), false);
}

/*! Returns the logarithm of the Quaternion. See also exp(). */
CGAL_INLINE_FUNCTION
Quaternion Quaternion::log() {
  qreal len = sqrt(q[0] * q[0] + q[1] * q[1] + q[2] * q[2]);

  if (len < 1E-6)
    return Quaternion(q[0], q[1], q[2], 0.0);
  else {
    qreal coef = acos(q[3]) / len;
    return Quaternion(q[0] * coef, q[1] * coef, q[2] * coef, 0.0);
  }
}

/*! Returns the exponential of the Quaternion. See also log(). */
CGAL_INLINE_FUNCTION
Quaternion Quaternion::exp() {
  qreal theta = sqrt(q[0] * q[0] + q[1] * q[1] + q[2] * q[2]);

  if (theta < 1E-6)
    return Quaternion(q[0], q[1], q[2], cos(theta));
  else {
    qreal coef = sin(theta) / theta;
    return Quaternion(q[0] * coef, q[1] * coef, q[2] * coef, cos(theta));
  }
}

/*! Returns log(a. inverse() * b). Useful for squadTangent(). */
CGAL_INLINE_FUNCTION
Quaternion Quaternion::lnDif(const Quaternion &a, const Quaternion &b) {
  Quaternion dif = a.inverse() * b;
  dif.normalize();
  return dif.log();
}

/*! Returns a tangent Quaternion for \p center, defined by \p before and \p
 after Quaternions.

 Useful for smooth spline interpolation of Quaternion with squad() and slerp().
 */
CGAL_INLINE_FUNCTION
Quaternion Quaternion::squadTangent(const Quaternion &before,
                                    const Quaternion &center,
                                    const Quaternion &after) {
  Quaternion l1 = Quaternion::lnDif(center, before);
  Quaternion l2 = Quaternion::lnDif(center, after);
  Quaternion e;
  for (int i = 0; i < 4; ++i)
    e.q[i] = -0.25 * (l1.q[i] + l2.q[i]);
  e = center * (e.exp());

  // if (Quaternion::dot(e,b) < 0.0)
  // e.negate();

  return e;
}

/*! Returns a random unit Quaternion.

You can create a randomly directed unit vector using:
\code
CGAL_INLINE_FUNCTION
Vec randomDir = Quaternion::randomQuaternion() * Vec(1.0, 0.0, 0.0); // or any
other Vec \endcode

\note This function uses rand() to create pseudo-random numbers and the random
number generator can be initialized using srand().*/
CGAL_INLINE_FUNCTION
Quaternion Quaternion::randomQuaternion() {
  // The rand() function is not very portable and may not be available on your
  // system. Add the appropriate include or replace by an other random function
  // in case of problem.
  qreal seed = rand() / (qreal)RAND_MAX;
  qreal r1 = sqrt(1.0 - seed);
  qreal r2 = sqrt(seed);
  qreal t1 = 2.0 * CGAL_PI * (rand() / (qreal)RAND_MAX);
  qreal t2 = 2.0 * CGAL_PI * (rand() / (qreal)RAND_MAX);
  return Quaternion(sin(t1) * r1, cos(t1) * r1, sin(t2) * r2, cos(t2) * r2);
}

CGAL_INLINE_FUNCTION
std::ostream &operator<<(std::ostream &o, const Quaternion &Q) {
  return o << Q[0] << '\t' << Q[1] << '\t' << Q[2] << '\t' << Q[3];
}

}} // namespace CGAL::qglviewer