Private Attributes
qglviewer::Quaternion Class Reference

The Quaternion class represents 3D rotations and orientations. More...

#include <QGLViewer/quaternion.h>

List of all members.

Private Attributes

double q [4]

Defining a Quaternion

 Quaternion ()
 Quaternion (const Vec &axis, double angle)
 Quaternion (const Vec &from, const Vec &to)
 Quaternion (double q0, double q1, double q2, double q3)
 Quaternion (const Quaternion &Q)
Quaternionoperator= (const Quaternion &Q)
void setAxisAngle (const Vec &axis, double angle)
void setValue (double q0, double q1, double q2, double q3)
void setFromRotationMatrix (const float m[3][3])
void setFromRotatedBase (const Vec &X, const Vec &Y, const Vec &Z)
void setFromRotationMatrix (const double m[3][3])
void setFromRotatedBasis (const Vec &X, const Vec &Y, const Vec &Z)

Accessing values

Vec axis () const
double angle () const
void getAxisAngle (Vec &axis, float &angle) const
double operator[] (int i) const
double & operator[] (int i)

Rotation computations

Quaternionoperator*= (const Quaternion &q)
Vec rotate (const Vec &v) const
Vec inverseRotate (const Vec &v) const
Quaternion operator* (const Quaternion &a, const Quaternion &b)
Vec operator* (const Quaternion &q, const Vec &v)

Inversion

Quaternion inverse () const
void invert ()
void negate ()
double normalize ()
Quaternion normalized () const

Associated matrix

const GLdouble * matrix () const
void getMatrix (GLdouble m[4][4]) const
void getMatrix (GLdouble m[16]) const
void getRotationMatrix (float m[3][3]) const
const GLdouble * inverseMatrix () const
void getInverseMatrix (GLdouble m[4][4]) const
void getInverseMatrix (GLdouble m[16]) const
void getInverseRotationMatrix (float m[3][3]) const

Slerp interpolation

Quaternion log ()
Quaternion exp ()
static Quaternion slerp (const Quaternion &a, const Quaternion &b, float t, bool allowFlip=true)
static Quaternion squad (const Quaternion &a, const Quaternion &tgA, const Quaternion &tgB, const Quaternion &b, float t)
static double dot (const Quaternion &a, const Quaternion &b)
static Quaternion lnDif (const Quaternion &a, const Quaternion &b)
static Quaternion squadTangent (const Quaternion &before, const Quaternion &center, const Quaternion &after)

Random Quaternion

static Quaternion randomQuaternion ()

XML representation

 Quaternion (const QDomElement &element)
QDomElement domElement (const QString &name, QDomDocument &document) const
void initFromDOMElement (const QDomElement &element)

Detailed Description

The Quaternion class represents 3D rotations and orientations.

The Quaternion is an appropriate (although not very intuitive) representation for 3D rotations and orientations. Many tools are provided to ease the definition of a Quaternion: see constructors, setAxisAngle(), setFromRotationMatrix(), setFromRotatedBasis().

You can apply the rotation represented by the Quaternion to 3D points using rotate() and inverseRotate(). See also the Frame class that represents a coordinate system and provides other conversion functions like Frame::coordinatesOf() and Frame::transformOf().

You can apply the Quaternion q rotation to the OpenGL matrices using:

        glMultMatrixd(q.matrix());
        // equvalent to glRotate(q.angle()*180.0/M_PI, q.axis().x, q.axis().y, q.axis().z);

Quaternion is part of the qglviewer namespace, specify qglviewer::Quaternion or use the qglviewer namespace:

 using namespace qglviewer; 

Internal representation

The internal representation of a Quaternion corresponding to a rotation around axis axis, with an angle alpha is made of four doubles q[i]:

        {q[0],q[1],q[2]} = sin(alpha/2) * {axis[0],axis[1],axis[2]}
        q[3] = cos(alpha/2)

Note that certain implementations place the cosine term in first position (instead of last here).

The Quaternion is always normalized, so that its inverse() is actually its conjugate.

See also the Vec and Frame classes' documentations.

Definition at line 66 of file quaternion.h.


Constructor & Destructor Documentation

Default constructor, builds an identity rotation.

Definition at line 72 of file quaternion.h.

qglviewer::Quaternion::Quaternion ( const Vec axis,
double  angle 
) [inline]

Constructor from rotation axis (non null) and angle (in radians). See also setAxisAngle().

Definition at line 76 of file quaternion.h.

Quaternion::Quaternion ( const Vec from,
const Vec to 
)

Constructs a Quaternion that will rotate from the from direction to the to direction.

Note that this rotation is not uniquely defined. The selected axis is usually orthogonal to from and to, minimizing the rotation angle. This method is robust and can handle small or almost identical vectors.

Definition at line 35 of file quaternion.cpp.

qglviewer::Quaternion::Quaternion ( double  q0,
double  q1,
double  q2,
double  q3 
) [inline]

Constructor from the four values of a Quaternion. First three values are axis*sin(angle/2) and last one is cos(angle/2).

Attention:
The identity Quaternion is Quaternion(0,0,0,1) and not Quaternion(0,0,0,0) (which is not unitary). The default Quaternion() creates such identity Quaternion.

Definition at line 88 of file quaternion.h.

Copy constructor.

Definition at line 92 of file quaternion.h.

Quaternion::Quaternion ( const QDomElement &  element) [explicit]

Constructs a Quaternion from a QDomElement representing an XML code of the form

 < anyTagName q0=".." q1=".." q2=".." q3=".." />

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().

Definition at line 286 of file quaternion.cpp.


Member Function Documentation

double Quaternion::angle ( ) const

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().

Definition at line 234 of file quaternion.cpp.

Vec Quaternion::axis ( ) const

Returns the normalized axis direction of the rotation represented by the Quaternion.

It is null for an identity Quaternion. See also angle() and getAxisAngle().

Definition at line 219 of file quaternion.cpp.

QDomElement Quaternion::domElement ( const QString &  name,
QDomDocument &  document 
) const

Returns an XML QDomElement that represents the Quaternion.

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

When output to a file, the resulting QDomElement will look like:

 <name q0=".." q1=".." q2=".." q3=".." />

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

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

See also Frame::domElement(), Camera::domElement(), KeyFrameInterpolator::domElement()...

Definition at line 256 of file quaternion.cpp.

static double qglviewer::Quaternion::dot ( const Quaternion a,
const Quaternion b 
) [inline, static]

Returns the "dot" product of a and b: a[0]*b[0] + a[1]*b[1] + a[2]*b[2] + a[3]*b[3].

Definition at line 270 of file quaternion.h.

Returns the exponential of the Quaternion. See also log().

Definition at line 493 of file quaternion.cpp.

void Quaternion::getAxisAngle ( Vec axis,
float &  angle 
) const

Returns the axis vector and the angle (in radians) of the rotation represented by the Quaternion. See the axis() and angle() documentations.

Definition at line 201 of file quaternion.cpp.

void Quaternion::getInverseMatrix ( GLdouble  m[4][4]) const

Fills 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().

Definition at line 404 of file quaternion.cpp.

void Quaternion::getInverseMatrix ( GLdouble  m[16]) const

Same as getInverseMatrix(), but with a GLdouble[16] parameter. See also getMatrix().

Definition at line 410 of file quaternion.cpp.

void Quaternion::getInverseRotationMatrix ( float  m[3][3]) const

m is set to the 3x3 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().

Definition at line 419 of file quaternion.cpp.

void Quaternion::getMatrix ( GLdouble  m[4][4]) const

Fills 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().

Definition at line 321 of file quaternion.cpp.

void Quaternion::getMatrix ( GLdouble  m[16]) const

Same as getMatrix(), but with a GLdouble[16] parameter. See also getInverseMatrix() and Frame::getMatrix().

Definition at line 359 of file quaternion.cpp.

void Quaternion::getRotationMatrix ( float  m[3][3]) const

Fills m with the 3x3 rotation matrix associated with the Quaternion.

See also getInverseRotationMatrix().

Attention:
m uses the European mathematical representation of the rotation matrix. Use matrix() and getMatrix() to retrieve the OpenGL transposed version.

Definition at line 375 of file quaternion.cpp.

void Quaternion::initFromDOMElement ( const QDomElement &  element)

Restores the Quaternion state from a QDomElement created by domElement().

The QDomElement should contain the q0, q1 , q2 and 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.

Definition at line 273 of file quaternion.cpp.

Returns the inverse Quaternion (inverse rotation).

Result has a negated axis() direction and the same angle(). A composition (see operator*()) of a Quaternion and its inverse() results in an identity function.

Use invert() to actually modify the Quaternion.

Definition at line 205 of file quaternion.h.

const GLdouble * Quaternion::inverseMatrix ( ) const

Returns the associated 4x4 OpenGL 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 glMultMatrixd(q.inverseMatrix())) or use getInverseMatrix() instead.
The matrix is given in OpenGL format (row-major order) and is the transpose of the actual mathematical European representation. Consider using getInverseRotationMatrix() instead.

Definition at line 393 of file quaternion.cpp.

Vec Quaternion::inverseRotate ( const Vec v) const

Returns the image of v by the Quaternion inverse() rotation.

rotate() performs an inverse transformation. Same as inverse().rotate(v).

Definition at line 68 of file quaternion.cpp.

void qglviewer::Quaternion::invert ( ) [inline]

Inverses the Quaternion (same rotation angle(), but negated axis()).

See also inverse().

Definition at line 210 of file quaternion.h.

Quaternion Quaternion::lnDif ( const Quaternion a,
const Quaternion b 
) [static]

Returns log(a. inverse() * b). Useful for squadTangent().

Definition at line 507 of file quaternion.cpp.

Returns the logarithm of the Quaternion. See also exp().

Definition at line 479 of file quaternion.cpp.

const GLdouble * Quaternion::matrix ( ) const

Returns the Quaternion associated 4x4 OpenGL rotation matrix.

Use glMultMatrixd(q.matrix()) to apply the rotation represented by Quaternion 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.
The matrix is given in OpenGL format (row-major order) and is the transpose of the actual mathematical European representation. Consider using getRotationMatrix() instead.

Definition at line 310 of file quaternion.cpp.

void qglviewer::Quaternion::negate ( ) [inline]

Negates all the coefficients of the Quaternion.

This results in an other representation of the same rotation (opposite rotation angle, but with a negated axis direction: the two cancel out). However, note that the results of axis() and angle() are unchanged after a call to this method since angle() always returns a value in [0,pi].

This method is mainly useful for Quaternion interpolation, so that the spherical interpolation takes the shortest path on the unit sphere. See slerp() for details.

Definition at line 220 of file quaternion.h.

double qglviewer::Quaternion::normalize ( ) [inline]

Normalizes the Quaternion coefficients.

This method should not need to be called since we only deal with unit Quaternions. This is however useful to prevent numerical drifts, especially with small rotational increments. See also normalized().

Definition at line 227 of file quaternion.h.

Returns a normalized version of the Quaternion.

See also normalize().

Definition at line 238 of file quaternion.h.

Quaternion& qglviewer::Quaternion::operator*= ( const Quaternion q) [inline]

Quaternion rotation is composed with q.

See operator*(), since this is equivalent to this = this * q.

Note:
For efficiency reasons, the resulting Quaternion is not normalized. You may normalize() it after each application in case of numerical drift.

Definition at line 178 of file quaternion.h.

Quaternion& qglviewer::Quaternion::operator= ( const Quaternion Q) [inline]

Equal operator.

Definition at line 96 of file quaternion.h.

double qglviewer::Quaternion::operator[] ( int  i) const [inline]

Bracket operator, with a constant return value. i must range in [0..3]. See the Quaternion(double, double, double, double) documentation.

Definition at line 144 of file quaternion.h.

double& qglviewer::Quaternion::operator[] ( int  i) [inline]

Bracket operator returning an l-value. i must range in [0..3]. See the Quaternion(double, double, double, double) documentation.

Definition at line 147 of file quaternion.h.

Returns a random unit Quaternion.

You can create a randomly directed unit vector using:

Vec randomDir = Quaternion::randomQuaternion() * Vec(1.0, 0.0, 0.0); // or any other Vec
Note:
This function uses rand() to create pseudo-random numbers and the random number generator can be initialized using srand().

Definition at line 546 of file quaternion.cpp.

Vec Quaternion::rotate ( const Vec v) const

Returns the image of v by the Quaternion rotation.

See also inverseRotate() and operator*(const Quaternion&, const Vec&).

Definition at line 76 of file quaternion.cpp.

void qglviewer::Quaternion::setAxisAngle ( const Vec axis,
double  angle 
) [inline]

Sets the Quaternion as a rotation of axis axis and angle angle (in radians).

axis does not need to be normalized. A null axis will result in an identity Quaternion.

Definition at line 106 of file quaternion.h.

void Quaternion::setFromRotatedBase ( const Vec X,
const Vec Y,
const Vec Z 
)

Definition at line 163 of file quaternion.cpp.

void Quaternion::setFromRotatedBasis ( const Vec X,
const Vec Y,
const Vec Z 
)

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).

  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).

See also setFromRotationMatrix() and Quaternion(const Vec&, const Vec&).

Definition at line 182 of file quaternion.cpp.

void Quaternion::setFromRotationMatrix ( const float  m[3][3])

Definition at line 151 of file quaternion.cpp.

void Quaternion::setFromRotationMatrix ( const double  m[3][3])

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

The matrix is expressed in European format: its three 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.

Definition at line 104 of file quaternion.cpp.

void qglviewer::Quaternion::setValue ( double  q0,
double  q1,
double  q2,
double  q3 
) [inline]

Sets the Quaternion value. See the Quaternion(double, double, double, double) constructor documentation.

Definition at line 125 of file quaternion.h.

Quaternion Quaternion::slerp ( const Quaternion a,
const Quaternion b,
float  t,
bool  allowFlip = true 
) [static]

Returns the slerp interpolation of Quaternions a and b, at time t.

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

When allowFlip is true (default) the slerp interpolation will always use the "shortest path" between the Quaternions' orientations, by "flipping" the source Quaternion if needed (see negate()).

Definition at line 437 of file quaternion.cpp.

Quaternion Quaternion::squad ( const Quaternion a,
const Quaternion tgA,
const Quaternion tgB,
const Quaternion b,
float  t 
) [static]

Returns the slerp interpolation of the two Quaternions a and b, at time t, using tangents tgA and tgB.

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

Use squadTangent() to define the Quaternion tangents tgA and tgB.

Definition at line 471 of file quaternion.cpp.

Quaternion Quaternion::squadTangent ( const Quaternion before,
const Quaternion center,
const Quaternion after 
) [static]

Returns a tangent Quaternion for center, defined by before and after Quaternions.

Useful for smooth spline interpolation of Quaternion with squad() and slerp().

Definition at line 517 of file quaternion.cpp.


Friends And Related Function Documentation

Quaternion operator* ( const Quaternion a,
const Quaternion b 
) [friend]

Returns the composition of the a and b rotations.

The order is important. When applied to a Vec v (see operator*(const Quaternion&, const Vec&) and rotate()) the resulting Quaternion acts as if b was applied first and then a was applied. This is obvious since the image v' of v by the composited rotation satisfies:

                v'= (a*b) * v = a * (b*v) 

Note that a*b usually differs from b*a.

Attention:
For efficiency reasons, the resulting Quaternion is not normalized. Use normalize() in case of numerical drift with small rotation composition.

Definition at line 164 of file quaternion.h.

Vec operator* ( const Quaternion q,
const Vec v 
) [friend]

Returns the image of v by the rotation q.

Same as q.rotate(v). See rotate() and inverseRotate().

Definition at line 187 of file quaternion.h.


Member Data Documentation

double qglviewer::Quaternion::q[4] [private]

The internal data representation is private, use operator[] to access values.

Definition at line 304 of file quaternion.h.


The documentation for this class was generated from the following files:


octovis
Author(s): Kai M. Wurm , Armin Hornung
autogenerated on Thu Feb 11 2016 23:51:21