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

#include <QGLViewer/quaternion.h>

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)
Quaternion &	operator= (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
Quaternion &	operator*= (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

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

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.

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

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 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.

Quaternion Quaternion::randomQuaternion ( ) [static]

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.

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).

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:

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

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

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

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

Private Attributes

Defining a Quaternion

Accessing values

Rotation computations

Inversion

Associated matrix

Slerp interpolation