Quaternions provide a representation of Orientation and rotations in three dimensions. More...

#include <Quaternion.h>

Public Member Functions
	Quaternion ()
	Construct an identity Quaternion.

	Quaternion (Vector< T, 3 > _axis, T _angle)
	Construct a Quaternion with an Axis and Angle of rotation.

	Quaternion (Vector< T, 4 > vector)

	Quaternion (std::initializer_list< T > ilst)
	An initializer list containing values for w, x, y, and z.

void	SetEpsilon (T epsilon)
	Set the comparison tolerance for this Quaternion.

Vector< T, 3 >	Axis () const
	Return the Axis of rotation of this Quaternion.

T	Angle () const
	Return the Angle of rotation of this Quaternion.

T	Dot (const Quaternion< T > &other) const
	Compute the Dot product of two quaternions.

T	Norm () const
	Compute the Norm of a Quaternion.

Quaternion	UnitQuaternion ()
	Return the unit Quaternion.

Quaternion	Conjugate () const
	Compute the Conjugate of a Quaternion.

Quaternion	Inverse () const
	Compute the Inverse of a Quaternion.

bool	IsIdentity () const
	Determine whether this is an identity Quaternion.

bool	IsUnitQuaternion () const
	Determine whether this is a unit Quaternion.

Vector< T, 4 >	AsVector () const
	Convert to Vector form.

Vector< T, 3 >	Rotate (Vector< T, 3 > vr) const
	Rotate Vector vr about this Quaternion.

Vector< T, 3 >	Euler () const
	Return Euler angles for this Quaternion.

Quaternion	operator+ (const Quaternion< T > &other) const
	Quaternion addition.

Quaternion	operator- (const Quaternion< T > &other) const
	Quaternion subtraction.

Quaternion	operator* (const T k) const
	Scalar multiplication.

Quaternion	operator/ (const T k) const
	Scalar division.

Quaternion	operator* (const Vector< T, 3 > &vector) const
	Quaternion Hamilton multiplication with a three- dimensional vector.

Quaternion	operator* (const Quaternion< T > &other) const
	Quaternion Hamilton multiplication.

bool	operator== (const Quaternion< T > &other) const
	Quaternion equivalence.

bool	operator!= (const Quaternion< T > &other) const
	Quaternion non-equivalence.

Friends
std::ostream &	operator<< (std::ostream &outs, const Quaternion< T > &q)
	Output a Quaternion to a stream in the form `a + <i, j, k>`.

Related Symbols
(Note that these are not member symbols.)
Quaternionf	MakeQuaternion (Vector3F axis, float angle)
	Convenience Quaternion construction function.

Quaterniond	MakeQuaternion (Vector3D axis, double angle)
	Convience Quaternion construction function.

template<typename T >
Quaternion< T >	MakeQuaternion (Vector< T, 3 > axis, T angle)
	Convience Quaternion construction function.

Quaternionf	FloatQuaternionFromEuler (Vector3F euler)
	COnstruct a Quaternion from Euler angles.

Quaterniond	DoubleQuaternionFromEuler (Vector3D euler)
	COnstruct a Quaternion from Euler angles.

Detailed Description

template<typename T>
class scmp::geom::Quaternion< T >

Quaternions provide a representation of Orientation and rotations in three dimensions.

Quaternions encode rotations in three-dimensional space. While technically a MakeQuaternion is comprised of a real element and a complex vector<3>, for the purposes of this library, it is modeled as a floating point 4D vector of the form <w, x, y, z>, where x, y, and z represent an Axis of rotation in R3 and w the Angle, in radians, of the rotation about that Axis. Where Euler angles are concerned, the ZYX (or yaw, pitch, roll) sequence is used.

For information on the underlying vector type, see the documentation for scmp::geom::Vector.

Like vectors, quaternions carry an internal tolerance value ε that is used for floating point comparisons. The math namespace contains the default values used for this; generally, a tolerance of 0.0001 is considered appropriate for the uses of this library. The tolerance can be explicitly set with the SetEpsilon method.

Constructor & Destructor Documentation

◆ Quaternion() [1/4]

template<typename T >

scmp::geom::Quaternion< T >::Quaternion ( )

inline

Construct an identity Quaternion.

◆ Quaternion() [2/4]

template<typename T >

scmp::geom::Quaternion< T >::Quaternion	(	Vector< T, 3 >	_axis,
		T	_angle
	)

inline

Construct a Quaternion with an Axis and Angle of rotation.

A Quaternion may be initialised with a Vector<T, 3> Axis of rotation and an Angle of rotation. This doesn't do the Angle transforms to simplify internal operations.

Parameters

_axis	A three-dimensional vector of the same type as the Quaternion.
_angle	The Angle of rotation about the Axis of rotation.

◆ Quaternion() [3/4]

template<typename T >

scmp::geom::Quaternion< T >::Quaternion ( Vector< T, 4 > vector )

inline

A Quaternion may be initialised with a Vector<T, 4> comprised of the Axis of rotation followed by the Angle of rotation.

Parameters

vector A vector in the form <w, x, y, z>.

◆ Quaternion() [4/4]

template<typename T >

scmp::geom::Quaternion< T >::Quaternion ( std::initializer_list< T > ilst )

inline

An initializer list containing values for w, x, y, and z.

Parameters

ilst	An initial set of values in the form <w, x, y, z>.

Member Function Documentation

◆ Angle()

template<typename T >

T scmp::geom::Quaternion< T >::Angle ( ) const

inline

Return the Angle of rotation of this Quaternion.

Returns: the Angle of rotation of this Quaternion.

◆ AsVector()

template<typename T >

Vector< T, 4 > scmp::geom::Quaternion< T >::AsVector ( ) const

inline

Convert to Vector form.

Return the Quaternion as a Vector<T, 4>, with the Axis of rotation followed by the Angle of rotation.

Returns: A vector representation of the Quaternion.

◆ Axis()

template<typename T >

Vector< T, 3 > scmp::geom::Quaternion< T >::Axis ( ) const

inline

Return the Axis of rotation of this Quaternion.

Returns: The Axis of rotation of this Quaternion.

◆ Conjugate()

template<typename T >

Quaternion scmp::geom::Quaternion< T >::Conjugate ( ) const

inline

Compute the Conjugate of a Quaternion.

Returns: The Conjugate of this Quaternion.

◆ Dot()

template<typename T >

T scmp::geom::Quaternion< T >::Dot ( const Quaternion< T > & other ) const

inline

Compute the Dot product of two quaternions.

Parameters

other Another Quaternion.

Returns: The Dot product between the two quaternions.

◆ Euler()

template<typename T >

Vector< T, 3 > scmp::geom::Quaternion< T >::Euler ( ) const

inline

Return Euler angles for this Quaternion.

Return the Euler angles for this Quaternion as a vector of <yaw, pitch, roll>.

Warning: Users of this function should watch out for gimbal lock.

Returns: A vector<T, 3> containing <yaw, pitch, roll>

◆ Inverse()

template<typename T >

Quaternion scmp::geom::Quaternion< T >::Inverse ( ) const

inline

Compute the Inverse of a Quaternion.

Returns: The Inverse of this Quaternion.

◆ IsIdentity()

template<typename T >

bool scmp::geom::Quaternion< T >::IsIdentity ( ) const

inline

Determine whether this is an identity Quaternion.

Returns: true if this is an identity Quaternion.

◆ IsUnitQuaternion()

template<typename T >

bool scmp::geom::Quaternion< T >::IsUnitQuaternion ( ) const

inline

Determine whether this is a unit Quaternion.

Returns: true if this is a unit Quaternion.

◆ Norm()

template<typename T >

T scmp::geom::Quaternion< T >::Norm ( ) const

inline

Compute the Norm of a Quaternion.

Treating the Quaternion as a Vector<T, 4>, this is the same process as computing the Magnitude.

Returns: A non-negative real number.

◆ operator!=()

template<typename T >

bool scmp::geom::Quaternion< T >::operator!= ( const Quaternion< T > & other ) const

inline

Quaternion non-equivalence.

Parameters

other The Quaternion to check inequality against.

Returns: True if the two quaternions are unequal within their tolerance.

◆ operator*() [1/3]

template<typename T >

Quaternion scmp::geom::Quaternion< T >::operator* ( const Quaternion< T > & other ) const

inline

Quaternion Hamilton multiplication.

Parameters

other The other Quaternion to multiply with this one.

Returns: The Hamilton product of the two quaternions.

◆ operator*() [2/3]

template<typename T >

Quaternion scmp::geom::Quaternion< T >::operator* ( const T k ) const

inline

Scalar multiplication.

Parameters

k	The scaling value.

Returns: A scaled Quaternion.

◆ operator*() [3/3]

template<typename T >

Quaternion scmp::geom::Quaternion< T >::operator* ( const Vector< T, 3 > & vector ) const

inline

Quaternion Hamilton multiplication with a three- dimensional vector.

This is done by treating the vector as a pure Quaternion (e.g. with an Angle of rotation of 0).

Parameters

vector The vector to multiply with this Quaternion.

Returns: The Hamilton product of the Quaternion and vector.

◆ operator+()

template<typename T >

Quaternion scmp::geom::Quaternion< T >::operator+ ( const Quaternion< T > & other ) const

inline

Quaternion addition.

Parameters

other The Quaternion to be added with this one.

Returns: The result of adding the two quaternions together.

◆ operator-()

template<typename T >

Quaternion scmp::geom::Quaternion< T >::operator- ( const Quaternion< T > & other ) const

inline

Quaternion subtraction.

Parameters

other The Quaternion to be subtracted from this one.

Returns: The result of subtracting the other Quaternion from this one.

◆ operator/()

template<typename T >

Quaternion scmp::geom::Quaternion< T >::operator/ ( const T k ) const

inline

Scalar division.

Parameters

k	The scalar divisor.

Returns: A scaled Quaternion.

◆ operator==()

template<typename T >

bool scmp::geom::Quaternion< T >::operator== ( const Quaternion< T > & other ) const

inline

Quaternion equivalence.

Parameters

other The Quaternion to check equality against.

Returns: True if the two quaternions are equal within their tolerance.

◆ Rotate()

template<typename T >

Vector< T, 3 > scmp::geom::Quaternion< T >::Rotate ( Vector< T, 3 > vr ) const

inline

Rotate Vector vr about this Quaternion.

Parameters

vr	The vector to be rotated.

Returns: The rotated vector.

◆ SetEpsilon()

template<typename T >

void scmp::geom::Quaternion< T >::SetEpsilon ( T epsilon )

inline

Set the comparison tolerance for this Quaternion.

Parameters

epsilon A tolerance value.

◆ UnitQuaternion()

template<typename T >

Quaternion scmp::geom::Quaternion< T >::UnitQuaternion ( )

inline

Return the unit Quaternion.

Returns: The unit Quaternion.

Friends And Related Symbol Documentation

◆ DoubleQuaternionFromEuler()

template<typename T >

Quaterniond DoubleQuaternionFromEuler ( Vector3D euler )

Given a vector of Euler angles in ZYX sequence (e.g. yaw, pitch, roll), return a Quaternion.

Parameters

euler A vector Euler Angle in ZYX sequence.

Returns: A Quaternion representation of the Orientation represented by the Euler angles.

◆ FloatQuaternionFromEuler()

template<typename T >

Quaternionf FloatQuaternionFromEuler ( Vector3F euler )

Given a vector of Euler angles in ZYX sequence (e.g. yaw, pitch, roll), return a Quaternion.

Parameters

euler A vector Euler Angle in ZYX sequence.

Returns: A Quaternion representation of the Orientation represented by the Euler angles.

◆ MakeQuaternion() [1/3]

template<typename T >

Quaterniond MakeQuaternion	(	Vector3D	axis,
		double	angle
	)

Return a double Quaternion scaled appropriately from a vector and Angle, e.g. Angle = cos(Angle / 2), Axis.UnitVector() * sin(Angle / 2).

Parameters

axis	The Axis of rotation.
angle	The Angle of rotation.

Returns: A Quaternion.

◆ MakeQuaternion() [2/3]

template<typename T >

Quaternionf MakeQuaternion	(	Vector3F	axis,
		float	angle
	)

Return a float Quaternion scaled appropriately from a vector and Angle, e.g. angle = cos(Angle / 2), Axis.UnitVector() * sin(Angle / 2).

Parameters

axis	The Axis of rotation.
angle	The Angle of rotation.

Returns: A Quaternion.

◆ MakeQuaternion() [3/3]

template<typename T >

Quaternion< T > MakeQuaternion	(	Vector< T, 3 >	axis,
		T	angle
	)

Return a double Quaternion scaled appropriately from a vector and Angle, e.g. Angle = cos(Angle / 2), Axis.UnitVector() * sin(Angle / 2).

Parameters

axis	The Axis of rotation.
angle	The Angle of rotation.

Returns: A Quaternion.

◆ operator<<

template<typename T >

std::ostream & operator<<	(	std::ostream &	outs,
		const Quaternion< T > &	q
	)

friend

Output a Quaternion to a stream in the form a + <i, j, k>.

Todo:: improve the formatting.

Parameters

outs	An output stream
q	A Quaternion

Returns: The output stream

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

include/scmp/geom/Quaternion.h

Public Member Functions

Friends

Related Symbols

Detailed Description

Constructor & Destructor Documentation

◆ Quaternion() [1/4]

◆ Quaternion() [2/4]

◆ Quaternion() [3/4]

◆ Quaternion() [4/4]

Member Function Documentation

◆ Angle()

◆ AsVector()

◆ Axis()

◆ Conjugate()

◆ Dot()

◆ Euler()

◆ Inverse()

◆ IsIdentity()

◆ IsUnitQuaternion()

◆ Norm()

◆ operator!=()

◆ operator*() [1/3]

◆ operator*() [2/3]

◆ operator*() [3/3]

◆ operator+()

◆ operator-()

◆ operator/()

◆ operator==()

◆ Rotate()

◆ SetEpsilon()

◆ UnitQuaternion()

Friends And Related Symbol Documentation

◆ DoubleQuaternionFromEuler()

◆ FloatQuaternionFromEuler()

◆ MakeQuaternion() [1/3]

◆ MakeQuaternion() [2/3]

◆ MakeQuaternion() [3/3]

◆ operator<<