Quaternion Class Reference

Quaternions. More...

#include <quaternion.h>

Public Member Functions
	Quaternion ()
	Empty.
	Quaternion (const double &, const double &, const double &, const double &)
	Create a quaternion.
	Quaternion (const Vector &, const double &)
	Create a quaternion from an axis vector and a rotation angle.
	Quaternion (const Vector &, const Vector &)
	Create a quaternion from two (normalized) vectors representing the rotation.
	Quaternion (const Matrix &)
	Create a quaternion from a rotation matrix.
Matrix	RotationMatrix () const
	Transforms a quaternion into a rotation matrix.
void	AngleAxis (double &, Vector &) const
	Transform a quaternion into an rotation defined by its axis and the angle of rotation angle.
double &	operator[] (int)
	Overloaded.
constexpr double	operator[] (int) const
	Returns the i-th coefficient of a quaternion.
Quaternion	operator+ (const Quaternion &) const
	Overloaded.
Quaternion	operator- (const Quaternion &) const
	Overloaded.
Quaternion	operator* (const Quaternion &) const
	Multiplication of two quaternion.
Quaternion	operator* (const double &) const
	Multiply a quaternion by a double value.
Quaternion	operator- () const
	Overloaded.
Quaternion	Inverse () const
	Compute the inverse of a quaternion.
Quaternion	Conjugate () const
	Computes the conjugatge of a quaternion.
Quaternion	Exp () const
	Compute the exponential of a quaternion.
Quaternion	Log () const
	Computes the logarithm of a quaternion.
Vector	operator* (const Vector &) const
	Compute the product between a quaternion and a vector.

Static Public Member Functions
static Quaternion	Lerp (const double &, const Quaternion &, const Quaternion &)
	Perform quaternion spherical interpolation, also known as slerping.
static Quaternion	FromAngles (const double &, const double &, const double &)
	Create a quaternion from Euler rotation angles.

Static Public Attributes
static const Quaternion	Null
	Null quaternion.
static const Quaternion	Identity
	Identity.

Protected Attributes
double	z = 0.0
	Quaternion components.

Static Protected Attributes
static const double	Epsilon = 1e-03
	Cutoff value for sine function whenever the angle gets near zero.

Friends
Quaternion	operator* (const double &c, const Quaternion &q)
	Left multiply a quaternion by a double value.
double	Norm (const Quaternion &q)
	Squared length of a quaternion.
std::ostream &	operator<< (std::ostream &s, const Quaternion &quaternion)
	Overloaded.

Detailed Description

Quaternions.

Constructor & Destructor Documentation

Quaternion() [1/4]

Quaternion::Quaternion ( const double & w, const double & x, const double & y, const double & z )

inlineexplicit

Create a quaternion.

Parameters

w	Angular parameter.
x,y,z	Axis coordinates.

Quaternion() [2/4]

Quaternion::Quaternion ( const Vector & axis, const double & a )

explicit

Create a quaternion from an axis vector and a rotation angle.

Axis should be unit length.

Parameters

axis	Normalized axis.
a	Angle.

Quaternion() [3/4]

Quaternion::Quaternion ( const Vector & a, const Vector & b )

explicit

Create a quaternion from two (normalized) vectors representing the rotation.

Simply coded as:

Vector a,b; // Should be unit
Quaternion q(a/b,acos(a*b));

Parameters

a,b	Vectors (should be unit).

Quaternion() [4/4]

Quaternion::Quaternion ( const Matrix & R )

explicit

Create a quaternion from a rotation matrix.

The algorithm can be found in Ken Shoemake's article in 1987 Siggraph course notes Quaternion Calculus and Fast Animation.

Parameters

R	Rotation matrix.

Member Function Documentation

AngleAxis()

void Quaternion::AngleAxis	(	double &	angle,
		Vector &	axis ) const

Transform a quaternion into an rotation defined by its axis and the angle of rotation angle.

The quaternion representing the rotation is : q=cos(a/2)+sin(a/2)*(x*i+y*j+z*k).

Parameters

angle	The angle.
axis	Vector axis.

Exp()

Quaternion Quaternion::Exp

(

)

const

Compute the exponential of a quaternion.

If q=a*(x,y,z) where (x,y,z) is unit length, then exp(q)=cos(a)+sin(a)*(x,y,z). If sin(a) is near zero, use exp(a)=cos(a)+a*(x,y,z).

FromAngles()

Quaternion Quaternion::FromAngles ( const double & a, const double & b, const double & c )

static

Create a quaternion from Euler rotation angles.

Using this function is simpler than calling the constructor from a rotation matrix.

Parameters

a,b,c

Rotation angles around x, y and z axes (should be in radian).

Inverse()

Quaternion Quaternion::Inverse

(

)

const

Compute the inverse of a quaternion.

This should be applied to non-zero quaternions only. If quaternion happens to be null, then the function return also null wich is an invalid result that flags the error.

Lerp()

Quaternion Quaternion::Lerp ( const double & t, const Quaternion & p, const Quaternion & q )

static

Perform quaternion spherical interpolation, also known as slerping.

The dot product of argument quaternions should be positive.

Parameters

p,q	Argument quaternions.
t	Interpolating value.

Log()

Quaternion Quaternion::Log

(

)

const

Computes the logarithm of a quaternion.

If quaternion is of unit length, then q=cos(a)+sin(a).v, then log(q)=a.v.

operator*() [1/3]

Quaternion Quaternion::operator* ( const double & c ) const

inline

Multiply a quaternion by a double value.

Parameters

c

Real.

operator*() [2/3]

Quaternion Quaternion::operator*

(

const Quaternion &

q

)

const

Multiplication of two quaternion.

Note that multiplication is not generally commutative.

operator*() [3/3]

Vector Quaternion::operator*

(

const Vector &

p

)

const

Compute the product between a quaternion and a vector.

This member calls Quaternion::RotationMatrix(). As such, the quaternion representation of a rotation matrix requires less space than the matrix and more time to compute the rotated vector. This is a typical space-time tradeoff.

operator[]()

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

inlineconstexpr

Returns the i-th coefficient of a quaternion.

Terms are x, y, and z if index is in [0,2], and w otherwise.

Friends And Related Symbol Documentation

Norm

double Norm ( const Quaternion & q )

friend

Squared length of a quaternion.

Parameters

q	Quaternion.

operator*

Quaternion operator* ( const double & c, const Quaternion & q )

friend

Left multiply a quaternion by a double value.

Parameters

c	Real.
q	Quaternion.

operator<<

std::ostream & operator<< ( std::ostream & s, const Quaternion & quaternion )

friend

Overloaded.

Parameters

s	Stream.
quaternion	The quaternion.