Quaternion Class Reference

The Quaternion class encapsulates quaternions (quelle surprise!), and provides a small set of operations for manipulating them. More...

#include <quaternion.h>

Public Types
enum	basis_quaternion { Unit, I, J, K }
	The set of basis quaternions. More...

Public Member Functions
	Quaternion ()
	The default constructor creates a K quaternion. More...
	Quaternion (const double q0, const double q1, const double q2, const double q3)
	The constructor creates a quaternion from four coefficients of the basis quaternions. More...
	Quaternion (const basis_quaternion)
	Produce one of the basis quaternions.
	Quaternion (const Position &)
	Create a quaternion vector which points in the direction of the given position.
Quaternion	operator- () const
	Produce a new quaternion which is the negative of this one.
Quaternion	operator+ (const Quaternion &) const
	Produces the sum of two quaternions.
Quaternion	operator- (const Quaternion &) const
	Produces the difference of two quaternions.
Quaternion	operator* (const Quaternion &) const
	Performs the quaternion product of two quaternions.
Quaternion	scale (double factor) const
	Return quaternion scaled by real factor.
Quaternion	rotate (const Quaternion &) const
	'Rotate' the quaternion. More...
Quaternion	reflect (const Quaternion &) const
	Reflect the quaternion through the line generated by the argument. More...
Quaternion	conj (void) const
	Returns a new quaternion which is the conjugate of this one.
const double *	cpts (void) const
	The components of the quaternion, as an array of doubles, in radians.
Position	coords (Position::System system=Position::TopoPolar, Position::System from_system=Position::TopoPolar) const
	Obtains the direction of a quaternion, by decomposing it into polar angles. More...
Quaternion	cross (const Quaternion &, bool norm=false) const
	Return the vector product of this quaternion with the argument. More...
double	dot (const Quaternion &) const
	Return scalar product of two quaternions in Vec(H). More...
Quaternion	rotation (const double angle) const
	Produce a quaternion which specifies a rotation about the current quaternion, by the given angle.
bool	is_unit_p (void)
	Test whether a quaternion is a unit quaternion. More...

Static Public Member Functions
static Quaternion	vector (const double longitude, double latitude)
	Creates a new vector quaternion which is pointing in the direction of the given (east-)longitude and latitude. More...
static const char *	coords_str (const Quaternion &, Position::System system=Position::TopoPolar, bool sexagesimal_p=true)
	Returns a pointer to a static buffer containing a formatted representation of the quaternion. More...

Detailed Description

The Quaternion class encapsulates quaternions (quelle surprise!), and provides a small set of operations for manipulating them.

The directions in this module are described either in terms of the basis quaternions 1, i, j, k, or in terms of longitude [0..2pi] and latitude [-pi/2..+pi/2], where the direction i is (long,lat) = (0,0), j is (pi/2, 0) and k is (0, pi/2). The longitude coordinate therefore increases from i to j (ie, moving eastwards).

Although it doesn't much matter to this class, when viewed as a topocentric coordinate system, i should be taken as pointing south, and j east.

The arithmetic operations in this class return new constant objects, rather than mutating existing ones.

Member Enumeration Documentation

enum Quaternion::basis_quaternion

The set of basis quaternions.

See also

Quaternion(const basis_quaternion)

Enumerator
Unit	The unit quaternion (sometimes used as a sentinel value; see is_unit_p())
I	the basis quaternion in the i direction
J	the basis quaternion in the j direction
K	the basis quaternion in the k direction

Constructor & Destructor Documentation

Quaternion::Quaternion

(

)

The default constructor creates a K quaternion.

Creating a Unit quaternion might be more intuitive, but since this isn't a vector, it has the potential to cause lots of trouble if used accidentally.

Quaternion::Quaternion	(	const double	q0,
		const double	q1,
		const double	q2,
		const double	q3
	)

The constructor creates a quaternion from four coefficients of the basis quaternions.

This is the only constructor that can create a non-unit quaternion.

Member Function Documentation

Position Quaternion::coords	(	Position::System	system = Position::TopoPolar,
		Position::System	from_system = Position::TopoPolar
	)		const

Obtains the direction of a quaternion, by decomposing it into polar angles.

The coordinates are returned as part of a Position object, so the specific result coordinate system may be chosen using the arguments.

Parameters

system	the system in which the result angles are to be expressed
from_system	the system in which the quaternion's rectangular components should be taken to be expressed

See also

Position::Position(double,double,Position::System,Position::System)

const char * Quaternion::coords_str ( const Quaternion &  q, Position::System  system = Position::TopoPolar, bool  sexagesimal_p = true  )

static

Returns a pointer to a static buffer containing a formatted representation of the quaternion.

Parameters

q	the quaternion to be formatted
system	the system to be used for the formatted output
sexagesimal_p	if true, output as sexagesimal, if false, in radians

Returns

a pointer to a common static buffer

Quaternion Quaternion::cross	(	const Quaternion &	arg,
		bool	norm = false
	)		const

Return the vector product of this quaternion with the argument.

We assume, but don't check, that the both the current and the argument quaternion are in Vec(H).

Parameters

arg	the quaternion to be crossed with this one (on the right)
norm	if true, the result is normalised to unit length

double Quaternion::dot

(

const Quaternion &

arg

)

const

Return scalar product of two quaternions in Vec(H).

We assume, but don't check, that the both the current and the argument quaternion are in Vec(H).

bool Quaternion::is_unit_p

(

void

)

Test whether a quaternion is a unit quaternion.

That is, test whether it is (1, 0, 0, 0). We sometimes use this quaternion, generated by Quaternion(const basis_quaternion), as a sentinel value.

Quaternion Quaternion::reflect

(

const Quaternion &

rq

)

const

Reflect the quaternion through the line generated by the argument.

This is equivalent to a rotation of pi radians around the argument, which is assumed to be a unit vector. The result of this method should be equivalent to the application of Quaternion::rotate, but should be faster because of the assumption that the argument is a vector.

Parameters

rq	the quaternion which generates the rotation, which is assumed to be a unit vector

Quaternion Quaternion::rotate

(

const Quaternion &

rq

)

const

'Rotate' the quaternion.

Given a quaternion, r, and taking the current quaternion to be A, return r*A*r.conj. This requires 23 multiplications and 24 additions, as opposed to 32 multiplications and 24 additions when done the 'long' way. This should also be more accurate.

Note that this is different from the rotation method, which constructs a quaternion which generates a rotation.

See quaternion-pointing notes, at note rotation.

Parameters

rq	the quaternion which generates the rotation, which is assumed to be a unit quaternion

Quaternion Quaternion::vector ( const double  longitude, double  latitude  )

static

Creates a new vector quaternion which is pointing in the direction of the given (east-)longitude and latitude.

Both of the angles are given in 'polar' radians, thus east of south.

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

• quaternion.h
• quaternion.cpp