4-dimensional normed division algebra over the Real Numbers

Every quaternion is uniquely expressible as a Real linear combination of basis quaternions \( 1, i, j, k \) satisfying the relation:

\[ i^{2} = j^{2} = k^{2} = ijk = -1 \label{ eq. 1 } \]

Quaternions satisfy vector space axioms over the Reals. The set \( \mathbf{H} \) of all quaternions has dimension 4. However, unlike Real and Complex numbers, multiplication of quaternions is not commutative. However, multiplication associates, and every non-zero element has a unique inverse.

The set of equations (eq. 1) is the fundamental formula for quaternion multiplication. The multiplication table of basis quaternions is easily derived from the relation \( ijk = -1 \).

Given quaternion set \( \mathbf{H} \), Real scalars \( a, b, c, d \), and quaternion \( z = a + bi + cj + dk \) the following additional definitions apply:

Conjugate of \( \mathit{z} \) is \( z^{*} = a - bi - cj - dk \)
Absolute value (magnitude) of \( \mathit{z} \) is the non-negative Real scalar value

\[ \vert z \vert = (zz^{*})^{\frac{1}{2}} = \sqrt{a^{2} + b^{2} + c^{2} + d^{2}} \]
Multiplicative inverse (reciprocal) \( \mathit{z} \) can be computed as

\[ \frac{1}{z} = \frac{z^{*}}{\vert z\vert^{2}} = \frac{z^{*}}{zz^{*}} \]
Given two distinct quaternions, \( q_1 = (a + \vec{u}) \) and \( q_2 = (b + \vec{v}) \), the product is not generally commutable and is defined as

\[ q_1 q_2 = (ab - \langle \vec{u}, \vec{v} \rangle) + (a\vec{v} + b\vec{u} + \vec{u} \times \vec{v}) \]
Let \( z = \cos{(\frac{\alpha}{2})} + \sin{(\frac{\alpha}{2})}\vec{U} \) be a unit quaternion. The mapping \( f(\vec{x}) = z\vec{x}z^{*} \) is known as conjugation by \( \mathit{z}, \) where the vector \( \vec{x} \) is considered a quaternion with its scalar component equal to zero. \( f(\vec{x}) \) rotates \( \vec{x} \) counterclockwise through an angle \( \alpha \) about an axis \( \vec{U} \). The composition of two rotations corresponds to quaternion multiplication.

#include <quaternion.h>

Public Member Functions
const bool	operator!= (const quaternion &q) const

const quaternion	operator* (const quaternion &q) const
	quaternion multiplication (Grassmann Product) More...

const quaternion	operator* (const scalar &s) const

const quaternion	operator+ (const quaternion &q) const

const quaternion	operator- (const quaternion &q) const

const quaternion	operator/ (const scalar &s) const

const bool	operator== (const quaternion &q) const

class construction and type conversion
	quaternion ()

	quaternion (const scalar &s, const vector3d &v)

	quaternion (const scalar &q1, const scalar &q2, const scalar &q3, const scalar &q4)

class public methods
/**
const quaternion	conjugate () const

const vector3d	cross (const quaternion &q) const

const scalar	dot (const quaternion &q) const
	dot-product (Euclidean inner-product) More...

const scalar	length () const
	modulus (absolute value or length from origin) More...

void	normalize ()

const quaternion	unit () const

const scalar &	Scalar () const

const vector3d &	Vector () const

const quaternion	sgn () const
	Sign, sgn(z), of a complex number finds the complex number of the same direction found on the unit circle. More...

const scalar	arg () const
	Argument, arg(z), finds the angle of the 4-vector quaternion from the unit scalar (i.e. More...

class member operators
const quaternion &	operator*= (const quaternion &q)

const quaternion &	operator+= (const quaternion &q)

const quaternion &	operator-= (const quaternion &q)

const quaternion &	operator*= (const scalar &s)

const quaternion &	operator/= (const scalar &s)

Friends
const quaternion	operator* (const scalar &s, const quaternion &q)

const quaternion	operator- (const quaternion &q)

math::quaternion::quaternion ( )

inline

math::quaternion::quaternion	(	const scalar &	s,
		const vector3d &	v
	)

inline

math::quaternion::quaternion	(	const scalar &	q1,
		const scalar &	q2,
		const scalar &	q3,
		const scalar &	q4
	)

inline

const scalar math::quaternion::arg ( ) const

inline

Argument, arg(z), finds the angle of the 4-vector quaternion from the unit scalar (i.e.

1).

const quaternion math::quaternion::conjugate ( ) const

inline

const vector3d math::quaternion::cross ( const quaternion & q ) const

inline

const scalar math::quaternion::dot ( const quaternion & q ) const

inline

dot-product (Euclidean inner-product)

const scalar math::quaternion::length ( ) const

inline

modulus (absolute value or length from origin)

void math::quaternion::normalize ( )

inline

const bool math::quaternion::operator!= ( const quaternion & q ) const

inline

const quaternion math::quaternion::operator* ( const quaternion & q ) const

inline

quaternion multiplication (Grassmann Product)

const quaternion math::quaternion::operator* ( const scalar & s ) const

inline

const quaternion& math::quaternion::operator*= ( const quaternion & q )

inline

const quaternion& math::quaternion::operator*= ( const scalar & s )

inline

const quaternion math::quaternion::operator+ ( const quaternion & q ) const

inline

const quaternion& math::quaternion::operator+= ( const quaternion & q )

inline

const quaternion math::quaternion::operator- ( const quaternion & q ) const

inline

const quaternion& math::quaternion::operator-= ( const quaternion & q )

inline

const quaternion math::quaternion::operator/ ( const scalar & s ) const

inline

const quaternion& math::quaternion::operator/= ( const scalar & s )

inline

const bool math::quaternion::operator== ( const quaternion & q ) const

inline

const scalar& math::quaternion::Scalar ( ) const

inline

const quaternion math::quaternion::sgn ( ) const

inline

Sign, sgn(z), of a complex number finds the complex number of the same direction found on the unit circle.

const quaternion math::quaternion::unit ( ) const

inline

const vector3d& math::quaternion::Vector ( ) const

inline

const quaternion operator*	(	const scalar &	s,
		const quaternion &	q
	)

friend

const quaternion operator- ( const quaternion & q )

friend

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Public Member Functions

Friends