#include <math/Quaternion.hpp>

Public Types
Types
using	value_type = T
	The underlying implementation type.

Public Member Functions
constexpr BasicQuaternion< T >	conjugate () const

BasicQuaternion< T >	pow (T exponent) const
	Computes this BasicQuaternion raised to a real power.

BasicQuaternion< T >	exp () const

BasicQuaternion< T >	log () const

T	normSquared () const

T	norm () const

T	magnitudeSquared () const

T	magnitude () const

constexpr BasicQuaternion< T >	normalized () const

BasicQuaternion< T >	inverse () const

BasicRadian< T >	angle () const

constexpr BasicVector3D< T >	axis () const

bool	isUnit () const

bool	isUnit (T tolerance) const

bool	isZero () const

bool	isZero (T tolerance) const

bool	isPure () const

Constructors
constexpr	BasicQuaternion ()=default

constexpr	BasicQuaternion (T real_number)
	Constructs a BasicQuaternion equivalent to the given real number.

constexpr	BasicQuaternion (T w, T i, T j, T k)

Element Access
constexpr const T &	w () const

constexpr const T &	real () const

constexpr const T &	i () const

constexpr const T &	j () const

constexpr const T &	k () const

constexpr BasicVector3D< T >	imaginary () const
	Extracts the imaginary part of a BasicQuaternion as a BasicVector3D.

Invalid Value Check
bool	isNaN () const

bool	isInf () const

Static Public Member Functions
Constants
static constexpr BasicQuaternion< T >	identity ()
	BasicQuaternion representation of the real number 1.

static constexpr BasicQuaternion< T >	zero ()
	BasicQuaternion representation of the real number 0.

static constexpr BasicQuaternion< T >	unit_real ()

static constexpr BasicQuaternion< T >	unit_i ()

static constexpr BasicQuaternion< T >	unit_j ()

static constexpr BasicQuaternion< T >	unit_k ()

Convenience Creation Functions
static constexpr BasicQuaternion< T >	make_pure (T x, T y, T z)

static constexpr BasicQuaternion< T >	make_pure (const BasicVector3D< T > &t)

static constexpr BasicQuaternion< T >	encode_point (T x, T y, T z)

static constexpr BasicQuaternion< T >	encode_point (const BasicVector3D< T > &point)

static constexpr BasicQuaternion< T >	make_rotation (const BasicRadian< T > &radians, T axis_x, T axis_y, T axis_z)

static constexpr BasicQuaternion< T >	make_rotation (const BasicRadian< T > &radians, const BasicVector3D< T > &axis)

Friends
Equality
constexpr bool	operator== (const BasicQuaternion< T > &left, const BasicQuaternion< T > &right)

template<std::floating_point OT = T> requires std::is_floating_point_v<T>
constexpr bool	approximately_equal_to (const BasicQuaternion< T > &value_to_test, const BasicQuaternion< T > &value_it_should_be, OT tolerance=OT{0.0002})

Operators
constexpr BasicQuaternion< T >	operator* (const BasicQuaternion< T > &left, const BasicQuaternion< T > &right)

template<std::floating_point OT>
constexpr BasicQuaternion< T >	operator* (const BasicQuaternion< T > &quaternion, OT scalar)

constexpr BasicQuaternion< T >	operator/ (const BasicQuaternion< T > &left, const BasicQuaternion< T > &right)

template<std::floating_point OT>
constexpr BasicQuaternion< T >	operator/ (const BasicQuaternion< T > &quaternion, OT scalar)

constexpr BasicQuaternion< T >	operator+ (const BasicQuaternion< T > &left, const BasicQuaternion< T > &right)

constexpr BasicQuaternion< T >	operator- (const BasicQuaternion< T > &left, const BasicQuaternion< T > &right)

constexpr BasicQuaternion< T >	operator- (const BasicQuaternion< T > &q)

Check
template<std::floating_point OT = T> requires std::is_floating_point_v<T>
bool	check_if_equal (const BasicQuaternion< T > &input, const BasicQuaternion< T > &near_to, OT tolerance=OT{0.0002})

template<std::floating_point OT = T> requires std::is_floating_point_v<T>
bool	check_if_not_equal (const BasicQuaternion< T > &input, const BasicQuaternion< T > &near_to, OT tolerance=OT{0.0002})

Assert
template<std::floating_point OT = T> requires std::is_floating_point_v<T>
void	CHECK_IF_EQUAL (const BasicQuaternion< T > &input, const BasicQuaternion< T > &near_to, OT tolerance=OT{0.0002})

template<std::floating_point OT = T> requires std::is_floating_point_v<T>
void	CHECK_IF_NOT_EQUAL (const BasicQuaternion< T > &input, const BasicQuaternion< T > &near_to, OT tolerance=OT{0.0002})

template<std::floating_point OT = T> requires std::is_floating_point_v<T>
void	CHECK_IF_ZERO (const BasicQuaternion< T > &input, OT tolerance=OT{0.0002})

Global Functions
constexpr T	dot (const BasicQuaternion< T > &left, const BasicQuaternion< T > &right)

constexpr T	dot_normalized (const BasicQuaternion< T > &left, const BasicQuaternion< T > &right)

constexpr BasicQuaternion< T >	passively_rotate_encoded_point (const BasicQuaternion< T > &rotation, const BasicQuaternion< T > &encoded_point)

constexpr BasicQuaternion< T >	locally_rotate_encoded_point (const BasicQuaternion< T > &rotation, const BasicQuaternion< T > &encoded_point)

constexpr BasicQuaternion< T >	actively_rotate_encoded_point (const BasicQuaternion< T > &rotation, const BasicQuaternion< T > &encoded_point)

constexpr BasicQuaternion< T >	globally_rotate_encoded_point (const BasicQuaternion< T > &rotation, const BasicQuaternion< T > &encoded_point)

constexpr BasicQuaternion< T >	compose_rotations (const BasicQuaternion< T > &rotation_1, const BasicQuaternion< T > &rotation_2)

constexpr BasicQuaternion< T >	normalized (const BasicQuaternion< T > &input)

constexpr T	arg (const BasicQuaternion< T > &input)

constexpr BasicQuaternion< T >	polar (const BasicVector3D< T > &axis, const BasicRadian< T > angle=BasicRadian< T >{})

constexpr T	accumulate (const BasicQuaternion< T > &input)

constexpr BasicQuaternion< T >	slerp (const BasicQuaternion< T > &begin, const BasicQuaternion< T > &end, T percent)

std::string	format (const BasicQuaternion< T > &input)

constexpr BasicQuaternion< T >	conjugate (const BasicQuaternion< T > &input)

constexpr BasicQuaternion< T >	log (const BasicQuaternion< T > &input)

BasicQuaternion< T >	exp (const BasicQuaternion< T > &input)

Related Symbols
(Note that these are not member symbols.)
Type Aliases
using	Quaternionf = BasicQuaternion< float >

using	Quaterniond = BasicQuaternion< double >

using	Quaternion = BasicQuaternion< double >

using	Quaternionl = BasicQuaternion< long double >

Detailed Description

template<class T>
class Math::BasicQuaternion< T >

A mathematical quaternion

Note: A quaternion is a 4-dimensional object that is an extension of the complex number system. Its main use is for encoding rotation in 3-dimensional space.

Definition at line 31 of file Quaternion.hpp.

Member Typedef Documentation

◆ value_type

template<class T >

using value_type = T

The underlying implementation type.

Definition at line 37 of file Quaternion.hpp.

Constructor & Destructor Documentation

◆ BasicQuaternion() [1/3]

template<class T >

constexpr BasicQuaternion ( )

constexprdefault

◆ BasicQuaternion() [2/3]

template<class T >

constexpr BasicQuaternion ( T real_number )

inlineconstexpr

Constructs a BasicQuaternion equivalent to the given real number.

Definition at line 44 of file Quaternion.hpp.

◆ BasicQuaternion() [3/3]

template<class T >

constexpr BasicQuaternion	(	T	w,
		T	i,
		T	j,
		T	k
	)

inlineconstexpr

Definition at line 45 of file Quaternion.hpp.

Member Function Documentation

◆ angle()

template<class T >

BasicRadian< T > angle ( ) const

inline

Definition at line 153 of file Quaternion.hpp.

◆ axis()

template<class T >

constexpr BasicVector3D< T > axis ( ) const

inlineconstexpr

Definition at line 158 of file Quaternion.hpp.

◆ conjugate()

template<class T >

constexpr BasicQuaternion< T > conjugate ( ) const

inlineconstexpr

Definition at line 62 of file Quaternion.hpp.

◆ encode_point() [1/2]

template<class T >

static constexpr BasicQuaternion< T > encode_point ( const BasicVector3D< T > & point )

inlinestaticconstexpr

Encode a 3D point as a pure BasicQuaternion

Postcondition: output.w() == 0 output.real() == 0

Note: A pure BasicQuaternion is one in which the w, or real, component is 0.

See also: make_pure

Definition at line 250 of file Quaternion.hpp.

◆ encode_point() [2/2]

template<class T >

static constexpr BasicQuaternion< T > encode_point	(	T	x,
		T	y,
		T	z
	)

inlinestaticconstexpr

Encode a 3D point as a pure BasicQuaternion

Postcondition: output.w() == 0 output.real() == 0

Note: A pure BasicQuaternion is one in which the w, or real, component is 0.

See also: make_pure

Definition at line 238 of file Quaternion.hpp.

◆ exp()

template<class T >

BasicQuaternion< T > exp ( ) const

inline

Computes the exponential form of this BasicQuaternion

Note: It is possible for this routine to output a non-unit BasicQuaternion when given a unit BasicQuaternion as input. It is for this reason that the implementation of log() has been adjusted to automatically handle non-unit Quaternions.

Definition at line 102 of file Quaternion.hpp.

◆ i()

template<class T >

constexpr const T & i ( ) const

inlineconstexpr

Definition at line 169 of file Quaternion.hpp.

◆ identity()

template<class T >

static constexpr BasicQuaternion< T > identity ( )

inlinestaticconstexpr

BasicQuaternion representation of the real number 1.

Definition at line 52 of file Quaternion.hpp.

◆ imaginary()

template<class T >

constexpr BasicVector3D< T > imaginary ( ) const

inlineconstexpr

Extracts the imaginary part of a BasicQuaternion as a BasicVector3D.

Definition at line 174 of file Quaternion.hpp.

◆ inverse()

template<class T >

BasicQuaternion< T > inverse ( ) const

inline

Definition at line 151 of file Quaternion.hpp.

◆ isInf()

template<class T >

bool isInf ( ) const

inline

Definition at line 196 of file Quaternion.hpp.

◆ isNaN()

template<class T >

bool isNaN ( ) const

inline

Definition at line 189 of file Quaternion.hpp.

◆ isPure()

template<class T >

bool isPure ( ) const

inline

Definition at line 184 of file Quaternion.hpp.

◆ isUnit() [1/2]

template<class T >

bool isUnit ( ) const

inline

Definition at line 177 of file Quaternion.hpp.

◆ isUnit() [2/2]

template<class T >

bool isUnit ( T tolerance ) const

inline

Definition at line 178 of file Quaternion.hpp.

◆ isZero() [1/2]

template<class T >

bool isZero ( ) const

inline

Definition at line 180 of file Quaternion.hpp.

◆ isZero() [2/2]

template<class T >

bool isZero ( T tolerance ) const

inline

Definition at line 181 of file Quaternion.hpp.

◆ j()

template<class T >

constexpr const T & j ( ) const

inlineconstexpr

Definition at line 170 of file Quaternion.hpp.

◆ k()

template<class T >

constexpr const T & k ( ) const

inlineconstexpr

Definition at line 171 of file Quaternion.hpp.

◆ log()

template<class T >

BasicQuaternion< T > log ( ) const

inline

Computes the log base e of this BasicQuaternion

Note: We handle non-unit Quaternions in this version so that we can satisfy the relationship: log( exp( x ) ) == x

Definition at line 120 of file Quaternion.hpp.

◆ magnitude()

template<class T >

T magnitude ( ) const

inline

Definition at line 142 of file Quaternion.hpp.

◆ magnitudeSquared()

template<class T >

T magnitudeSquared ( ) const

inline

Definition at line 141 of file Quaternion.hpp.

◆ make_pure() [1/2]

template<class T >

static constexpr BasicQuaternion< T > make_pure ( const BasicVector3D< T > & t )

inlinestaticconstexpr

Construct a pure BasicQuaternion

Postcondition: output.w() == 0 output.real() == 0

Note: A pure BasicQuaternion is one in which the w, or real, component is 0.

Definition at line 226 of file Quaternion.hpp.

◆ make_pure() [2/2]

template<class T >

static constexpr BasicQuaternion< T > make_pure	(	T	x,
		T	y,
		T	z
	)

inlinestaticconstexpr

Construct a pure BasicQuaternion

Postcondition: output.w() == 0 output.real() == 0

Note: A pure BasicQuaternion is one in which the w, or real, component is 0.

Definition at line 216 of file Quaternion.hpp.

◆ make_rotation() [1/2]

template<class T >

static constexpr BasicQuaternion< T > make_rotation	(	const BasicRadian< T > &	radians,
		const BasicVector3D< T > &	axis
	)

inlinestaticconstexpr

Enocde a rotation into a BasicQuaternion

Parameters

radians	The amount of rotation to apply (in radians)
axis	The axis to rotate around

Postcondition: output.isUnit() == true

Definition at line 273 of file Quaternion.hpp.

◆ make_rotation() [2/2]

template<class T >

static constexpr BasicQuaternion< T > make_rotation	(	const BasicRadian< T > &	radians,
		T	axis_x,
		T	axis_y,
		T	axis_z
	)

inlinestaticconstexpr

Enocde a rotation into a BasicQuaternion

Parameters

radians	The amount of rotation to apply (in radians)
axis_x	The X component of the vector to rotate around
axis_y	The Y component of the vector to rotate around
axis_z	The Z component of the vector to rotate around

Postcondition: output.isUnit() == true

Definition at line 261 of file Quaternion.hpp.

◆ norm()

template<class T >

T norm ( ) const

inline

Definition at line 139 of file Quaternion.hpp.

◆ normalized()

template<class T >

constexpr BasicQuaternion< T > normalized ( ) const

inlineconstexpr

Definition at line 144 of file Quaternion.hpp.

◆ normSquared()

template<class T >

T normSquared ( ) const

inline

Definition at line 138 of file Quaternion.hpp.

◆ pow()

template<class T >

BasicQuaternion< T > pow ( T exponent ) const

inline

Computes this BasicQuaternion raised to a real power.

Definition at line 71 of file Quaternion.hpp.

◆ real()

template<class T >

constexpr const T & real ( ) const

inlineconstexpr

Definition at line 167 of file Quaternion.hpp.

◆ unit_i()

template<class T >

static constexpr BasicQuaternion< T > unit_i ( )

inlinestaticconstexpr

Definition at line 57 of file Quaternion.hpp.

◆ unit_j()

template<class T >

static constexpr BasicQuaternion< T > unit_j ( )

inlinestaticconstexpr

Definition at line 58 of file Quaternion.hpp.

◆ unit_k()

template<class T >

static constexpr BasicQuaternion< T > unit_k ( )

inlinestaticconstexpr

Definition at line 59 of file Quaternion.hpp.

◆ unit_real()

template<class T >

static constexpr BasicQuaternion< T > unit_real ( )

inlinestaticconstexpr

Definition at line 56 of file Quaternion.hpp.

◆ w()

template<class T >

constexpr const T & w ( ) const

inlineconstexpr

Definition at line 166 of file Quaternion.hpp.

◆ zero()

template<class T >

static constexpr BasicQuaternion< T > zero ( )

inlinestaticconstexpr

BasicQuaternion representation of the real number 0.

Definition at line 55 of file Quaternion.hpp.

Friends And Related Symbol Documentation

◆ accumulate

template<class T >

constexpr T accumulate ( const BasicQuaternion< T > & input )

friend

Sums up the components of input

Parameters

input The BasicQuaternion to operate on

Returns: The sum of all the components

Definition at line 687 of file Quaternion.hpp.

◆ actively_rotate_encoded_point

template<class T >

constexpr BasicQuaternion< T > actively_rotate_encoded_point	(	const BasicQuaternion< T > &	rotation,
		const BasicQuaternion< T > &	encoded_point
	)

friend

Rotates the encoded_point using the given rotation

Parameters

rotation	The input rotation
encoded_point	The input point to be rotated

Returns: The actively rotated encoded point

Precondition: rotation is a unit BasicQuaternion. encoded_point is a pure BasicQuaternion.

Postcondition: The output is a pure BasicQuaternion

Note: This is an active rotation, meaning that the point is rotated with respect to the coordinate system. This is also known as a global rotation.

Definition at line 619 of file Quaternion.hpp.

◆ approximately_equal_to

template<class T >

template<std::floating_point OT = T>
requires std::is_floating_point_v<T>

constexpr bool approximately_equal_to	(	const BasicQuaternion< T > &	value_to_test,
		const BasicQuaternion< T > &	value_it_should_be,
		OT	tolerance = `OT{0.0002}`
	)

friend

Compares two BasicQuaternion inputs equal, component-wise, to within a tolerance

Parameters

value_to_test
value_it_should_be
tolerance	How close they should be to be considered equal

Returns: true if they are equal

See also: Equality

Definition at line 324 of file Quaternion.hpp.

◆ arg

template<class T >

constexpr T arg ( const BasicQuaternion< T > & input )

friend

Computes the phase-angle (in radians of a Quaternion)

Note: This is meant to mirror the behavior of std::arg( std::complex )

Definition at line 664 of file Quaternion.hpp.

◆ compose_rotations

template<class T >

constexpr BasicQuaternion< T > compose_rotations	(	const BasicQuaternion< T > &	rotation_1,
		const BasicQuaternion< T > &	rotation_2
	)

friend

Performs a concatenation of two rotations

Parameters

rotation_1	The first rotation to perform
rotation_2	The second rotation to perform

Definition at line 643 of file Quaternion.hpp.

◆ conjugate

template<class T >

constexpr BasicQuaternion< T > conjugate ( const BasicQuaternion< T > & input )

friend

Computes the conjugate of the input

Note: This will just call input.conjugate()

Definition at line 716 of file Quaternion.hpp.

◆ dot

template<class T >

constexpr T dot	(	const BasicQuaternion< T > &	left,
		const BasicQuaternion< T > &	right
	)

friend

Calculates the dot product of two Quaternions

Note: In this case, the Quaternions are just treated as separate 4-tuples and the dot product of those are calculated.

Definition at line 553 of file Quaternion.hpp.

◆ dot_normalized

template<class T >

constexpr T dot_normalized	(	const BasicQuaternion< T > &	left,
		const BasicQuaternion< T > &	right
	)

friend

Definition at line 561 of file Quaternion.hpp.

◆ exp

template<class T >

BasicQuaternion< T > exp ( const BasicQuaternion< T > & input )

friend

Computes the exponential of the input

Note: This will just call input.exp()

Definition at line 734 of file Quaternion.hpp.

◆ format

template<class T >

std::string format ( const BasicQuaternion< T > & input )

friend

Definition at line 707 of file Quaternion.hpp.

◆ globally_rotate_encoded_point

template<class T >

constexpr BasicQuaternion< T > globally_rotate_encoded_point	(	const BasicQuaternion< T > &	rotation,
		const BasicQuaternion< T > &	encoded_point
	)

friend

See also: actively_rotate_encoded_point

Definition at line 632 of file Quaternion.hpp.

◆ locally_rotate_encoded_point

template<class T >

constexpr BasicQuaternion< T > locally_rotate_encoded_point	(	const BasicQuaternion< T > &	rotation,
		const BasicQuaternion< T > &	encoded_point
	)

friend

See also: passively_rotate_encoded_point

Definition at line 598 of file Quaternion.hpp.

◆ log

template<class T >

constexpr BasicQuaternion< T > log ( const BasicQuaternion< T > & input )

friend

Computes the log of the input

Note: This will just call input.log()

Definition at line 725 of file Quaternion.hpp.

◆ normalized

template<class T >

constexpr BasicQuaternion< T > normalized ( const BasicQuaternion< T > & input )

friend

Creates the normalized form of a Quaternion

Parameters

input The Quaternion to normalize

Returns: The normalized version of input

Definition at line 655 of file Quaternion.hpp.

◆ operator==

template<class T >

constexpr bool operator==	(	const BasicQuaternion< T > &	left,
		const BasicQuaternion< T > &	right
	)

friend

Defines equality of two Quaternions

Note: Uses approximately_equal_to under-the-hood; Use C++20's ability to generate the operator !=() from operator ==()

See also: Equality

Definition at line 307 of file Quaternion.hpp.

◆ passively_rotate_encoded_point

template<class T >

constexpr BasicQuaternion< T > passively_rotate_encoded_point	(	const BasicQuaternion< T > &	rotation,
		const BasicQuaternion< T > &	encoded_point
	)

friend

Rotates the encoded point using the given rotation

Parameters

rotation	The input rotation
encoded_point	The input point to be rotated

Returns: The passively rotated encoded point

Precondition: rotation is a unit BasicQuaternion. encoded_point is a pure Auaternion.

Postcondition: The output is a pure BasicQuaternion

Note: This is a passive rotation, meaning that the coordinate system is rotated with respect to the point. This is also known as a local rotation.

Definition at line 585 of file Quaternion.hpp.

◆ polar

template<class T >

constexpr BasicQuaternion< T > polar	(	const BasicVector3D< T > &	axis,
		const BasicRadian< T >	angle = `BasicRadian<T>{}`
	)

friend

Constructs a unit Quaternion from the given axis and angle

Note: This is meant to mirror the behavior of the std::complex version of std::polar()

Definition at line 673 of file Quaternion.hpp.

◆ slerp

template<class T >

constexpr BasicQuaternion< T > slerp	(	const BasicQuaternion< T > &	begin,
		const BasicQuaternion< T > &	end,
		T	percent
	)

friend

Calculates the Spherical Linear Interpolation betwee two Quaternions

Parameters

begin	Origin value
end	Destination value
percent	[0..1] Represents the percentage to interpolate

Definition at line 698 of file Quaternion.hpp.

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

lib/include/Math/math/Quaternion.hpp

Public Types

Public Member Functions

Static Public Member Functions

Friends

Related Symbols

Detailed Description

Member Typedef Documentation

◆ value_type

Constructor & Destructor Documentation

◆ BasicQuaternion() [1/3]

◆ BasicQuaternion() [2/3]

◆ BasicQuaternion() [3/3]

Member Function Documentation

◆ angle()

◆ axis()

◆ conjugate()

◆ encode_point() [1/2]

◆ encode_point() [2/2]

◆ exp()

◆ i()

◆ identity()

◆ imaginary()

◆ inverse()

◆ isInf()

◆ isNaN()

◆ isPure()

◆ isUnit() [1/2]

◆ isUnit() [2/2]

◆ isZero() [1/2]

◆ isZero() [2/2]

◆ j()

◆ k()

◆ log()

◆ magnitude()

◆ magnitudeSquared()

◆ make_pure() [1/2]

◆ make_pure() [2/2]

◆ make_rotation() [1/2]

◆ make_rotation() [2/2]

◆ norm()

◆ normalized()

◆ normSquared()

◆ pow()

◆ real()

◆ unit_i()

◆ unit_j()

◆ unit_k()

◆ unit_real()

◆ w()

◆ zero()

Friends And Related Symbol Documentation

◆ accumulate

◆ actively_rotate_encoded_point

◆ approximately_equal_to

◆ arg

◆ compose_rotations

◆ conjugate

◆ dot

◆ dot_normalized

◆ exp

◆ format

◆ globally_rotate_encoded_point

◆ locally_rotate_encoded_point

◆ log

◆ normalized

◆ operator==

◆ passively_rotate_encoded_point

◆ polar

◆ slerp