Quaternion class. More...

#include <geomc/linalg/Quaternion.h>

Inheritance diagram for Quat< T >:

Public Types
using	elem_t
	The coordinate type of this object.

using	point_t
	The type of a point in this object's space.

typedef Quat< T >	self_t
	Self type. I.e., `Vec<T,N>` if a vector, `Quat<T>` if a quaternion.

Public Member Functions
	Quat ()
	Construct a quaternion with all elements 0.

	Quat (const std::initializer_list< T > &items)

	Quat (const T v[4])
	Construct a quaternion from the contents of the 4-element array `v`

	Quat (const Vec< T, 3 > &v, T w)
	Construct a quaternion with vector part `v` and real part `w`.

	Quat (const Vec< T, 4 > &v)
	Construct a quaternion from the 4D vector `v`.

	Quat (T x, T y, T z, T w)
	Construct a quaternion with elements `(x, y, z, w)`

self_t	abs () const
	Element-wise absolute value.

self_t	add (const self_t &v) const
	Vector addition.

self_t	align (const self_t &from, const self_t &to) const
	Apply a rotation to `this` which aligns the unit vectors `from` with `to`.

T	angle () const

T	angle_to (const self_t &v) const
	Angle between vectors.

index_t	argmax () const
	Return the index of the coordinate with the largest absolute value.

index_t	argmin () const
	Return the index of the coordinate with the smallest absolute value.

T *	begin ()

const T *	begin () const

self_t	bounce_on (const self_t &normal) const
	Elastic collision.

self_t	ceil () const
	Element-wise ceiling function.

self_t	clamp (const self_t &lo, const self_t &hi) const
	Element-wise clamp.

Quat< T >	conj () const
	The complex conjugate of this quaternion; also the inverse rotation.

T	dist (const self_t &pt) const
	Distance between points.

T	dist2 (const self_t &pt) const
	Distance squared to a point.

T	dot (const self_t &v) const
	Dot product.

T *	end ()

const T *	end () const

Quat< T >	exp () const
	Quaternion exponential `e`^q.

self_t	floor () const
	Element-wise floor function.

T	fraction_on (const self_t &axis) const
	Return the component of `this` that projects to `axis`, as a fraction of axis's length.

T &	get (index_t idx)

const T &	get (index_t idx) const

Vec< T, 3 >	imag () const
	The vector part of this quaternion.

bool	is_zero () const
	Return `true` if all elements are zero.

Quat< T >	log () const
	Quaternion natural log.

T	mag () const
	Euclidean norm (magnitude).

T	mag2 () const
	Squared magnitude.

T	max () const
	Maximum element.

self_t	max (const self_t &v) const
	Element-wise maximum of two `Vec`s.

T	min () const
	Minimum element.

self_t	min (const self_t &v) const
	Element-wise minimum of two `Vec`s.

self_t	mix (const self_t &v, T mix) const
	Linear interpolation.

Quat< T >	mult (const Quat< T > &q) const
	Quaternion multiplication.

template<typename U>
	operator Quat< U > () const
	Cast this Quat to a new coordinate type `U`.

	operator Vec< U, N > () const
	Element typecast.

bool	operator!= (const self_t &vv) const
	Inequality test.

self_t &	*operator=** (const self_t &vv)
	Element-wise multiplication and assignment.

self_t &	*operator=** (T s)
	Scalar multiplication and assignment.

self_t	operator+ (const self_t &v) const
	Element-wise addition.

self_t &	operator+= (const self_t &vv)
	Element-wise addition and assignment.

self_t	operator- () const
	Negation.

self_t	operator- (const self_t &v) const
	Element-wise subtraction.

self_t &	operator-= (const self_t &vv)
	Subtraction and assignment.

self_t &	operator/= (T s)
	Scalar division and assignment.

bool	operator== (const self_t &vv) const
	Equality test.

T &	operator[] (index_t idx)
	Vector element access.

const T &	operator[] (index_t idx) const
	Vector element access.

T	product () const
	Multiply all elements together.

self_t	project_on (const self_t &axis) const
	Orthogonal projection to an axis.

T	real () const
	The scalar part of this quaternion.

self_t	reflect_about (self_t normal) const
	Reflection about a normal.

PointType< T, M >::point_t	resized () const
	Resized copy of a vector.

Quat< T >	rotation_scale (T t) const
	Interpolate this unit quaternion with the null rotation according to the interpolation parameter `0 <= t <= 1`.

self_t	round () const
	Round each element to the nearest integer.

self_t	scale (const self_t &v) const
	Element-wise multiplication.

self_t	scale (T a) const
	Scalar multiple.

index_t	size () const
	The number of elements in this vector. Always equal to `N`.

Quat< T >	slerp (const Quat< T > &q1, T t) const
	Spherical linear interpolation, per Ken Shoemake. Interpolate smoothly between two quaternion orientation states.

self_t	sub (const self_t &v) const
	Vector subtraction.

T	sum () const
	Sum the elements of the vector.

Vec< T, 3 >	to_angular_velocity () const
	Convert this unit quaternion to an angular velocity representation, with the magnitude of the axis representing the rate of rotation in radians.

Vec< T, 4 >	to_axis_angle_rotation () const
	Convert this unit quaternion to an axis-angle rotation representation `(x, y, z, radians)`.

self_t	unit () const
	Vector normalization.

self_t	with_length (T mag) const
	Compute a vector with the direction of `this` and a new magnitude `mag`.

Static Public Member Functions
static Quat< T >	identity ()

static Quat< T >	rotation_direction_align (const Vec< T, 3 > &unit_v, const Vec< T, 3 > &unit_align_with)
	Rotation to align one vector with another.

static Quat< T >	rotation_from_axis_angle (const Vec< T, 3 > &axis, T radians)
	Rotation quaternion from an axis and angle.

Static Public Attributes
static constexpr index_t	N
	The dimension of this object.

static const self_t	ones

static const self_t	unit_x

static const self_t	zeros

Protected Attributes
T	v [N]

Friends
Vec< T, 3 >	operator* (const Quat< T > &q, const Vec< T, 3 > &v)
	Quaternion conjugation, ‘q * v * q’`.

Quat< T >	operator* (const Quat< T > &q1, const Quat< T > &q2)
	Quaternion multiplication.

Vec< T, 3 >	operator* (const Vec< T, 3 > &v, const Quat< T > &q)
	Quaternion inverse conjugation, q' * v * q.

Detailed Description

template<typename T>
class geom::Quat< T >

Quaternion class.

(x, y, z) is the vector part, while w is the real part. This class differs slightly from convention in that the real part is the last coordinate rather than the first; this scheme was chosen to maintain naming consistency with the rest of the library.

Member Typedef Documentation

◆ point_t

using point_t

inherited

The type of a point in this object's space.

An N-vector of T if N > 1, otherwise a T.

Constructor & Destructor Documentation

◆ Quat()

template<typename T>

Quat ( const std::initializer_list< T > & items )

inline

Construct a quaternion from a brace-initialization list. (c++11)

Example: Quat<float> q = {0, 0, 0, 1};

Parameters

items A brace-initializer list.

Member Function Documentation

◆ abs()

self_t abs ( ) const

inlineinherited

Element-wise absolute value.

Returns: A new vector x such that x[i] = abs(this[i]).

◆ add()

self_t add ( const self_t & v ) const

inlineinherited

Vector addition.

Parameters

v	Another vector.

Returns: A new vector x such that x[i] = this[i] + v[i].

◆ angle_to()

T angle_to ( const self_t & v ) const

inlineinherited

Angle between vectors.

Parameters

v	Another vector.

Returns: Angle in radians between this and v, between 0 and pi.

◆ begin() [1/2]

template<typename T, index_t N>

T * begin ( )

inlineinherited

Returns: A writeable iterator pointing at the first element.

◆ begin() [2/2]

template<typename T, index_t N>

const T * begin ( ) const

inlineinherited

Returns: A read-only iterator pointing at the first element.

◆ bounce_on()

self_t bounce_on ( const self_t & normal ) const

inlineinherited

Elastic collision.

Treat this as a velocity vector or incident ray; this function returns the velocity reflected off of a surface with normal normal. Convenience for -reflect_about(normal).

Parameters

normal Normal of surface to "bounce" on.

Returns: The "bounced" direction vector.

◆ ceil()

self_t ceil ( ) const

inlineinherited

Element-wise ceiling function.

Returns: A new vector x such that x[i] = ceil(this[i]).

◆ clamp()

self_t clamp	(	const self_t &	lo,
		const self_t &	hi ) const

inlineinherited

Element-wise clamp.

Parameters

lo	Element-wise lower extremes.
hi	Element-wise upper extremes.

Returns: A new vector such that each element x[i] is clamped between lo[i] and hi[i].

◆ dist()

T dist ( const self_t & pt ) const

inlineinherited

Distance between points.

Parameters

pt	Another point.

Returns: The distance between this and pt.

◆ dist2()

T dist2 ( const self_t & pt ) const

inlineinherited

Distance squared to a point.

Parameters

pt	Another point.

Returns: The square of the distance between this and pt.

◆ dot()

T dot ( const self_t & v ) const

inlineinherited

Dot product.

Parameters

v	Another vector.

Returns: The dot product of this with v.

◆ end() [1/2]

template<typename T, index_t N>

T * end ( )

inlineinherited

Returns: A writeable iterator pointing just beyond the last element.

◆ end() [2/2]

template<typename T, index_t N>

const T * end ( ) const

inlineinherited

Returns: A read-only iterator pointing just beyond the last element.

◆ exp()

template<typename T>

Quat< T > exp ( ) const

inline

Quaternion exponential e^q.

Returns: A quaternion representing a rotation about q.imag() by angle 2 * |q| and a scaling by e^q.real().

◆ floor()

self_t floor ( ) const

inlineinherited

Element-wise floor function.

Returns: A new vector x such that x[i] = floor(this[i]).

◆ fraction_on()

T fraction_on ( const self_t & axis ) const

inlineinherited

Return the component of this that projects to axis, as a fraction of axis's length.

Parameters

axis	An arbitrary basis vector.

◆ get() [1/2]

template<typename T, index_t N>

T & get ( index_t idx )

inlineinherited

Get the element at index idx.

Parameters

idx	Index of element.

Returns: A reference to the element at idx.

◆ get() [2/2]

template<typename T, index_t N>

const T & get ( index_t idx ) const

inlineinherited

Get the element at index idx.

Parameters

idx	Index of element.

Returns: A const reference to the element at idx.

◆ mag()

T mag ( ) const

inlineinherited

Euclidean norm (magnitude).

Returns: The Euclidean magnitude (geometric length) of this vector.

◆ mag2()

T mag2 ( ) const

inlineinherited

Squared magnitude.

Returns: The square of the magnitude of this vector.

◆ max() [1/2]

T max ( ) const

inlineinherited

Maximum element.

Returns: The value of the component with the highest value.

◆ max() [2/2]

self_t max ( const self_t & v ) const

inlineinherited

Element-wise maximum of two Vecs.

Parameters

v	Another vector.

Returns: A new vector x such that x[i] = max(this[i], v[i]).

◆ min() [1/2]

T min ( ) const

inlineinherited

Minimum element.

Returns: The value of the component with the lowest value.

◆ min() [2/2]

self_t min ( const self_t & v ) const

inlineinherited

Element-wise minimum of two Vecs.

Parameters

v	Another vector.

Returns: A new vector x such that x[i] = min(this[i], v[i]).

◆ mix()

self_t mix	(	const self_t &	v,
		T	mix ) const

inlineinherited

Linear interpolation.

A mix parameter of 0 evaluates to this, while 1 is v.

Parameters

v	Another vector.
mix	A mixing factor between 0 and 1.

Returns: A linear mixing of this with v.

◆ operator Vec< U, N >()

operator Vec< U, N > ( ) const

inlineexplicitinherited

Element typecast.

Returns: A new vector with all elements cast to type U.

◆ operator!=()

bool operator!= ( const self_t & vv ) const

inlineinherited

Inequality test.

Returns: true if any corresponding elements of this and vv are unequal, false otherwise.

◆ operator-()

self_t operator- ( ) const

inlineinherited

Negation.

Returns: A copy of this vector with all elements negated (i.e. a vector pointing in the opposite direction).

◆ operator==()

bool operator== ( const self_t & vv ) const

inlineinherited

Equality test.

Returns: true if all corresponding elements of this and vv are equal, false otherwise.

◆ operator[]() [1/2]

T & operator[] ( index_t idx )

inlineinherited

Vector element access.

Parameters

idx	Index of element to retrieve.

Returns: A reference to the element at index idx.

◆ operator[]() [2/2]

const T & operator[] ( index_t idx ) const

inlineinherited

Vector element access.

Parameters

idx	Index of element to retrieve.

Returns: A read-only reference to the element at index idx.

◆ project_on()

self_t project_on ( const self_t & axis ) const

inlineinherited

Orthogonal projection to an axis.

Parameters

axis	A direction vector.

Returns: A vector in direction axis with magnitude equal to the component of this aligned with axis.

◆ reflect_about()

self_t reflect_about ( self_t normal ) const

inlineinherited

Reflection about a normal.

Parameters

normal Axis of reflection.

Returns: A copy of this vector reflected across the given axis.

◆ resized()

PointType< T, M >::point_t resized ( ) const

inlineinherited

Resized copy of a vector.

Template Parameters

M	Dimension of new vector.

Returns: A new vector with size M. If M is larger than N, the new elements will be set to zero. If M is 1, then the return type is T.

◆ rotation_direction_align()

template<typename T>

static Quat< T > rotation_direction_align	(	const Vec< T, 3 > &	unit_v,
		const Vec< T, 3 > &	unit_align_with )

inlinestatic

Rotation to align one vector with another.

Parameters

v	Unit vector to align.
alignWith	Unit direction to align with.

Returns: A quaternion rotating v into alignment with alignWith.

◆ rotation_from_axis_angle()

template<typename T>

static Quat< T > rotation_from_axis_angle	(	const Vec< T, 3 > &	axis,
		T	radians )

inlinestatic

Rotation quaternion from an axis and angle.

Parameters

axis	Axis of rotation (not necessesarily a unit vector)
angle	Angle of rotation in radians

Returns: A unit quaternion representing a rotation of angle radians about axis.

◆ scale() [1/2]

self_t scale ( const self_t & v ) const

inlineinherited

Element-wise multiplication.

Parameters

v	Another vector.

Returns: A new vector x such that x[i] = this[i] * v[i].

◆ scale() [2/2]

self_t scale ( T a ) const

inlineinherited

Scalar multiple.

Parameters

a	A constant scale factor.

Returns: A new vector x such that x[i] = this[i] * a.

◆ slerp()

template<typename T>

Quat< T > slerp	(	const Quat< T > &	q1,
		T	t ) const

inline

Spherical linear interpolation, per Ken Shoemake. Interpolate smoothly between two quaternion orientation states.

Parameters

q1	The rotation at `t = 1`.
t	Interpolation parameter between 0 and 1.

Returns

◆ sub()

self_t sub ( const self_t & v ) const

inlineinherited

Vector subtraction.

Parameters

v	Another vector.

Returns: A new vector x such that x[i] = this[i] - v[i].

◆ unit()

self_t unit ( ) const

inlineinherited

Vector normalization.

Returns: A copy of this vector with unit length.

◆ with_length()

self_t with_length ( T mag ) const

inlineinherited

Compute a vector with the direction of this and a new magnitude mag.

If this is the zero vector, it will remain unchanged.

Friends And Related Symbol Documentation

◆ operator* [1/2]

template<typename T>

Vec< T, 3 > operator*	(	const Quat< T > &	q,
		const Vec< T, 3 > &	v )

friend

Quaternion conjugation, ‘q * v * q’`.

If q is a unit quaternion, then q * v is a rotation of v by q.

◆ operator* [2/2]

template<typename T>

Vec< T, 3 > operator*	(	const Vec< T, 3 > &	v,
		const Quat< T > &	q )

friend

Quaternion inverse conjugation, q' * v * q.

If q is a unit quaternion, then v * q is a rotation of v by q^-1.

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

geomc/linalg/LinalgTypes.h
geomc/linalg/Quaternion.h

Public Types

Public Member Functions

Static Public Member Functions

Static Public Attributes

Protected Attributes

Friends

Detailed Description

Member Typedef Documentation

◆ point_t

Constructor & Destructor Documentation

◆ Quat()

Member Function Documentation

◆ abs()

◆ add()

◆ angle_to()

◆ begin() [1/2]

◆ begin() [2/2]

◆ bounce_on()

◆ ceil()

◆ clamp()

◆ dist()

◆ dist2()

◆ dot()

◆ end() [1/2]

◆ end() [2/2]

◆ exp()

◆ floor()

◆ fraction_on()

◆ get() [1/2]

◆ get() [2/2]

◆ mag()

◆ mag2()

◆ max() [1/2]

◆ max() [2/2]

◆ min() [1/2]

◆ min() [2/2]

◆ mix()

◆ operator Vec< U, N >()

◆ operator!=()

◆ operator-()

◆ operator==()

◆ operator[]() [1/2]

◆ operator[]() [2/2]

◆ project_on()

◆ reflect_about()

◆ resized()

◆ rotation_direction_align()

◆ rotation_from_axis_angle()

◆ scale() [1/2]

◆ scale() [2/2]

◆ slerp()

◆ sub()

◆ unit()

◆ with_length()

Friends And Related Symbol Documentation

◆ operator* [1/2]

◆ operator* [2/2]