Struct na::geometry::Quaternion

#[repr(C)]
pub struct Quaternion<N> where
    N: RealField,  {
    pub coords: Matrix<N, U4, U1, <DefaultAllocator as Allocator<N, U4, U1>>::Buffer>,
}

A quaternion. See the type alias UnitQuaternion = Unit<Quaternion> for a quaternion that may be used as a rotation.

Fields

coords: Matrix<N, U4, U1, <DefaultAllocator as Allocator<N, U4, U1>>::Buffer>

This quaternion as a 4D vector of coordinates in the [ x, y, z, w ] storage order.

Methods

impl<N> Quaternion<N> where
    N: RealField, 

pub fn into_owned(self) -> Quaternion<N>

Deprecated:

This method is a no-op and will be removed in a future release.

Moves this unit quaternion into one that owns its data.

pub fn clone_owned(&self) -> Quaternion<N>

Deprecated:

This method is a no-op and will be removed in a future release.

Clones this unit quaternion into one that owns its data.

pub fn normalize(&self) -> Quaternion<N>

Normalizes this quaternion.

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let q_normalized = q.normalize();
relative_eq!(q_normalized.norm(), 1.0);

pub fn imag(
    &self
) -> Matrix<N, U3, U1, <DefaultAllocator as Allocator<N, U3, U1>>::Buffer>

The imaginary part of this quaternion.

pub fn conjugate(&self) -> Quaternion<N>

The conjugate of this quaternion.

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let conj = q.conjugate();
assert!(conj.i == -2.0 && conj.j == -3.0 && conj.k == -4.0 && conj.w == 1.0);

pub fn try_inverse(&self) -> Option<Quaternion<N>>

Inverts this quaternion if it is not zero.

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let inv_q = q.try_inverse();

assert!(inv_q.is_some());
assert_relative_eq!(inv_q.unwrap() * q, Quaternion::identity());

//Non-invertible case
let q = Quaternion::new(0.0, 0.0, 0.0, 0.0);
let inv_q = q.try_inverse();

assert!(inv_q.is_none());

pub fn lerp(&self, other: &Quaternion<N>, t: N) -> Quaternion<N>

Linear interpolation between two quaternion.

Computes self * (1 - t) + other * t.

Example

let q1 = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let q2 = Quaternion::new(10.0, 20.0, 30.0, 40.0);

assert_eq!(q1.lerp(&q2, 0.1), Quaternion::new(1.9, 3.8, 5.7, 7.6));

pub fn vector(
    &self
) -> Matrix<N, U3, U1, SliceStorage<N, U3, U1, <<DefaultAllocator as Allocator<N, U4, U1>>::Buffer as Storage<N, U4, U1>>::RStride, <<DefaultAllocator as Allocator<N, U4, U1>>::Buffer as Storage<N, U4, U1>>::CStride>>

The vector part (i, j, k) of this quaternion.

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
assert_eq!(q.vector()[0], 2.0);
assert_eq!(q.vector()[1], 3.0);
assert_eq!(q.vector()[2], 4.0);

pub fn scalar(&self) -> N

The scalar part w of this quaternion.

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
assert_eq!(q.scalar(), 1.0);

pub fn as_vector(
    &self
) -> &Matrix<N, U4, U1, <DefaultAllocator as Allocator<N, U4, U1>>::Buffer>

Reinterprets this quaternion as a 4D vector.

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
// Recall that the quaternion is stored internally as (i, j, k, w)
// while the crate::new constructor takes the arguments as (w, i, j, k).
assert_eq!(*q.as_vector(), Vector4::new(2.0, 3.0, 4.0, 1.0));

pub fn norm(&self) -> N

The norm of this quaternion.

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
assert_relative_eq!(q.norm(), 5.47722557, epsilon = 1.0e-6);

pub fn magnitude(&self) -> N

A synonym for the norm of this quaternion.

Aka the length. This is the same as .norm()

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
assert_relative_eq!(q.magnitude(), 5.47722557, epsilon = 1.0e-6);

pub fn norm_squared(&self) -> N

The squared norm of this quaternion.

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
assert_eq!(q.magnitude_squared(), 30.0);

pub fn magnitude_squared(&self) -> N

A synonym for the squared norm of this quaternion.

Aka the squared length. This is the same as .norm_squared()

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
assert_eq!(q.magnitude_squared(), 30.0);

pub fn dot(&self, rhs: &Quaternion<N>) -> N

The dot product of two quaternions.

Example

let q1 = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let q2 = Quaternion::new(5.0, 6.0, 7.0, 8.0);
assert_eq!(q1.dot(&q2), 70.0);

pub fn inner(&self, other: &Quaternion<N>) -> Quaternion<N>

Calculates the inner product (also known as the dot product). See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel Formula 4.89.

Example

let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
let expected = Quaternion::new(-20.0, 0.0, 0.0, 0.0);
let result = a.inner(&b);
assert_relative_eq!(expected, result, epsilon = 1.0e-5);

pub fn outer(&self, other: &Quaternion<N>) -> Quaternion<N>

Calculates the outer product (also known as the wedge product). See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel Formula 4.89.

Example

let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
let expected = Quaternion::new(0.0, -5.0, 18.0, -11.0);
let result = a.outer(&b);
assert_relative_eq!(expected, result, epsilon = 1.0e-5);

pub fn project(&self, other: &Quaternion<N>) -> Option<Quaternion<N>>

Calculates the projection of self onto other (also known as the parallel). See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel Formula 4.94.

Example

let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
let expected = Quaternion::new(0.0, 3.333333333333333, 1.3333333333333333, 0.6666666666666666);
let result = a.project(&b).unwrap();
assert_relative_eq!(expected, result, epsilon = 1.0e-5);

pub fn reject(&self, other: &Quaternion<N>) -> Option<Quaternion<N>>

Calculates the rejection of self from other (also known as the perpendicular). See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel Formula 4.94.

Example

let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
let expected = Quaternion::new(0.0, -1.3333333333333333, 1.6666666666666665, 3.3333333333333335);
let result = a.reject(&b).unwrap();
assert_relative_eq!(expected, result, epsilon = 1.0e-5);

pub fn polar_decomposition(
    &self
) -> (N, N, Option<Unit<Matrix<N, U3, U1, <DefaultAllocator as Allocator<N, U3, U1>>::Buffer>>>)

The polar decomposition of this quaternion.

Returns, from left to right: the quaternion norm, the half rotation angle, the rotation axis. If the rotation angle is zero, the rotation axis is set to None.

Example

let q = Quaternion::new(0.0, 5.0, 0.0, 0.0);
let (norm, half_ang, axis) = q.polar_decomposition();
assert_eq!(norm, 5.0);
assert_eq!(half_ang, f32::consts::FRAC_PI_2);
assert_eq!(axis, Some(Vector3::x_axis()));

pub fn ln(&self) -> Quaternion<N>

Compute the natural logarithm of a quaternion.

Example

let q = Quaternion::new(2.0, 5.0, 0.0, 0.0);
assert_relative_eq!(q.ln(), Quaternion::new(1.683647, 1.190289, 0.0, 0.0), epsilon = 1.0e-6)

pub fn exp(&self) -> Quaternion<N>

Compute the exponential of a quaternion.

Example

let q = Quaternion::new(1.683647, 1.190289, 0.0, 0.0);
assert_relative_eq!(q.exp(), Quaternion::new(2.0, 5.0, 0.0, 0.0), epsilon = 1.0e-5)

pub fn exp_eps(&self, eps: N) -> Quaternion<N>

Compute the exponential of a quaternion. Returns the identity if the vector part of this quaternion has a norm smaller than eps.

Example

let q = Quaternion::new(1.683647, 1.190289, 0.0, 0.0);
assert_relative_eq!(q.exp_eps(1.0e-6), Quaternion::new(2.0, 5.0, 0.0, 0.0), epsilon = 1.0e-5);

// Singular case.
let q = Quaternion::new(0.0000001, 0.0, 0.0, 0.0);
assert_eq!(q.exp_eps(1.0e-6), Quaternion::identity());

pub fn powf(&self, n: N) -> Quaternion<N>

Raise the quaternion to a given floating power.

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
assert_relative_eq!(q.powf(1.5), Quaternion::new( -6.2576659, 4.1549037, 6.2323556, 8.3098075), epsilon = 1.0e-6);

pub fn as_vector_mut(
    &mut self
) -> &mut Matrix<N, U4, U1, <DefaultAllocator as Allocator<N, U4, U1>>::Buffer>

Transforms this quaternion into its 4D vector form (Vector part, Scalar part).

Example

let mut q = Quaternion::identity();
*q.as_vector_mut() = Vector4::new(1.0, 2.0, 3.0, 4.0);
assert!(q.i == 1.0 && q.j == 2.0 && q.k == 3.0 && q.w == 4.0);

pub fn vector_mut(
    &mut self
) -> Matrix<N, U3, U1, SliceStorageMut<N, U3, U1, <<DefaultAllocator as Allocator<N, U4, U1>>::Buffer as Storage<N, U4, U1>>::RStride, <<DefaultAllocator as Allocator<N, U4, U1>>::Buffer as Storage<N, U4, U1>>::CStride>>

The mutable vector part (i, j, k) of this quaternion.

Example

let mut q = Quaternion::identity();
{
    let mut v = q.vector_mut();
    v[0] = 2.0;
    v[1] = 3.0;
    v[2] = 4.0;
}
assert!(q.i == 2.0 && q.j == 3.0 && q.k == 4.0 && q.w == 1.0);

pub fn conjugate_mut(&mut self)

Replaces this quaternion by its conjugate.

Example

let mut q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
q.conjugate_mut();
assert!(q.i == -2.0 && q.j == -3.0 && q.k == -4.0 && q.w == 1.0);

pub fn try_inverse_mut(&mut self) -> bool

Inverts this quaternion in-place if it is not zero.

Example

let mut q = Quaternion::new(1.0, 2.0, 3.0, 4.0);

assert!(q.try_inverse_mut());
assert_relative_eq!(q * Quaternion::new(1.0, 2.0, 3.0, 4.0), Quaternion::identity());

//Non-invertible case
let mut q = Quaternion::new(0.0, 0.0, 0.0, 0.0);
assert!(!q.try_inverse_mut());

pub fn normalize_mut(&mut self) -> N

Normalizes this quaternion.

Example

let mut q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
q.normalize_mut();
assert_relative_eq!(q.norm(), 1.0);

pub fn squared(&self) -> Quaternion<N>

Calculates square of a quaternion.

pub fn half(&self) -> Quaternion<N>

Divides quaternion into two.

pub fn sqrt(&self) -> Quaternion<N>

Calculates square root.

pub fn is_pure(&self) -> bool

Check if the quaternion is pure.

pub fn pure(&self) -> Quaternion<N>

Convert quaternion to pure quaternion.

pub fn left_div(&self, other: &Quaternion<N>) -> Option<Quaternion<N>>

Left quaternionic division.

Calculates B^-1 * A where A = self, B = other.

pub fn right_div(&self, other: &Quaternion<N>) -> Option<Quaternion<N>>

Right quaternionic division.

Calculates A * B^-1 where A = self, B = other.

Example

let a = Quaternion::new(0.0, 1.0, 2.0, 3.0);
let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
let result = a.right_div(&b).unwrap();
let expected = Quaternion::new(0.4, 0.13333333333333336, -0.4666666666666667, 0.26666666666666666);
assert_relative_eq!(expected, result, epsilon = 1.0e-7);

pub fn cos(&self) -> Quaternion<N>

Calculates the quaternionic cosinus.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let expected = Quaternion::new(58.93364616794395, -34.086183690465596, -51.1292755356984, -68.17236738093119);
let result = input.cos();
assert_relative_eq!(expected, result, epsilon = 1.0e-7);

pub fn acos(&self) -> Quaternion<N>

Calculates the quaternionic arccosinus.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let result = input.cos().acos();
assert_relative_eq!(input, result, epsilon = 1.0e-7);

pub fn sin(&self) -> Quaternion<N>

Calculates the quaternionic sinus.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let expected = Quaternion::new(91.78371578403467, 21.886486853029176, 32.82973027954377, 43.77297370605835);
let result = input.sin();
assert_relative_eq!(expected, result, epsilon = 1.0e-7);

pub fn asin(&self) -> Quaternion<N>

Calculates the quaternionic arcsinus.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let result = input.sin().asin();
assert_relative_eq!(input, result, epsilon = 1.0e-7);

pub fn tan(&self) -> Quaternion<N>

Calculates the quaternionic tangent.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let expected = Quaternion::new(0.00003821631725009489, 0.3713971716439371, 0.5570957574659058, 0.7427943432878743);
let result = input.tan();
assert_relative_eq!(expected, result, epsilon = 1.0e-7);

pub fn atan(&self) -> Quaternion<N>

Calculates the quaternionic arctangent.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let result = input.tan().atan();
assert_relative_eq!(input, result, epsilon = 1.0e-7);

pub fn sinh(&self) -> Quaternion<N>

Calculates the hyperbolic quaternionic sinus.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let expected = Quaternion::new(0.7323376060463428, -0.4482074499805421, -0.6723111749708133, -0.8964148999610843);
let result = input.sinh();
assert_relative_eq!(expected, result, epsilon = 1.0e-7);

pub fn asinh(&self) -> Quaternion<N>

Calculates the hyperbolic quaternionic arcsinus.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let expected = Quaternion::new(2.385889902585242, 0.514052600662788, 0.7710789009941821, 1.028105201325576);
let result = input.asinh();
assert_relative_eq!(expected, result, epsilon = 1.0e-7);

pub fn cosh(&self) -> Quaternion<N>

Calculates the hyperbolic quaternionic cosinus.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let expected = Quaternion::new(0.9615851176369566, -0.3413521745610167, -0.5120282618415251, -0.6827043491220334);
let result = input.cosh();
assert_relative_eq!(expected, result, epsilon = 1.0e-7);

pub fn acosh(&self) -> Quaternion<N>

Calculates the hyperbolic quaternionic arccosinus.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let expected = Quaternion::new(2.4014472020074007, 0.5162761016176176, 0.7744141524264264, 1.0325522032352352);
let result = input.acosh();
assert_relative_eq!(expected, result, epsilon = 1.0e-7);

pub fn tanh(&self) -> Quaternion<N>

Calculates the hyperbolic quaternionic tangent.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let expected = Quaternion::new(1.0248695360556623, -0.10229568178876419, -0.1534435226831464, -0.20459136357752844);
let result = input.tanh();
assert_relative_eq!(expected, result, epsilon = 1.0e-7);

pub fn atanh(&self) -> Quaternion<N>

Calculates the hyperbolic quaternionic arctangent.

Example

let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let expected = Quaternion::new(0.03230293287000163, 0.5173453683196951, 0.7760180524795426, 1.0346907366393903);
let result = input.atanh();
assert_relative_eq!(expected, result, epsilon = 1.0e-7);

impl<N> Quaternion<N> where
    N: RealField, 

pub fn from_vector(
    vector: Matrix<N, U4, U1, <DefaultAllocator as Allocator<N, U4, U1>>::Buffer>
) -> Quaternion<N>

Deprecated:

Use ::from instead.

Creates a quaternion from a 4D vector. The quaternion scalar part corresponds to the w vector component.

pub fn new(w: N, i: N, j: N, k: N) -> Quaternion<N>

Creates a new quaternion from its individual components. Note that the arguments order does not follow the storage order.

The storage order is [ i, j, k, w ] while the arguments for this functions are in the order (w, i, j, k).

Example

let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
assert!(q.i == 2.0 && q.j == 3.0 && q.k == 4.0 && q.w == 1.0);
assert_eq!(*q.as_vector(), Vector4::new(2.0, 3.0, 4.0, 1.0));

pub fn from_imag(
    vector: Matrix<N, U3, U1, <DefaultAllocator as Allocator<N, U3, U1>>::Buffer>
) -> Quaternion<N>

Constructs a pure quaternion.

pub fn from_parts<SB>(scalar: N, vector: Matrix<N, U3, U1, SB>) -> Quaternion<N> where
    SB: Storage<N, U3, U1>, 

Creates a new quaternion from its scalar and vector parts. Note that the arguments order does not follow the storage order.

The storage order is [ vector, scalar ].

Example

let w = 1.0;
let ijk = Vector3::new(2.0, 3.0, 4.0);
let q = Quaternion::from_parts(w, ijk);
assert!(q.i == 2.0 && q.j == 3.0 && q.k == 4.0 && q.w == 1.0);
assert_eq!(*q.as_vector(), Vector4::new(2.0, 3.0, 4.0, 1.0));

pub fn from_real(r: N) -> Quaternion<N>

Constructs a real quaternion.

pub fn from_polar_decomposition<SB>(
    scale: N,
    theta: N,
    axis: Unit<Matrix<N, U3, U1, SB>>
) -> Quaternion<N> where
    SB: Storage<N, U3, U1>, 

Creates a new quaternion from its polar decomposition.

Note that axis is assumed to be a unit vector.

pub fn identity() -> Quaternion<N>

The quaternion multiplicative identity.

Example

let q = Quaternion::identity();
let q2 = Quaternion::new(1.0, 2.0, 3.0, 4.0);

assert_eq!(q * q2, q2);
assert_eq!(q2 * q, q2);

Trait Implementations

impl<N> TwoSidedInverse<Additive> for Quaternion<N> where
    N: RealField, 

fn two_sided_inverse(&self) -> Quaternion<N>

Returns the two_sided_inverse of self, relative to the operator O.

default fn two_sided_inverse_mut(&mut self)

In-place inversion of self, relative to the operator O.

impl<'b, N> AddAssign<&'b Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

fn add_assign(&mut self, rhs: &'b Quaternion<N>)

Performs the += operation.

impl<N> AddAssign<Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

fn add_assign(&mut self, rhs: Quaternion<N>)

Performs the += operation.

impl<N> VectorSpace for Quaternion<N> where
    N: RealField, 

type Field = N

The underlying scalar field.

impl<N> MulAssign<Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

fn mul_assign(&mut self, rhs: Quaternion<N>)

Performs the *= operation.

impl<'b, N> MulAssign<&'b Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

fn mul_assign(&mut self, rhs: &'b Quaternion<N>)

Performs the *= operation.

impl<N> MulAssign<N> for Quaternion<N> where
    N: RealField, 

fn mul_assign(&mut self, n: N)

Performs the *= operation.

impl<N> AbsDiffEq<Quaternion<N>> for Quaternion<N> where
    N: AbsDiffEq<N, Epsilon = N> + RealField, 

type Epsilon = N

Used for specifying relative comparisons.

fn default_epsilon() -> <Quaternion<N> as AbsDiffEq<Quaternion<N>>>::Epsilon

The default tolerance to use when testing values that are close together.

fn abs_diff_eq(
    &self,
    other: &Quaternion<N>,
    epsilon: <Quaternion<N> as AbsDiffEq<Quaternion<N>>>::Epsilon
) -> bool

A test for equality that uses the absolute difference to compute the approximate equality of two numbers.

default fn abs_diff_ne(&self, other: &Rhs, epsilon: Self::Epsilon) -> bool

The inverse of ApproxEq::abs_diff_eq.

impl<N> Copy for Quaternion<N> where
    N: RealField, 

impl<N> Clone for Quaternion<N> where
    N: RealField, 

fn clone(&self) -> Quaternion<N>

Returns a copy of the value.

default fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl<N> One for Quaternion<N> where
    N: RealField, 

fn one() -> Quaternion<N>

Returns the multiplicative identity element of Self, 1.

default fn is_one(&self) -> bool where
    Self: PartialEq<Self>, 

Returns true if self is equal to the multiplicative identity.

impl<N> AbstractGroup<Additive> for Quaternion<N> where
    N: RealField, 

impl<N> AbstractSemigroup<Multiplicative> for Quaternion<N> where
    N: RealField, 

default fn prop_is_associative_approx(args: (Self, Self, Self)) -> bool where
    Self: RelativeEq<Self>, 

Returns true if associativity holds for the given arguments. Approximate equality is used for verifications.

default fn prop_is_associative(args: (Self, Self, Self)) -> bool where
    Self: Eq, 

Returns true if associativity holds for the given arguments.

impl<N> AbstractSemigroup<Additive> for Quaternion<N> where
    N: RealField, 

default fn prop_is_associative_approx(args: (Self, Self, Self)) -> bool where
    Self: RelativeEq<Self>, 

Returns true if associativity holds for the given arguments. Approximate equality is used for verifications.

default fn prop_is_associative(args: (Self, Self, Self)) -> bool where
    Self: Eq, 

Returns true if associativity holds for the given arguments.

impl<N> Module for Quaternion<N> where
    N: RealField, 

type Ring = N

The underlying scalar field.

impl<N> PartialEq<Quaternion<N>> for Quaternion<N> where
    N: RealField, 

fn eq(&self, rhs: &Quaternion<N>) -> bool

This method tests for self and other values to be equal, and is used by ==.

#[must_use]

default fn ne(&self, other: &Rhs) -> bool

This method tests for !=.

impl<N> RelativeEq<Quaternion<N>> for Quaternion<N> where
    N: RelativeEq<N, Epsilon = N> + RealField, 

fn default_max_relative(
) -> <Quaternion<N> as AbsDiffEq<Quaternion<N>>>::Epsilon

The default relative tolerance for testing values that are far-apart.

fn relative_eq(
    &self,
    other: &Quaternion<N>,
    epsilon: <Quaternion<N> as AbsDiffEq<Quaternion<N>>>::Epsilon,
    max_relative: <Quaternion<N> as AbsDiffEq<Quaternion<N>>>::Epsilon
) -> bool

A test for equality that uses a relative comparison if the values are far apart.

default fn relative_ne(
    &self,
    other: &Rhs,
    epsilon: Self::Epsilon,
    max_relative: Self::Epsilon
) -> bool

The inverse of ApproxEq::relative_eq.

impl<N> DivAssign<N> for Quaternion<N> where
    N: RealField, 

fn div_assign(&mut self, n: N)

Performs the /= operation.

impl<N> Identity<Multiplicative> for Quaternion<N> where
    N: RealField, 

fn identity() -> Quaternion<N>

The identity element.

default fn id(O) -> Self

Specific identity.

impl<N> Identity<Additive> for Quaternion<N> where
    N: RealField, 

fn identity() -> Quaternion<N>

The identity element.

default fn id(O) -> Self

Specific identity.

impl<N> Neg for Quaternion<N> where
    N: RealField, 

type Output = Quaternion<N>

The resulting type after applying the - operator.

fn neg(self) -> <Quaternion<N> as Neg>::Output

Performs the unary - operation.

impl<'a, N> Neg for &'a Quaternion<N> where
    N: RealField, 

type Output = Quaternion<N>

The resulting type after applying the - operator.

fn neg(self) -> <&'a Quaternion<N> as Neg>::Output

Performs the unary - operation.

impl<N> NormedSpace for Quaternion<N> where
    N: RealField, 

type RealField = N

The result of the norm (not necessarily the same same as the field used by this vector space).

type ComplexField = N

The field of this space must be this complex number.

fn norm_squared(&self) -> N

The squared norm of this vector.

fn norm(&self) -> N

The norm of this vector.

fn normalize(&self) -> Quaternion<N>

Returns a normalized version of this vector.

fn normalize_mut(&mut self) -> N

Normalizes this vector in-place and returns its norm.

fn try_normalize(&self, min_norm: N) -> Option<Quaternion<N>>

Returns a normalized version of this vector unless its norm as smaller or equal to eps.

fn try_normalize_mut(&mut self, min_norm: N) -> Option<N>

Normalizes this vector in-place or does nothing if its norm is smaller or equal to eps.

impl<N> AbstractModule<Additive, Additive, Multiplicative> for Quaternion<N> where
    N: RealField, 

type AbstractRing = N

The underlying scalar field.

fn multiply_by(&self, n: N) -> Quaternion<N>

Multiplies an element of the ring with an element of the module.

impl<N> AbstractLoop<Additive> for Quaternion<N> where
    N: RealField, 

impl<'a, 'b, N> Mul<&'b Quaternion<N>> for &'a Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the * operator.

fn mul(
    self,
    rhs: &'b Quaternion<N>
) -> <&'a Quaternion<N> as Mul<&'b Quaternion<N>>>::Output

Performs the * operation.

impl<'b, N> Mul<&'b Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the * operator.

fn mul(
    self,
    rhs: &'b Quaternion<N>
) -> <Quaternion<N> as Mul<&'b Quaternion<N>>>::Output

Performs the * operation.

impl<N> Mul<Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the * operator.

fn mul(
    self,
    rhs: Quaternion<N>
) -> <Quaternion<N> as Mul<Quaternion<N>>>::Output

Performs the * operation.

impl<'a, N> Mul<N> for &'a Quaternion<N> where
    N: RealField, 

type Output = Quaternion<N>

The resulting type after applying the * operator.

fn mul(self, n: N) -> <&'a Quaternion<N> as Mul<N>>::Output

Performs the * operation.

impl<N> Mul<N> for Quaternion<N> where
    N: RealField, 

type Output = Quaternion<N>

The resulting type after applying the * operator.

fn mul(self, n: N) -> <Quaternion<N> as Mul<N>>::Output

Performs the * operation.

impl<'a, N> Mul<Quaternion<N>> for &'a Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the * operator.

fn mul(
    self,
    rhs: Quaternion<N>
) -> <&'a Quaternion<N> as Mul<Quaternion<N>>>::Output

Performs the * operation.

impl<N> Display for Quaternion<N> where
    N: Display + RealField, 

fn fmt(&self, f: &mut Formatter) -> Result<(), Error>

Formats the value using the given formatter.

impl<N> Debug for Quaternion<N> where
    N: Debug + RealField, 

fn fmt(&self, f: &mut Formatter) -> Result<(), Error>

Formats the value using the given formatter.

impl<N> Hash for Quaternion<N> where
    N: Hash + RealField, 

fn hash<H>(&self, state: &mut H) where
    H: Hasher, 

Feeds this value into the given [Hasher].

default fn hash_slice<H>(data: &[Self], state: &mut H) where
    H: Hasher, 

Feeds a slice of this type into the given [Hasher].

impl<N1, N2> SubsetOf<Quaternion<N2>> for Quaternion<N1> where
    N1: RealField,
    N2: RealField + SupersetOf<N1>, 

fn to_superset(&self) -> Quaternion<N2>

The inclusion map: converts self to the equivalent element of its superset.

fn is_in_subset(q: &Quaternion<N2>) -> bool

Checks if element is actually part of the subset Self (and can be converted to it).

unsafe fn from_superset_unchecked(q: &Quaternion<N2>) -> Quaternion<N1>

Use with care! Same as self.to_superset but without any property checks. Always succeeds.

default fn from_superset(element: &T) -> Option<Self>

The inverse inclusion map: attempts to construct self from the equivalent element of its superset.

impl<N> AbstractGroupAbelian<Additive> for Quaternion<N> where
    N: RealField, 

default fn prop_is_commutative_approx(args: (Self, Self)) -> bool where
    Self: RelativeEq<Self>, 

Returns true if the operator is commutative for the given argument tuple. Approximate equality is used for verifications.

default fn prop_is_commutative(args: (Self, Self)) -> bool where
    Self: Eq, 

Returns true if the operator is commutative for the given argument tuple.

impl<'a, N> Add<Quaternion<N>> for &'a Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the + operator.

fn add(
    self,
    rhs: Quaternion<N>
) -> <&'a Quaternion<N> as Add<Quaternion<N>>>::Output

Performs the + operation.

impl<'b, N> Add<&'b Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the + operator.

fn add(
    self,
    rhs: &'b Quaternion<N>
) -> <Quaternion<N> as Add<&'b Quaternion<N>>>::Output

Performs the + operation.

impl<'a, 'b, N> Add<&'b Quaternion<N>> for &'a Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the + operator.

fn add(
    self,
    rhs: &'b Quaternion<N>
) -> <&'a Quaternion<N> as Add<&'b Quaternion<N>>>::Output

Performs the + operation.

impl<N> Add<Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the + operator.

fn add(
    self,
    rhs: Quaternion<N>
) -> <Quaternion<N> as Add<Quaternion<N>>>::Output

Performs the + operation.

impl<N> Zero for Quaternion<N> where
    N: RealField, 

fn zero() -> Quaternion<N>

Returns the additive identity element of Self, 0.

fn is_zero(&self) -> bool

Returns true if self is equal to the additive identity.

impl<N> AbstractMagma<Additive> for Quaternion<N> where
    N: RealField, 

fn operate(&self, rhs: &Quaternion<N>) -> Quaternion<N>

Performs an operation.

default fn op(&self, O, lhs: &Self) -> Self

Performs specific operation.

impl<N> AbstractMagma<Multiplicative> for Quaternion<N> where
    N: RealField, 

fn operate(&self, rhs: &Quaternion<N>) -> Quaternion<N>

Performs an operation.

default fn op(&self, O, lhs: &Self) -> Self

Performs specific operation.

impl<N> Deref for Quaternion<N> where
    N: RealField, 

type Target = IJKW<N>

The resulting type after dereferencing.

fn deref(&self) -> &<Quaternion<N> as Deref>::Target

Dereferences the value.

impl<N> DerefMut for Quaternion<N> where
    N: RealField, 

fn deref_mut(&mut self) -> &mut <Quaternion<N> as Deref>::Target

Mutably dereferences the value.

impl<'b, N> SubAssign<&'b Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

fn sub_assign(&mut self, rhs: &'b Quaternion<N>)

Performs the -= operation.

impl<N> SubAssign<Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

fn sub_assign(&mut self, rhs: Quaternion<N>)

Performs the -= operation.

impl<N> UlpsEq<Quaternion<N>> for Quaternion<N> where
    N: UlpsEq<N, Epsilon = N> + RealField, 

fn default_max_ulps() -> u32

The default ULPs to tolerate when testing values that are far-apart.

fn ulps_eq(
    &self,
    other: &Quaternion<N>,
    epsilon: <Quaternion<N> as AbsDiffEq<Quaternion<N>>>::Epsilon,
    max_ulps: u32
) -> bool

A test for equality that uses units in the last place (ULP) if the values are far apart.

default fn ulps_ne(
    &self,
    other: &Rhs,
    epsilon: Self::Epsilon,
    max_ulps: u32
) -> bool

The inverse of ApproxEq::ulps_eq.

impl<'a, N> Div<N> for &'a Quaternion<N> where
    N: RealField, 

type Output = Quaternion<N>

The resulting type after applying the / operator.

fn div(self, n: N) -> <&'a Quaternion<N> as Div<N>>::Output

Performs the / operation.

impl<N> Div<N> for Quaternion<N> where
    N: RealField, 

type Output = Quaternion<N>

The resulting type after applying the / operator.

fn div(self, n: N) -> <Quaternion<N> as Div<N>>::Output

Performs the / operation.

impl<N> AbstractMonoid<Additive> for Quaternion<N> where
    N: RealField, 

default fn prop_operating_identity_element_is_noop_approx(args: (Self,)) -> bool where
    Self: RelativeEq<Self>, 

Checks whether operating with the identity element is a no-op for the given argument. Approximate equality is used for verifications.

default fn prop_operating_identity_element_is_noop(args: (Self,)) -> bool where
    Self: Eq, 

Checks whether operating with the identity element is a no-op for the given argument.

impl<N> AbstractMonoid<Multiplicative> for Quaternion<N> where
    N: RealField, 

default fn prop_operating_identity_element_is_noop_approx(args: (Self,)) -> bool where
    Self: RelativeEq<Self>, 

Checks whether operating with the identity element is a no-op for the given argument. Approximate equality is used for verifications.

default fn prop_operating_identity_element_is_noop(args: (Self,)) -> bool where
    Self: Eq, 

Checks whether operating with the identity element is a no-op for the given argument.

impl<N> FiniteDimVectorSpace for Quaternion<N> where
    N: RealField, 

fn dimension() -> usize

The vector space dimension.

fn canonical_basis_element(i: usize) -> Quaternion<N>

The i-the canonical basis element.

fn dot(&self, other: &Quaternion<N>) -> N

The dot product between two vectors.

unsafe fn component_unchecked(&self, i: usize) -> &N

Same as &self[i] but without bound-checking.

unsafe fn component_unchecked_mut(&mut self, i: usize) -> &mut N

Same as &mut self[i] but without bound-checking.

default fn canonical_basis<F>(f: F) where
    F: FnMut(&Self) -> bool, 

Applies the given closule to each element of this vector space's canonical basis. Stops if f returns false.

impl<N> IndexMut<usize> for Quaternion<N> where
    N: RealField, 

fn index_mut(&mut self, i: usize) -> &mut N

Performs the mutable indexing (container[index]) operation.

impl<N> AbstractQuasigroup<Additive> for Quaternion<N> where
    N: RealField, 

default fn prop_inv_is_latin_square_approx(args: (Self, Self)) -> bool where
    Self: RelativeEq<Self>, 

Returns true if latin squareness holds for the given arguments. Approximate equality is used for verifications.

default fn prop_inv_is_latin_square(args: (Self, Self)) -> bool where
    Self: Eq, 

Returns true if latin squareness holds for the given arguments.

impl<N> Index<usize> for Quaternion<N> where
    N: RealField, 

type Output = N

The returned type after indexing.

fn index(&self, i: usize) -> &<Quaternion<N> as Index<usize>>::Output

Performs the indexing (container[index]) operation.

impl<N> Eq for Quaternion<N> where
    N: Eq + RealField, 

impl<'a, 'b, N> Sub<&'b Quaternion<N>> for &'a Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the - operator.

fn sub(
    self,
    rhs: &'b Quaternion<N>
) -> <&'a Quaternion<N> as Sub<&'b Quaternion<N>>>::Output

Performs the - operation.

impl<N> Sub<Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the - operator.

fn sub(
    self,
    rhs: Quaternion<N>
) -> <Quaternion<N> as Sub<Quaternion<N>>>::Output

Performs the - operation.

impl<'b, N> Sub<&'b Quaternion<N>> for Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the - operator.

fn sub(
    self,
    rhs: &'b Quaternion<N>
) -> <Quaternion<N> as Sub<&'b Quaternion<N>>>::Output

Performs the - operation.

impl<'a, N> Sub<Quaternion<N>> for &'a Quaternion<N> where
    N: RealField,
    DefaultAllocator: Allocator<N, U4, U1>,
    DefaultAllocator: Allocator<N, U4, U1>, 

type Output = Quaternion<N>

The resulting type after applying the - operator.

fn sub(
    self,
    rhs: Quaternion<N>
) -> <&'a Quaternion<N> as Sub<Quaternion<N>>>::Output

Performs the - operation.

impl<N> From<Matrix<N, U4, U1, <DefaultAllocator as Allocator<N, U4, U1>>::Buffer>> for Quaternion<N> where
    N: RealField, 

fn from(
    coords: Matrix<N, U4, U1, <DefaultAllocator as Allocator<N, U4, U1>>::Buffer>
) -> Quaternion<N>

Performs the conversion.