Struct lnkit::prelude::geometry::Quaternion

#[repr(C)]
pub struct Quaternion<T> {
    pub coords: Matrix<T, Const<{_: usize}>, Const<1_usize>, ArrayStorage<T, 4_usize, 1_usize>>,
}

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

Fields

coords: Matrix<T, Const<{_: usize}>, Const<1_usize>, ArrayStorage<T, 4_usize, 1_usize>>

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

Implementations

impl<T> Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

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

👎 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<T>

👎 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.

#[must_use = "Did you mean to use normalize_mut()?"]

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

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<T, Const<{_: usize}>, Const<1_usize>, ArrayStorage<T, 3_usize, 1_usize>>

The imaginary part of this quaternion.

#[must_use = "Did you mean to use conjugate_mut()?"]

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

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 lerp(&self, other: &Quaternion<T>, t: T) -> Quaternion<T>

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<T, Const<{_: usize}>, Const<1_usize>, SliceStorage<'_, T, Const<{_: usize}>, Const<1_usize>, <<DefaultAllocator as Allocator<T, Const<{_: usize}>, Const<1_usize>>>::Buffer as Storage<T, Const<{_: usize}>, Const<1_usize>>>::RStride, <<DefaultAllocator as Allocator<T, Const<{_: usize}>, Const<1_usize>>>::Buffer as Storage<T, Const<{_: usize}>, Const<1_usize>>>::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) -> T

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<T, Const<{_: usize}>, Const<1_usize>, ArrayStorage<T, 4_usize, 1_usize>>

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) -> T

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) -> T

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) -> T

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) -> T

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<T>) -> T

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);

impl<T> Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

#[must_use = "Did you mean to use try_inverse_mut()?"]

pub fn try_inverse(&self) -> Option<Quaternion<T>> where
    T: RealField, 

Inverts this quaternion if it is not zero.

This method also does not works with SIMD components (see simd_try_inverse instead).

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());

#[must_use = "Did you mean to use try_inverse_mut()?"]

pub fn simd_try_inverse(&self) -> SimdOption<Quaternion<T>>

Attempt to inverse this quaternion.

This method also works with SIMD components.

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

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<T>) -> Quaternion<T>

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<T>) -> Option<Quaternion<T>> where
    T: RealField, 

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<T>) -> Option<Quaternion<T>> where
    T: RealField, 

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
) -> (T, T, Option<Unit<Matrix<T, Const<{_: usize}>, Const<1_usize>, ArrayStorage<T, 3_usize, 1_usize>>>>) where
    T: RealField, 

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<T>

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<T>

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: T) -> Quaternion<T>

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: T) -> Quaternion<T>

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<T, Const<{_: usize}>, Const<1_usize>, ArrayStorage<T, 4_usize, 1_usize>>

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<T, Const<{_: usize}>, Const<1_usize>, SliceStorageMut<'_, T, Const<{_: usize}>, Const<1_usize>, <<DefaultAllocator as Allocator<T, Const<{_: usize}>, Const<1_usize>>>::Buffer as Storage<T, Const<{_: usize}>, Const<1_usize>>>::RStride, <<DefaultAllocator as Allocator<T, Const<{_: usize}>, Const<1_usize>>>::Buffer as Storage<T, Const<{_: usize}>, Const<1_usize>>>::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) -> <T as SimdValue>::SimdBool

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

Example

let mut q = Quaternion::new(1.0f32, 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(0.0f32, 0.0, 0.0, 0.0);
assert!(!q.try_inverse_mut());

pub fn normalize_mut(&mut self) -> T

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<T>

Calculates square of a quaternion.

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

Divides quaternion into two.

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

Calculates square root.

pub fn is_pure(&self) -> bool

Check if the quaternion is pure.

A quaternion is pure if it has no real part (self.w == 0.0).

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

Convert quaternion to pure quaternion.

pub fn left_div(&self, other: &Quaternion<T>) -> Option<Quaternion<T>> where
    T: RealField, 

Left quaternionic division.

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

pub fn right_div(&self, other: &Quaternion<T>) -> Option<Quaternion<T>> where
    T: RealField, 

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<T>

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<T>

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<T>

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<T>

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<T> where
    T: RealField, 

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<T> where
    T: RealField, 

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<T>

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<T>

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<T>

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<T>

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<T> where
    T: RealField, 

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<T>

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<T> Quaternion<T>

pub const fn from_vector(
    vector: Matrix<T, Const<{_: usize}>, Const<1_usize>, ArrayStorage<T, 4_usize, 1_usize>>
) -> Quaternion<T>

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

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

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 cast<To>(self) -> Quaternion<To> where
    T: Scalar,
    To: Scalar + SupersetOf<T>, 

Cast the components of self to another type.

Example

let q = Quaternion::new(1.0f64, 2.0, 3.0, 4.0);
let q2 = q.cast::<f32>();
assert_eq!(q2, Quaternion::new(1.0f32, 2.0, 3.0, 4.0));

impl<T> Quaternion<T> where
    T: SimdRealField, 

pub fn from_imag(
    vector: Matrix<T, Const<{_: usize}>, Const<1_usize>, ArrayStorage<T, 3_usize, 1_usize>>
) -> Quaternion<T>

Constructs a pure quaternion.

pub fn from_parts<SB>(
    scalar: T,
    vector: Matrix<T, Const<{_: usize}>, Const<1_usize>, SB>
) -> Quaternion<T> where
    SB: Storage<T, Const<{_: usize}>, Const<1_usize>>, 

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: T) -> Quaternion<T>

Constructs a real quaternion.

pub fn identity() -> Quaternion<T>

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);

impl<T> Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

pub fn from_polar_decomposition<SB>(
    scale: T,
    theta: T,
    axis: Unit<Matrix<T, Const<{_: usize}>, Const<1_usize>, SB>>
) -> Quaternion<T> where
    SB: Storage<T, Const<{_: usize}>, Const<1_usize>>, 

Creates a new quaternion from its polar decomposition.

Note that axis is assumed to be a unit vector.

Trait Implementations

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

type Epsilon = T

Used for specifying relative comparisons.

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

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

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

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

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

The inverse of [AbsDiffEq::abs_diff_eq].

impl<'a, 'b, T> Add<&'b Quaternion<T>> for &'a Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the + operator.

pub fn add(
    self,
    rhs: &'b Quaternion<T>
) -> <&'a Quaternion<T> as Add<&'b Quaternion<T>>>::Output

Performs the + operation.

impl<'b, T> Add<&'b Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the + operator.

pub fn add(
    self,
    rhs: &'b Quaternion<T>
) -> <Quaternion<T> as Add<&'b Quaternion<T>>>::Output

Performs the + operation.

impl<'a, T> Add<Quaternion<T>> for &'a Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the + operator.

pub fn add(
    self,
    rhs: Quaternion<T>
) -> <&'a Quaternion<T> as Add<Quaternion<T>>>::Output

Performs the + operation.

impl<T> Add<Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the + operator.

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

Performs the + operation.

impl<'b, T> AddAssign<&'b Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

pub fn add_assign(&mut self, rhs: &'b Quaternion<T>)

Performs the += operation.

impl<T> AddAssign<Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

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

Performs the += operation.

impl<T> Clone for Quaternion<T> where
    T: Clone, 

pub fn clone(&self) -> Quaternion<T>

Returns a copy of the value.

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

Performs copy-assignment from source.

impl<T> Copy for Quaternion<T> where
    T: Copy, 

impl<T> Debug for Quaternion<T> where
    T: Debug, 

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

Formats the value using the given formatter.

impl<T> Default for Quaternion<T> where
    T: Scalar + Zero, 

pub fn default() -> Quaternion<T>

Returns the “default value” for a type.

impl<T> Deref for Quaternion<T> where
    T: Scalar + SimdValue, 

type Target = IJKW<T>

The resulting type after dereferencing.

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

Dereferences the value.

impl<T> DerefMut for Quaternion<T> where
    T: Scalar + SimdValue, 

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

Mutably dereferences the value.

impl<'a, T> Deserialize<'a> for Quaternion<T> where
    T: Scalar,
    <DefaultAllocator as Allocator<T, Const<{_: usize}>, Const<1_usize>>>::Buffer: Deserialize<'a>, 

pub fn deserialize<Des>(
    deserializer: Des
) -> Result<Quaternion<T>, <Des as Deserializer<'a>>::Error> where
    Des: Deserializer<'a>, 

Deserialize this value from the given Serde deserializer.

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

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

Formats the value using the given formatter.

impl<'a, T> Div<T> for &'a Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the / operator.

pub fn div(self, n: T) -> <&'a Quaternion<T> as Div<T>>::Output

Performs the / operation.

impl<T> Div<T> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the / operator.

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

Performs the / operation.

impl<T> DivAssign<T> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

pub fn div_assign(&mut self, n: T)

Performs the /= operation.

impl<T> Eq for Quaternion<T> where
    T: Scalar + Eq, 

impl<T> From<[Quaternion<<T as SimdValue>::Element>; 16]> for Quaternion<T> where
    T: Scalar + PrimitiveSimdValue + From<[<T as SimdValue>::Element; 16]>,
    <T as SimdValue>::Element: Scalar,
    <T as SimdValue>::Element: Copy, 

pub fn from(arr: [Quaternion<<T as SimdValue>::Element>; 16]) -> Quaternion<T>

Performs the conversion.

impl<T> From<[Quaternion<<T as SimdValue>::Element>; 2]> for Quaternion<T> where
    T: Scalar + PrimitiveSimdValue + From<[<T as SimdValue>::Element; 2]>,
    <T as SimdValue>::Element: Scalar,
    <T as SimdValue>::Element: Copy, 

pub fn from(arr: [Quaternion<<T as SimdValue>::Element>; 2]) -> Quaternion<T>

Performs the conversion.

impl<T> From<[Quaternion<<T as SimdValue>::Element>; 4]> for Quaternion<T> where
    T: Scalar + PrimitiveSimdValue + From<[<T as SimdValue>::Element; 4]>,
    <T as SimdValue>::Element: Scalar,
    <T as SimdValue>::Element: Copy, 

pub fn from(arr: [Quaternion<<T as SimdValue>::Element>; 4]) -> Quaternion<T>

Performs the conversion.

impl<T> From<[Quaternion<<T as SimdValue>::Element>; 8]> for Quaternion<T> where
    T: Scalar + PrimitiveSimdValue + From<[<T as SimdValue>::Element; 8]>,
    <T as SimdValue>::Element: Scalar,
    <T as SimdValue>::Element: Copy, 

pub fn from(arr: [Quaternion<<T as SimdValue>::Element>; 8]) -> Quaternion<T>

Performs the conversion.

impl<T> From<[T; 4]> for Quaternion<T> where
    T: Scalar, 

pub fn from(coords: [T; 4]) -> Quaternion<T>

Performs the conversion.

impl<T> From<Matrix<T, Const<{_: usize}>, Const<1_usize>, ArrayStorage<T, 4_usize, 1_usize>>> for Quaternion<T> where
    T: Scalar, 

pub fn from(
    coords: Matrix<T, Const<{_: usize}>, Const<1_usize>, ArrayStorage<T, 4_usize, 1_usize>>
) -> Quaternion<T>

Performs the conversion.

impl<T> Hash for Quaternion<T> where
    T: Scalar + Hash, 

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

Feeds this value into the given Hasher.

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

Feeds a slice of this type into the given Hasher.

impl<T> Index<usize> for Quaternion<T> where
    T: Scalar, 

type Output = T

The returned type after indexing.

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

Performs the indexing (container[index]) operation.

impl<T> IndexMut<usize> for Quaternion<T> where
    T: Scalar, 

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

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

impl<'a, 'b, T> Mul<&'b Quaternion<T>> for &'a Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the * operator.

pub fn mul(
    self,
    rhs: &'b Quaternion<T>
) -> <&'a Quaternion<T> as Mul<&'b Quaternion<T>>>::Output

Performs the * operation.

impl<'b, T> Mul<&'b Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the * operator.

pub fn mul(
    self,
    rhs: &'b Quaternion<T>
) -> <Quaternion<T> as Mul<&'b Quaternion<T>>>::Output

Performs the * operation.

impl<'a, T> Mul<Quaternion<T>> for &'a Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the * operator.

pub fn mul(
    self,
    rhs: Quaternion<T>
) -> <&'a Quaternion<T> as Mul<Quaternion<T>>>::Output

Performs the * operation.

impl<T> Mul<Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the * operator.

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

Performs the * operation.

impl<'a, T> Mul<T> for &'a Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the * operator.

pub fn mul(self, n: T) -> <&'a Quaternion<T> as Mul<T>>::Output

Performs the * operation.

impl<T> Mul<T> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the * operator.

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

Performs the * operation.

impl<'b, T> MulAssign<&'b Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

pub fn mul_assign(&mut self, rhs: &'b Quaternion<T>)

Performs the *= operation.

impl<T> MulAssign<Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

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

Performs the *= operation.

impl<T> MulAssign<T> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

pub fn mul_assign(&mut self, n: T)

Performs the *= operation.

impl<T> Neg for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the - operator.

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

Performs the unary - operation.

impl<'a, T> Neg for &'a Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the - operator.

pub fn neg(self) -> <&'a Quaternion<T> as Neg>::Output

Performs the unary - operation.

impl<T> Normed for Quaternion<T> where
    T: SimdRealField, 

type Norm = <T as SimdComplexField>::SimdRealField

The type of the norm.

pub fn norm(&self) -> <T as SimdComplexField>::SimdRealField

Computes the norm.

pub fn norm_squared(&self) -> <T as SimdComplexField>::SimdRealField

Computes the squared norm.

pub fn scale_mut(&mut self, n: <Quaternion<T> as Normed>::Norm)

Multiply self by n.

pub fn unscale_mut(&mut self, n: <Quaternion<T> as Normed>::Norm)

Divides self by n.

impl<T> One for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

pub fn one() -> Quaternion<T>

Returns the multiplicative identity element of Self, 1.

pub fn set_one(&mut self)

Sets self to the multiplicative identity element of Self, 1.

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

Returns true if self is equal to the multiplicative identity.

impl<T> PartialEq<Quaternion<T>> for Quaternion<T> where
    T: Scalar, 

pub fn eq(&self, right: &Quaternion<T>) -> bool

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

#[must_use]

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

This method tests for !=.

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

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

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

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

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

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

The inverse of [RelativeEq::relative_eq].

impl<T> Serialize for Quaternion<T> where
    T: Scalar,
    <DefaultAllocator as Allocator<T, Const<{_: usize}>, Const<1_usize>>>::Buffer: Serialize, 

pub fn serialize<S>(
    &self,
    serializer: S
) -> Result<<S as Serializer>::Ok, <S as Serializer>::Error> where
    S: Serializer, 

Serialize this value into the given Serde serializer.

impl<T> SimdValue for Quaternion<T> where
    T: Scalar + SimdValue,
    <T as SimdValue>::Element: Scalar, 

type Element = Quaternion<<T as SimdValue>::Element>

The type of the elements of each lane of this SIMD value.

type SimdBool = <T as SimdValue>::SimdBool

Type of the result of comparing two SIMD values like self.

pub fn lanes() -> usize

The number of lanes of this SIMD value.

pub fn splat(val: <Quaternion<T> as SimdValue>::Element) -> Quaternion<T>

Initializes an SIMD value with each lanes set to val.

pub fn extract(&self, i: usize) -> <Quaternion<T> as SimdValue>::Element

Extracts the i-th lane of self.

pub unsafe fn extract_unchecked(
    &self,
    i: usize
) -> <Quaternion<T> as SimdValue>::Element

Extracts the i-th lane of self without bound-checking.

pub fn replace(&mut self, i: usize, val: <Quaternion<T> as SimdValue>::Element)

Replaces the i-th lane of self by val.

pub unsafe fn replace_unchecked(
    &mut self,
    i: usize,
    val: <Quaternion<T> as SimdValue>::Element
)

Replaces the i-th lane of self by val without bound-checking.

pub fn select(
    self,
    cond: <Quaternion<T> as SimdValue>::SimdBool,
    other: Quaternion<T>
) -> Quaternion<T>

Merges self and other depending on the lanes of cond.

pub fn map_lanes(self, f: impl Fn(Self::Element) -> Self::Element) -> Self where
    Self: Clone, 

Applies a function to each lane of self.

pub fn zip_map_lanes(
    self,
    b: Self,
    f: impl Fn(Self::Element, Self::Element) -> Self::Element
) -> Self where
    Self: Clone, 

Applies a function to each lane of self paired with the corresponding lane of b.

impl<'a, 'b, T> Sub<&'b Quaternion<T>> for &'a Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the - operator.

pub fn sub(
    self,
    rhs: &'b Quaternion<T>
) -> <&'a Quaternion<T> as Sub<&'b Quaternion<T>>>::Output

Performs the - operation.

impl<'b, T> Sub<&'b Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the - operator.

pub fn sub(
    self,
    rhs: &'b Quaternion<T>
) -> <Quaternion<T> as Sub<&'b Quaternion<T>>>::Output

Performs the - operation.

impl<T> Sub<Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the - operator.

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

Performs the - operation.

impl<'a, T> Sub<Quaternion<T>> for &'a Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

type Output = Quaternion<T>

The resulting type after applying the - operator.

pub fn sub(
    self,
    rhs: Quaternion<T>
) -> <&'a Quaternion<T> as Sub<Quaternion<T>>>::Output

Performs the - operation.

impl<'b, T> SubAssign<&'b Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

pub fn sub_assign(&mut self, rhs: &'b Quaternion<T>)

Performs the -= operation.

impl<T> SubAssign<Quaternion<T>> for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

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

Performs the -= operation.

impl<T1, T2> SubsetOf<Quaternion<T2>> for Quaternion<T1> where
    T1: Scalar,
    T2: Scalar + SupersetOf<T1>, 

pub fn to_superset(&self) -> Quaternion<T2>

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

pub fn is_in_subset(q: &Quaternion<T2>) -> bool

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

pub fn from_superset_unchecked(q: &Quaternion<T2>) -> Quaternion<T1>

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

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

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

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

pub fn default_max_ulps() -> u32

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

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

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

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

The inverse of [UlpsEq::ulps_eq].

impl<T> Zero for Quaternion<T> where
    T: SimdRealField,
    <T as SimdValue>::Element: SimdRealField, 

pub fn zero() -> Quaternion<T>

Returns the additive identity element of Self, 0.

pub fn is_zero(&self) -> bool

Returns true if self is equal to the additive identity.

pub fn set_zero(&mut self)

Sets self to the additive identity element of Self, 0.