Struct nalgebra::geometry::Quaternion

#[repr(C)]
pub struct Quaternion<T> {
    pub coords: Vector4<T>,
}

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

Fields

coords: Vector4<T>

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

Implementations

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

pub fn into_owned(self) -> Self

👎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) -> Self

👎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) -> Self

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) -> Vector3<T>

The imaginary part of this quaternion.

pub fn conjugate(&self) -> Self

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: &Self, t: T) -> Self

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
) -> MatrixSlice<'_, T, U3, U1, RStride<T, U4, U1>, CStride<T, U4, U1>>

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) -> &Vector4<T>

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: &Self) -> 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: SimdRealField> Quaternion<T>where
    T::Element: SimdRealField,

pub fn try_inverse(&self) -> Option<Self>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());

pub fn simd_try_inverse(&self) -> SimdOption<Self>

Attempt to inverse this quaternion.

This method also works with SIMD components.

pub fn inner(&self, other: &Self) -> Self

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: &Self) -> Self

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: &Self) -> Option<Self>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: &Self) -> Option<Self>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<Vector3<T>>>)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) -> Self

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) -> Self

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) -> Self

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) -> Self

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 Vector4<T>

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
) -> MatrixSliceMut<'_, T, U3, U1, RStride<T, U4, U1>, CStride<T, U4, U1>>

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::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) -> Self

Calculates square of a quaternion.

pub fn half(&self) -> Self

Divides quaternion into two.

pub fn sqrt(&self) -> Self

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) -> Self

Convert quaternion to pure quaternion.

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

Left quaternionic division.

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

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

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) -> Self

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) -> Self

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) -> Self

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) -> Selfwhere
    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) -> Selfwhere
    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) -> Self

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) -> Self

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) -> Self

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) -> Self

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) -> Selfwhere
    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) -> Self

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: Vector4<T>) -> Self

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) -> Self

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: Scalar>(self) -> Quaternion<To>where
    T: Scalar,
    To: 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: SimdRealField> Quaternion<T>

pub fn from_imag(vector: Vector3<T>) -> Self

Constructs a pure quaternion.

pub fn from_parts<SB>(scalar: T, vector: Vector<T, U3, SB>) -> Selfwhere
    SB: Storage<T, U3>,

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) -> Self

Constructs a real quaternion.

pub fn identity() -> Self

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

pub fn from_polar_decomposition<SB>(
    scale: T,
    theta: T,
    axis: Unit<Vector<T, U3, SB>>
) -> Selfwhere
    SB: Storage<T, U3>,

Creates a new quaternion from its polar decomposition.

Note that axis is assumed to be a unit vector.

Trait Implementations

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

type Epsilon = T

Used for specifying relative comparisons.

fn default_epsilon() -> Self::Epsilon

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

fn abs_diff_eq(&self, other: &Self, epsilon: Self::Epsilon) -> bool

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

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

The inverse of AbsDiffEq::abs_diff_eq.

impl<T: RealField + RealField> AbstractMagma<Additive> for Quaternion<T>

fn operate(&self, rhs: &Self) -> Self

Performs an operation.

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

Performs specific operation.

impl<T: RealField + RealField> AbstractMagma<Multiplicative> for Quaternion<T>

fn operate(&self, rhs: &Self) -> Self

Performs an operation.

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

Performs specific operation.

impl<T: RealField + RealField> AbstractModule<Additive, Additive, Multiplicative> for Quaternion<T>

type AbstractRing = T

The underlying scalar field.

fn multiply_by(&self, n: T) -> Self

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

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

type Output = Quaternion<T>

The resulting type after applying the + operator.

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

Performs the + operation.

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

type Output = Quaternion<T>

The resulting type after applying the + operator.

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

Performs the + operation.

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

type Output = Quaternion<T>

The resulting type after applying the + operator.

fn add(self, rhs: Quaternion<T>) -> Self::Output

Performs the + operation.

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

type Output = Quaternion<T>

The resulting type after applying the + operator.

fn add(self, rhs: Quaternion<T>) -> Self::Output

Performs the + operation.

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

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

Performs the += operation.

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

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

Performs the += operation.

impl<T: SimdRealField + Arbitrary> Arbitrary for Quaternion<T>where
    Owned<T, U4>: Send,

fn arbitrary(g: &mut Gen) -> Self

Return an arbitrary value.

fn shrink(&self) -> Box<dyn Iterator<Item = Self> + 'static, Global>

Return an iterator of values that are smaller than itself.

impl<T> Archive for Quaternion<T>where
    Vector4<T>: Archive,

type Archived = ArchivedQuaternion<T>

The archived representation of this type.

type Resolver = QuaternionResolver<T>

The resolver for this type. It must contain all the additional information from serializing needed to make the archived type from the normal type.

unsafe fn resolve(
    &self,
    pos: usize,
    resolver: Self::Resolver,
    out: *mut Self::Archived
)

Creates the archived version of this value at the given position and writes it to the given output.

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

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

Returns a copy of the value.

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

Performs copy-assignment from source.

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

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

Formats the value using the given formatter.

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

fn default() -> Self

Returns the “default value” for a type.

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

type Target = IJKW<T>

The resulting type after dereferencing.

fn deref(&self) -> &Self::Target

Dereferences the value.

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

fn deref_mut(&mut self) -> &mut Self::Target

Mutably dereferences the value.

impl<'a, T: Scalar> Deserialize<'a> for Quaternion<T>where
    Owned<T, U4>: Deserialize<'a>,

fn deserialize<Des>(deserializer: Des) -> Result<Self, Des::Error>where
    Des: Deserializer<'a>,

Deserialize this value from the given Serde deserializer.

impl<__D: Fallible + ?Sized, T> Deserialize<Quaternion<T>, __D> for Archived<Quaternion<T>>where
    Vector4<T>: Archive,
    Archived<Vector4<T>>: Deserialize<Vector4<T>, __D>,

fn deserialize(
    &self,
    deserializer: &mut __D
) -> Result<Quaternion<T>, __D::Error>

Deserializes using the given deserializer

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

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

Formats the value using the given formatter.

impl<T: SimdRealField> Distribution<Quaternion<T>> for Standardwhere
    Standard: Distribution<T>,

fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Quaternion<T>

Generate a random value of T, using rng as the source of randomness.

fn sample_iter<R>(self, rng: R) -> DistIter<Self, R, T>where
    R: Rng,
    Self: Sized,

Create an iterator that generates random values of T, using rng as the source of randomness.

fn map<F, S>(self, func: F) -> DistMap<Self, F, T, S>where
    F: Fn(T) -> S,
    Self: Sized,

Create a distribution of values of ‘S’ by mapping the output of Self through the closure F

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

type Output = Quaternion<T>

The resulting type after applying the / operator.

fn div(self, n: T) -> Self::Output

Performs the / operation.

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

type Output = Quaternion<T>

The resulting type after applying the / operator.

fn div(self, n: T) -> Self::Output

Performs the / operation.

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

fn div_assign(&mut self, n: T)

Performs the /= operation.

impl<T: RealField + RealField> FiniteDimVectorSpace for Quaternion<T>

fn dimension() -> usize

The vector space dimension.

fn canonical_basis_element(i: usize) -> Self

The i-the canonical basis element.

fn dot(&self, other: &Self) -> T

The dot product between two vectors.

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

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

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

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

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

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

Converts to this type from the input type.

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

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

Converts to this type from the input type.

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

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

Converts to this type from the input type.

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

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

Converts to this type from the input type.

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

fn from(coords: [T; 4]) -> Self

Converts to this type from the input type.

impl From<DQuat> for Quaternion<f64>

fn from(e: DQuat) -> Quaternion<f64>

Converts to this type from the input type.

impl From<DQuat> for Quaternion<f64>

fn from(e: DQuat) -> Quaternion<f64>

Converts to this type from the input type.

impl From<DQuat> for Quaternion<f64>

fn from(e: DQuat) -> Quaternion<f64>

Converts to this type from the input type.

impl From<DQuat> for Quaternion<f64>

fn from(e: DQuat) -> Quaternion<f64>

Converts to this type from the input type.

impl From<DQuat> for Quaternion<f64>

fn from(e: DQuat) -> Quaternion<f64>

Converts to this type from the input type.

impl From<DQuat> for Quaternion<f64>

fn from(e: DQuat) -> Quaternion<f64>

Converts to this type from the input type.

impl From<DQuat> for Quaternion<f64>

fn from(e: DQuat) -> Quaternion<f64>

Converts to this type from the input type.

impl From<DQuat> for Quaternion<f64>

fn from(e: DQuat) -> Quaternion<f64>

Converts to this type from the input type.

impl From<DQuat> for Quaternion<f64>

fn from(e: DQuat) -> Quaternion<f64>

Converts to this type from the input type.

impl<T: Scalar> From<Matrix<T, Const<4>, Const<1>, ArrayStorage<T, 4, 1>>> for Quaternion<T>

fn from(coords: Vector4<T>) -> Self

Converts to this type from the input type.

impl From<Quat> for Quaternion<f32>

fn from(e: Quat) -> Quaternion<f32>

Converts to this type from the input type.

impl From<Quat> for Quaternion<f32>

fn from(e: Quat) -> Quaternion<f32>

Converts to this type from the input type.

impl From<Quat> for Quaternion<f32>

fn from(e: Quat) -> Quaternion<f32>

Converts to this type from the input type.

impl From<Quat> for Quaternion<f32>

fn from(e: Quat) -> Quaternion<f32>

Converts to this type from the input type.

impl From<Quat> for Quaternion<f32>

fn from(e: Quat) -> Quaternion<f32>

Converts to this type from the input type.

impl From<Quat> for Quaternion<f32>

fn from(e: Quat) -> Quaternion<f32>

Converts to this type from the input type.

impl From<Quat> for Quaternion<f32>

fn from(e: Quat) -> Quaternion<f32>

Converts to this type from the input type.

impl From<Quat> for Quaternion<f32>

fn from(e: Quat) -> Quaternion<f32>

Converts to this type from the input type.

impl From<Quat> for Quaternion<f32>

fn from(e: Quat) -> Quaternion<f32>

Converts to this type from the input type.

impl<T: Scalar> From<Quaternion<T>> for Quaternion<T>

fn from(q: Quaternion<T>) -> Self

Converts to this type from the input type.

impl From<Quaternion<f32>> for Quat

fn from(e: Quaternion<f32>) -> Quat

Converts to this type from the input type.

impl From<Quaternion<f32>> for Quat

fn from(e: Quaternion<f32>) -> Quat

Converts to this type from the input type.

impl From<Quaternion<f32>> for Quat

fn from(e: Quaternion<f32>) -> Quat

Converts to this type from the input type.

impl From<Quaternion<f32>> for Quat

fn from(e: Quaternion<f32>) -> Quat

Converts to this type from the input type.

impl From<Quaternion<f32>> for Quat

fn from(e: Quaternion<f32>) -> Quat

Converts to this type from the input type.

impl From<Quaternion<f32>> for Quat

fn from(e: Quaternion<f32>) -> Quat

Converts to this type from the input type.

impl From<Quaternion<f32>> for Quat

fn from(e: Quaternion<f32>) -> Quat

Converts to this type from the input type.

impl From<Quaternion<f32>> for Quat

fn from(e: Quaternion<f32>) -> Quat

Converts to this type from the input type.

impl From<Quaternion<f32>> for Quat

fn from(e: Quaternion<f32>) -> Quat

Converts to this type from the input type.

impl From<Quaternion<f64>> for DQuat

fn from(e: Quaternion<f64>) -> DQuat

Converts to this type from the input type.

impl From<Quaternion<f64>> for DQuat

fn from(e: Quaternion<f64>) -> DQuat

Converts to this type from the input type.

impl From<Quaternion<f64>> for DQuat

fn from(e: Quaternion<f64>) -> DQuat

Converts to this type from the input type.

impl From<Quaternion<f64>> for DQuat

fn from(e: Quaternion<f64>) -> DQuat

Converts to this type from the input type.

impl From<Quaternion<f64>> for DQuat

fn from(e: Quaternion<f64>) -> DQuat

Converts to this type from the input type.

impl From<Quaternion<f64>> for DQuat

fn from(e: Quaternion<f64>) -> DQuat

Converts to this type from the input type.

impl From<Quaternion<f64>> for DQuat

fn from(e: Quaternion<f64>) -> DQuat

Converts to this type from the input type.

impl From<Quaternion<f64>> for DQuat

fn from(e: Quaternion<f64>) -> DQuat

Converts to this type from the input type.

impl From<Quaternion<f64>> for DQuat

fn from(e: Quaternion<f64>) -> DQuat

Converts to this type from the input type.

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

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

Feeds this value into the given Hasher.

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

Feeds a slice of this type into the given Hasher.

impl<T: RealField + RealField> Identity<Additive> for Quaternion<T>

fn identity() -> Self

The identity element.

fn id(_: O) -> Selfwhere
    Self: Sized,

Specific identity.

impl<T: RealField + RealField> Identity<Multiplicative> for Quaternion<T>

fn identity() -> Self

The identity element.

fn id(_: O) -> Selfwhere
    Self: Sized,

Specific identity.

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

type Output = T

The returned type after indexing.

fn index(&self, i: usize) -> &Self::Output

Performs the indexing (container[index]) operation.

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

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

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

impl<T: Scalar> Into<Quaternion<T>> for Quaternion<T>

fn into(self) -> Quaternion<T>

Converts this type into the (usually inferred) input type.

impl<T: RealField + RealField> Module for Quaternion<T>

type Ring = T

The underlying scalar field.

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

type Output = Quaternion<T>

The resulting type after applying the * operator.

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

Performs the * operation.

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

type Output = Quaternion<T>

The resulting type after applying the * operator.

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

Performs the * operation.

impl<'b> Mul<&'b Quaternion<f32>> for f32

type Output = Quaternion<f32>

The resulting type after applying the * operator.

fn mul(self, right: &'b Quaternion<f32>) -> Self::Output

Performs the * operation.

impl<'b> Mul<&'b Quaternion<f64>> for f64

type Output = Quaternion<f64>

The resulting type after applying the * operator.

fn mul(self, right: &'b Quaternion<f64>) -> Self::Output

Performs the * operation.

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

type Output = Quaternion<T>

The resulting type after applying the * operator.

fn mul(self, rhs: Quaternion<T>) -> Self::Output

Performs the * operation.

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

type Output = Quaternion<T>

The resulting type after applying the * operator.

fn mul(self, rhs: Quaternion<T>) -> Self::Output

Performs the * operation.

impl Mul<Quaternion<f32>> for f32

type Output = Quaternion<f32>

The resulting type after applying the * operator.

fn mul(self, right: Quaternion<f32>) -> Self::Output

Performs the * operation.

impl Mul<Quaternion<f64>> for f64

type Output = Quaternion<f64>

The resulting type after applying the * operator.

fn mul(self, right: Quaternion<f64>) -> Self::Output

Performs the * operation.

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

type Output = Quaternion<T>

The resulting type after applying the * operator.

fn mul(self, n: T) -> Self::Output

Performs the * operation.

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

type Output = Quaternion<T>

The resulting type after applying the * operator.

fn mul(self, n: T) -> Self::Output

Performs the * operation.

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

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

Performs the *= operation.

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

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

Performs the *= operation.

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

fn mul_assign(&mut self, n: T)

Performs the *= operation.

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

type Output = Quaternion<T>

The resulting type after applying the - operator.

fn neg(self) -> Self::Output

Performs the unary - operation.

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

type Output = Quaternion<T>

The resulting type after applying the - operator.

fn neg(self) -> Self::Output

Performs the unary - operation.

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

type Norm = <T as SimdComplexField>::SimdRealField

The type of the norm.

fn norm(&self) -> T::SimdRealField

Computes the norm.

fn norm_squared(&self) -> T::SimdRealField

Computes the squared norm.

fn scale_mut(&mut self, n: Self::Norm)

Multiply self by n.

fn unscale_mut(&mut self, n: Self::Norm)

Divides self by n.

impl<T: RealField + RealField> NormedSpace for Quaternion<T>

type RealField = T

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

type ComplexField = T

The field of this space must be this complex number.

fn norm_squared(&self) -> T

The squared norm of this vector.

fn norm(&self) -> T

The norm of this vector.

fn normalize(&self) -> Self

Returns a normalized version of this vector.

fn normalize_mut(&mut self) -> T

Normalizes this vector in-place and returns its norm.

fn try_normalize(&self, min_norm: T) -> Option<Self>

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

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

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

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

fn one() -> Self

Returns the multiplicative identity element of Self, 1.

fn set_one(&mut self)

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

fn is_one(&self) -> boolwhere
    Self: PartialEq<Self>,

Returns true if self is equal to the multiplicative identity.

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

fn eq(&self, right: &Self) -> bool

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

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

This method tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

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

fn default_max_relative() -> Self::Epsilon

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

fn relative_eq(
    &self,
    other: &Self,
    epsilon: Self::Epsilon,
    max_relative: Self::Epsilon
) -> bool

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

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

The inverse of RelativeEq::relative_eq.

impl<__S: Fallible + ?Sized, T> Serialize<__S> for Quaternion<T>where
    Vector4<T>: Serialize<__S>,

fn serialize(&self, serializer: &mut __S) -> Result<Self::Resolver, __S::Error>

Writes the dependencies for the object and returns a resolver that can create the archived type.

impl<T: Scalar> Serialize for Quaternion<T>where
    Owned<T, U4>: Serialize,

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

Serialize this value into the given Serde serializer.

impl<T: Scalar + SimdValue> SimdValue for Quaternion<T>where
    T::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.

fn lanes() -> usize

The number of lanes of this SIMD value.

fn splat(val: Self::Element) -> Self

Initializes an SIMD value with each lanes set to val.

fn extract(&self, i: usize) -> Self::Element

Extracts the i-th lane of self.

unsafe fn extract_unchecked(&self, i: usize) -> Self::Element

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

fn replace(&mut self, i: usize, val: Self::Element)

Replaces the i-th lane of self by val.

unsafe fn replace_unchecked(&mut self, i: usize, val: Self::Element)

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

fn select(self, cond: Self::SimdBool, other: Self) -> Self

Merges self and other depending on the lanes of cond.

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

Applies a function to each lane of self.

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

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

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

type Output = Quaternion<T>

The resulting type after applying the - operator.

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

Performs the - operation.

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

type Output = Quaternion<T>

The resulting type after applying the - operator.

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

Performs the - operation.

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

type Output = Quaternion<T>

The resulting type after applying the - operator.

fn sub(self, rhs: Quaternion<T>) -> Self::Output

Performs the - operation.

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

type Output = Quaternion<T>

The resulting type after applying the - operator.

fn sub(self, rhs: Quaternion<T>) -> Self::Output

Performs the - operation.

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

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

Performs the -= operation.

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

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>,

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

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

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

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

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

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

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

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

impl<T: RealField + RealField> TwoSidedInverse<Additive> for Quaternion<T>

fn two_sided_inverse(&self) -> Self

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

fn two_sided_inverse_mut(&mut self)

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

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

fn default_max_ulps() -> u32

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

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

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

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

The inverse of UlpsEq::ulps_eq.

impl<T: RealField + RealField> VectorSpace for Quaternion<T>

type Field = T

The underlying scalar field.

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

fn zero() -> Self

Returns the additive identity element of Self, 0.

fn is_zero(&self) -> bool

Returns true if self is equal to the additive identity.

fn set_zero(&mut self)

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

impl<T: Scalar> Zeroable for Quaternion<T>where
    Vector4<T>: Zeroable,

fn zeroed() -> Self

Calls zeroed.

impl<T: RealField + RealField> AbstractGroup<Additive> for Quaternion<T>

impl<T: RealField + RealField> AbstractGroupAbelian<Additive> for Quaternion<T>

impl<T: RealField + RealField> AbstractLoop<Additive> for Quaternion<T>

impl<T: RealField + RealField> AbstractMonoid<Additive> for Quaternion<T>

impl<T: RealField + RealField> AbstractMonoid<Multiplicative> for Quaternion<T>

impl<T: RealField + RealField> AbstractQuasigroup<Additive> for Quaternion<T>

impl<T: RealField + RealField> AbstractSemigroup<Additive> for Quaternion<T>

impl<T: RealField + RealField> AbstractSemigroup<Multiplicative> for Quaternion<T>

impl<T: Copy> Copy for Quaternion<T>

impl<T: DeviceCopy> DeviceCopy for Quaternion<T>

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

impl<T: Scalar> Pod for Quaternion<T>where
    Vector4<T>: Pod,
    T: Copy,