Struct vek::quaternion::repr_c::Quaternion

#[repr(C)]
pub struct Quaternion<T> {
    pub x: T,
    pub y: T,
    pub z: T,
    pub w: T,
}

Quaternions are a convenient representation for rotations in 3D spaces.

IMPORTANT: Quaternions are only valid as rotations as long as they are normalized (i.e their magnitude is 1). Most operations assume this, instead of normalizing inputs behind your back, so be careful.

They essentially consist of a vector part (x, y, z), and scalar part (w). For unit quaternions, the vector part is the unit axis of rotation scaled by the sine of the half-angle of the rotation, and the scalar part is the cosine of the half-angle.

Fields

x: T

y: T

z: T

w: T

Implementations

impl<T> Quaternion<T>

pub fn from_xyzw(x: T, y: T, z: T, w: T) -> Self

Creates a new quaternion with x, y, z and w elements in order.

You are responsible for ensuring that the resulting quaternion is normalized.

pub fn from_scalar_and_vec3<V: Into<Vec3<T>>>(pair: (T, V)) -> Self

Creates a new quaternion from a scalar-and-vector pair.

You are responsible for ensuring that the resulting quaternion is normalized.

pub fn into_scalar_and_vec3(self) -> (T, Vec3<T>)

Converts this quaternion into a scalar-and-vector pair by destructuring.

Not to be confused with into_angle_axis().

pub fn zero() -> Self where
    T: Zero, 

Creates a new quaternion with all elements set to zero.

Be careful: since it has a magnitude equal to zero, it is not valid to use for most operations.

pub fn identity() -> Self where
    T: Zero + One, 

Creates the identity quaternion.

use std::f32::consts::PI;

let id = Quaternion::<f32>::identity();
assert_eq!(id, Default::default());
assert_relative_eq!(id, id.conjugate());
assert_relative_eq!(id, id.inverse());

let q = Quaternion::rotation_y(PI);
assert_relative_eq!(id * q, q);
assert_relative_eq!(q * id, q);

pub fn conjugate(self) -> Self where
    T: Neg<Output = T>, 

Gets this quaternion’s conjugate (copy with negated vector part).

On normalized quaternions, the conjugate also happens to be the inverse.

use std::f32::consts::PI;

let p = Quaternion::rotation_x(PI);
let q = Quaternion::rotation_z(PI);
assert_relative_eq!((p*q).conjugate(), q.conjugate() * p.conjugate());

// Rotation quaternions are normalized, so their conjugate is also their inverse.
assert_relative_eq!(q.conjugate(), q.inverse());

pub fn inverse(self) -> Self where
    T: Neg<Output = T> + Copy + Add<T, Output = T> + Mul<Output = T> + Div<Output = T>, 

Gets this quaternion’s inverse, i.e the one that reverses its effect.

On normalized quaternions, the inverse happens to be the conjugate.

use std::f32::consts::PI;

let rot = Quaternion::rotation_y(PI);
let inv = rot.inverse();
assert_relative_eq!(rot*inv, Quaternion::identity());
assert_relative_eq!(inv*rot, Quaternion::identity());

let p = Quaternion::rotation_x(PI);
let q = Quaternion::rotation_z(PI);
assert_relative_eq!((p*q).inverse(), q.inverse() * p.inverse());

pub fn dot(self, q: Self) -> T where
    T: Copy + Add<T, Output = T> + Mul<Output = T>, 

Gets the dot product between two quaternions.

pub fn normalized(self) -> Self where
    T: Real + Add<T, Output = T>, 

Gets a normalized copy of this quaternion.

pub fn magnitude_squared(self) -> T where
    T: Real + Add<T, Output = T>, 

Gets this quaternion’s magnitude, squared.

pub fn magnitude(self) -> T where
    T: Real + Add<T, Output = T>, 

Gets this quaternion’s magnitude.

pub fn rotation_from_to_3d<V: Into<Vec3<T>>>(from: V, to: V) -> Self where
    T: Real + Add<T, Output = T>, 

Creates a quaternion that would rotate a from direction to to.

let (from, to) = (Vec4::<f32>::unit_x(), Vec4::<f32>::unit_y());
let q = Quaternion::<f32>::rotation_from_to_3d(from, to);
assert_relative_eq!(q * from, to);
assert_relative_eq!(q * Vec4::unit_y(), -Vec4::unit_x());

let (from, to) = (Vec4::<f32>::unit_x(), -Vec4::<f32>::unit_x());
let q = Quaternion::<f32>::rotation_from_to_3d(from, to);
assert_relative_eq!(q * from, to);

pub fn rotation_3d<V: Into<Vec3<T>>>(angle_radians: T, axis: V) -> Self where
    T: Real + Add<T, Output = T>, 

Creates a quaternion from an angle and axis. The axis is not required to be normalized.

pub fn rotation_x(angle_radians: T) -> Self where
    T: Real + Add<T, Output = T>, 

Creates a quaternion from an angle for a rotation around the X axis.

pub fn rotation_y(angle_radians: T) -> Self where
    T: Real + Add<T, Output = T>, 

Creates a quaternion from an angle for a rotation around the Y axis.

pub fn rotation_z(angle_radians: T) -> Self where
    T: Real + Add<T, Output = T>, 

Creates a quaternion from an angle for a rotation around the Y axis.

pub fn rotated_3d<V: Into<Vec3<T>>>(self, angle_radians: T, axis: V) -> Self where
    T: Real + Add<T, Output = T>, 

Returns this quaternion rotated around the given axis with given angle. The axis is not required to be normalized.

pub fn rotated_x(self, angle_radians: T) -> Self where
    T: Real + Add<T, Output = T>, 

Returns this quaternion rotated around the X axis with given angle.

pub fn rotated_y(self, angle_radians: T) -> Self where
    T: Real + Add<T, Output = T>, 

Returns this quaternion rotated around the Y axis with given angle.

pub fn rotated_z(self, angle_radians: T) -> Self where
    T: Real + Add<T, Output = T>, 

Returns this quaternion rotated around the Z axis with given angle.

pub fn rotate_3d<V: Into<Vec3<T>>>(&mut self, angle_radians: T, axis: V) where
    T: Real + Add<T, Output = T>, 

Rotates this quaternion around the given axis with given angle. The axis is not required to be normalized.

pub fn rotate_x(&mut self, angle_radians: T) where
    T: Real + Add<T, Output = T>, 

Rotates this quaternion around the X axis with given angle.

pub fn rotate_y(&mut self, angle_radians: T) where
    T: Real + Add<T, Output = T>, 

Rotates this quaternion around the Y axis with given angle.

pub fn rotate_z(&mut self, angle_radians: T) where
    T: Real + Add<T, Output = T>, 

Rotates this quaternion around the Z axis with given angle.

pub fn into_angle_axis(self) -> (T, Vec3<T>) where
    T: Real, 

Convert this quaternion to angle-axis representation, assuming the quaternion is normalized.

use std::f32::consts::PI;

let q = Quaternion::rotation_x(PI/2.);
let (angle, axis) = q.into_angle_axis();
assert_relative_eq!(angle, PI/2.);
assert_relative_eq!(axis, Vec3::unit_x());

let angle = PI*4./5.;
let axis = Vec3::new(1_f32, 2., 3.);
let q = Quaternion::rotation_3d(angle, axis);
let (a, v) = q.into_angle_axis();
assert_relative_eq!(a, angle);
assert_relative_eq!(v, axis.normalized());

pub fn into_vec4(self) -> Vec4<T>

Converts this quaternion to a Vec4 by destructuring.

pub fn from_vec4(v: Vec4<T>) -> Self

Creates a quaternion from a Vec4 by destructuring. You are responsible for ensuring that the resulting quaternion is normalized.

pub fn into_vec3(self) -> Vec3<T>

Converts this quaternion to a Vec3 by destructuring, dropping the w element.

impl<T> Quaternion<T> where
    T: Copy + One + Mul<Output = T> + Sub<Output = T> + MulAdd<T, T, Output = T>, 

pub fn lerp_precise_unnormalized(from: Self, to: Self, factor: T) -> Self where
    T: Clamp + Zero, 

Performs linear interpolation without normalizing the result, using an implementation that supposedly yields a more precise result.

This is probably not what you’re looking for. For an implementation that normalizes the result (which is more commonly wanted), see the Lerp implementation.

pub fn lerp_unclamped_precise_unnormalized(
    from: Self,
    to: Self,
    factor: T
) -> Self

Performs linear interpolation without normalizing the result and without implicitly constraining factor to be between 0 and 1, using an implementation that supposedly yields a more precise result.

This is probably not what you’re looking for. For an implementation that normalizes the result (which is more commonly wanted), see the Lerp implementation.

impl<T> Quaternion<T> where
    T: Copy + Sub<Output = T> + MulAdd<T, T, Output = T>, 

pub fn lerp_unnormalized(from: Self, to: Self, factor: T) -> Self where
    T: Clamp + Zero + One, 

Performs linear interpolation without normalizing the result.

This is probably not what you’re looking for. For an implementation that normalizes the result (which is more commonly wanted), see the Lerp implementation.

pub fn lerp_unclamped_unnormalized(from: Self, to: Self, factor: T) -> Self

Performs linear interpolation without normalizing the result and without implicitly constraining factor to be between 0 and 1.

This is probably not what you’re looking for. For an implementation that normalizes the result (which is more commonly wanted), see the Lerp implementation.

impl<T> Quaternion<T> where
    T: Lerp<T, Output = T> + Add<T, Output = T> + Real, 

pub fn slerp_unclamped(from: Self, to: Self, factor: T) -> Self

Performs spherical linear interpolation without implictly constraining factor to be between 0 and 1.

use std::f32::consts::PI;

let from = Quaternion::rotation_z(0_f32);
let to = Quaternion::rotation_z(PI*9./10.);

let angles = 32;
for i in 0..angles {
    let factor = (i as f32) / (angles as f32);
    let expected = Quaternion::rotation_z(factor * PI*9./10.);
    let slerp = Quaternion::slerp(from, to, factor);
    assert_relative_eq!(slerp, expected);
}

pub fn slerp(from: Self, to: Self, factor: T) -> Self where
    T: Clamp, 

Perform spherical linear interpolation, constraining factor to be between 0 and 1.

Trait Implementations

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

type Epsilon = T::Epsilon

Used for specifying relative comparisons.

fn default_epsilon() -> T::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> Add<Quaternion<T>> for Quaternion<T> where
    T: Add<Output = T>, 

type Output = Self

The resulting type after applying the + operator.

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

Performs the + operation.

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, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

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

The default value for a quaternion is the identity.

assert_eq!(Quaternion::<f32>::identity(), Quaternion::default());

fn default() -> Self

Returns the “default value” for a type.

impl<'de, T> Deserialize<'de> for Quaternion<T> where
    T: Deserialize<'de>, 

fn deserialize<__D>(__deserializer: __D) -> Result<Self, __D::Error> where
    __D: Deserializer<'de>, 

Deserialize this value from the given Serde deserializer.

impl<T> Div<T> for Quaternion<T> where
    T: Copy + Div<Output = T>, 

type Output = Self

The resulting type after applying the / operator.

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

Performs the / operation.

impl<T> From<Quaternion<T>> for Mat4<T> where
    T: Copy + Zero + One + Mul<Output = T> + Add<Output = T> + Sub<Output = T>, 

Rotation matrices can be obtained from quaternions. This implementation only works properly if the quaternion is normalized.

use std::f32::consts::PI;

let angles = 32;
for i in 0..angles {
    let theta = PI * 2. * (i as f32) / (angles as f32);

    assert_relative_eq!(Mat4::rotation_x(theta), Mat4::from(Quaternion::rotation_x(theta)), epsilon = 0.000001);
    assert_relative_eq!(Mat4::rotation_y(theta), Mat4::from(Quaternion::rotation_y(theta)), epsilon = 0.000001);
    assert_relative_eq!(Mat4::rotation_z(theta), Mat4::from(Quaternion::rotation_z(theta)), epsilon = 0.000001);

    assert_relative_eq!(Mat4::rotation_x(theta), Mat4::rotation_3d(theta, Vec4::unit_x()));
    assert_relative_eq!(Mat4::rotation_y(theta), Mat4::rotation_3d(theta, Vec4::unit_y()));
    assert_relative_eq!(Mat4::rotation_z(theta), Mat4::rotation_3d(theta, Vec4::unit_z()));

    // See what rotating unit vectors do for most angles between 0 and 2*PI.
    // It's helpful to picture this as a right-handed coordinate system.

    let v = Vec4::unit_y();
    let m = Mat4::rotation_x(theta);
    assert_relative_eq!(m * v, Vec4::new(0., theta.cos(), theta.sin(), 0.));

    let v = Vec4::unit_z();
    let m = Mat4::rotation_y(theta);
    assert_relative_eq!(m * v, Vec4::new(theta.sin(), 0., theta.cos(), 0.));

    let v = Vec4::unit_x();
    let m = Mat4::rotation_z(theta);
    assert_relative_eq!(m * v, Vec4::new(theta.cos(), theta.sin(), 0., 0.));
}

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

Performs the conversion.

impl<T> From<Quaternion<T>> for Mat3<T> where
    T: Copy + Zero + One + Mul<Output = T> + Add<Output = T> + Sub<Output = T>, 

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

Performs the conversion.

impl<T> From<Quaternion<T>> for Mat4<T> where
    T: Copy + Zero + One + Mul<Output = T> + Add<Output = T> + Sub<Output = T>, 

Rotation matrices can be obtained from quaternions. This implementation only works properly if the quaternion is normalized.

use std::f32::consts::PI;

let angles = 32;
for i in 0..angles {
    let theta = PI * 2. * (i as f32) / (angles as f32);

    assert_relative_eq!(Mat4::rotation_x(theta), Mat4::from(Quaternion::rotation_x(theta)), epsilon = 0.000001);
    assert_relative_eq!(Mat4::rotation_y(theta), Mat4::from(Quaternion::rotation_y(theta)), epsilon = 0.000001);
    assert_relative_eq!(Mat4::rotation_z(theta), Mat4::from(Quaternion::rotation_z(theta)), epsilon = 0.000001);

    assert_relative_eq!(Mat4::rotation_x(theta), Mat4::rotation_3d(theta, Vec4::unit_x()));
    assert_relative_eq!(Mat4::rotation_y(theta), Mat4::rotation_3d(theta, Vec4::unit_y()));
    assert_relative_eq!(Mat4::rotation_z(theta), Mat4::rotation_3d(theta, Vec4::unit_z()));

    // See what rotating unit vectors do for most angles between 0 and 2*PI.
    // It's helpful to picture this as a right-handed coordinate system.

    let v = Vec4::unit_y();
    let m = Mat4::rotation_x(theta);
    assert_relative_eq!(m * v, Vec4::new(0., theta.cos(), theta.sin(), 0.));

    let v = Vec4::unit_z();
    let m = Mat4::rotation_y(theta);
    assert_relative_eq!(m * v, Vec4::new(theta.sin(), 0., theta.cos(), 0.));

    let v = Vec4::unit_x();
    let m = Mat4::rotation_z(theta);
    assert_relative_eq!(m * v, Vec4::new(theta.cos(), theta.sin(), 0., 0.));
}

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

Performs the conversion.

impl<T> From<Quaternion<T>> for Mat3<T> where
    T: Copy + Zero + One + Mul<Output = T> + Add<Output = T> + Sub<Output = T>, 

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

Performs the conversion.

impl<T> From<Quaternion<T>> for Vec4<T>

A Vec4 can be created directly from a quaternion’s x, y, z and w elements.

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

Performs the conversion.

impl<T> From<Quaternion<T>> for Vec3<T>

A Vec3 can be created directly from a quaternion’s x, y and z elements.

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

Performs the conversion.

impl<T> From<Quaternion<T>> for SimdVec4<T>

A Vec4 can be created directly from a quaternion’s x, y, z and w elements.

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

Performs the conversion.

impl<T> From<Quaternion<T>> for SimdVec3<T>

A Vec3 can be created directly from a quaternion’s x, y and z elements.

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

Performs the conversion.

impl<T> From<Vec4<T>> for Quaternion<T>

A quaternion can be created directly from a Vec4’s x, y, z and w elements. You are responsible for ensuring that the resulting quaternion is normalized.

fn from(v: Vec4<T>) -> Self

Performs the conversion.

impl<T> From<Vec4<T>> for Quaternion<T>

A quaternion can be created directly from a Vec4’s x, y, z and w elements. You are responsible for ensuring that the resulting quaternion is normalized.

fn from(v: SimdVec4<T>) -> Self

Performs the conversion.

impl<T: 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, 

Feeds a slice of this type into the given Hasher.

impl<T, Factor> Lerp<Factor> for Quaternion<T> where
    T: Lerp<Factor, Output = T> + Add<T, Output = T> + Real,
    Factor: Copy, 

The Lerp implementation for quaternion is the “Normalized LERP”.

type Output = Self

The resulting type after performing the LERP operation.

fn lerp_unclamped_precise(from: Self, to: Self, factor: Factor) -> Self

Returns the linear interpolation of from to to with factor unconstrained, using a possibly slower but more precise operation.

fn lerp_unclamped(from: Self, to: Self, factor: Factor) -> Self

Returns the linear interpolation of from to to with factor unconstrained, using the supposedly fastest but less precise implementation.

fn lerp_unclamped_inclusive_range(
    range: RangeInclusive<Self>,
    factor: Factor
) -> Self::Output

Version of lerp_unclamped() that used a single RangeInclusive parameter instead of two values.

fn lerp_unclamped_precise_inclusive_range(
    range: RangeInclusive<Self>,
    factor: Factor
) -> Self::Output

Version of lerp_unclamped_precise() that used a single RangeInclusive parameter instead of two values.

fn lerp(from: Self, to: Self, factor: Factor) -> Self::Output where
    Factor: Clamp + Zero + One, 

Alias to lerp_unclamped which constrains factor to be between 0 and 1 (inclusive).

fn lerp_inclusive_range(
    range: RangeInclusive<Self>,
    factor: Factor
) -> Self::Output where
    Factor: Clamp + Zero + One, 

Version of lerp() that used a single RangeInclusive parameter instead of two values.

fn lerp_precise(from: Self, to: Self, factor: Factor) -> Self::Output where
    Factor: Clamp + Zero + One, 

Alias to lerp_unclamped_precise which constrains factor to be between 0 and 1 (inclusive).

fn lerp_precise_inclusive_range(
    range: RangeInclusive<Self>,
    factor: Factor
) -> Self::Output where
    Factor: Clamp + Zero + One, 

Version of lerp_precise() that used a single RangeInclusive parameter instead of two values.

impl<'a, T, Factor> Lerp<Factor> for &'a Quaternion<T> where
    T: Lerp<Factor, Output = T> + Add<T, Output = T> + Real,
    Factor: Copy, 

The Lerp implementation for quaternion is the “Normalized LERP”.

type Output = Quaternion<T>

The resulting type after performing the LERP operation.

fn lerp_unclamped_precise(from: Self, to: Self, factor: Factor) -> Quaternion<T>

Returns the linear interpolation of from to to with factor unconstrained, using a possibly slower but more precise operation.

fn lerp_unclamped(from: Self, to: Self, factor: Factor) -> Quaternion<T>

Returns the linear interpolation of from to to with factor unconstrained, using the supposedly fastest but less precise implementation.

fn lerp_unclamped_inclusive_range(
    range: RangeInclusive<Self>,
    factor: Factor
) -> Self::Output

Version of lerp_unclamped() that used a single RangeInclusive parameter instead of two values.

fn lerp_unclamped_precise_inclusive_range(
    range: RangeInclusive<Self>,
    factor: Factor
) -> Self::Output

Version of lerp_unclamped_precise() that used a single RangeInclusive parameter instead of two values.

fn lerp(from: Self, to: Self, factor: Factor) -> Self::Output where
    Factor: Clamp + Zero + One, 

Alias to lerp_unclamped which constrains factor to be between 0 and 1 (inclusive).

fn lerp_inclusive_range(
    range: RangeInclusive<Self>,
    factor: Factor
) -> Self::Output where
    Factor: Clamp + Zero + One, 

Version of lerp() that used a single RangeInclusive parameter instead of two values.

fn lerp_precise(from: Self, to: Self, factor: Factor) -> Self::Output where
    Factor: Clamp + Zero + One, 

Alias to lerp_unclamped_precise which constrains factor to be between 0 and 1 (inclusive).

fn lerp_precise_inclusive_range(
    range: RangeInclusive<Self>,
    factor: Factor
) -> Self::Output where
    Factor: Clamp + Zero + One, 

Version of lerp_precise() that used a single RangeInclusive parameter instead of two values.

impl<T> Mul<Quaternion<T>> for Quaternion<T> where
    T: Copy + Mul<Output = T> + Sub<Output = T> + Zero + Add<T, Output = T>, 

The Mul implementation for quaternions is concatenation, a.k.a Grassman product.

use std::f32::consts::PI;

let v = Vec4::unit_x();
let p = Quaternion::rotation_y(PI/2.);
let q = Quaternion::rotation_z(PI/2.);
assert_relative_eq!((p*q)*v, p*(q*v));
assert_relative_eq!(p*q*v, Vec4::unit_y());
assert_relative_eq!(q*p*v, -Vec4::unit_z());

type Output = Self

The resulting type after applying the * operator.

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

Performs the * operation.

impl<T> Mul<T> for Quaternion<T> where
    T: Mul<Output = T> + Copy, 

type Output = Self

The resulting type after applying the * operator.

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

Performs the * operation.

impl<T: Real + Add<T, Output = T>> Mul<Vec3<T>> for Quaternion<T>

3D vectors can be rotated by being premultiplied by a quaternion, assuming the quaternion is normalized.

type Output = Vec3<T>

The resulting type after applying the * operator.

fn mul(self, rhs: Vec3<T>) -> Self::Output

Performs the * operation.

impl<T: Real + Add<T, Output = T>> Mul<Vec3<T>> for Quaternion<T>

3D vectors can be rotated by being premultiplied by a quaternion, assuming the quaternion is normalized.

type Output = SimdVec3<T>

The resulting type after applying the * operator.

fn mul(self, rhs: SimdVec3<T>) -> Self::Output

Performs the * operation.

impl<T: Real + Add<T, Output = T>> Mul<Vec4<T>> for Quaternion<T>

3D vectors can be rotated by being premultiplied by a quaternion, assuming the quaternion is normalized. On Vec4s, the w element is preserved, so you can safely rotate points and directions.

use std::f32::consts::PI;

let v = Vec4::unit_x();

let q = Quaternion::<f32>::identity();
assert_relative_eq!(q * v, v);

let q = Quaternion::rotation_z(PI);
assert_relative_eq!(q * v, -v);

let q = Quaternion::rotation_z(PI * 0.5);
assert_relative_eq!(q * v, Vec4::unit_y());

let q = Quaternion::rotation_z(PI * 1.5);
assert_relative_eq!(q * v, -Vec4::unit_y());

let angles = 32;
for i in 0..angles {
    let theta = PI * 2. * (i as f32) / (angles as f32);

    // See what rotating unit vectors do for most angles between 0 and 2*PI.
    // It's helpful to picture this as a right-handed coordinate system.

    let v = Vec4::unit_y();
    let q = Quaternion::rotation_x(theta);
    assert_relative_eq!(q * v, Vec4::new(0., theta.cos(), theta.sin(), 0.));

    let v = Vec4::unit_z();
    let q = Quaternion::rotation_y(theta);
    assert_relative_eq!(q * v, Vec4::new(theta.sin(), 0., theta.cos(), 0.));

    let v = Vec4::unit_x();
    let q = Quaternion::rotation_z(theta);
    assert_relative_eq!(q * v, Vec4::new(theta.cos(), theta.sin(), 0., 0.));
}

type Output = Vec4<T>

The resulting type after applying the * operator.

fn mul(self, rhs: Vec4<T>) -> Self::Output

Performs the * operation.

impl<T: Real + Add<T, Output = T>> Mul<Vec4<T>> for Quaternion<T>

3D vectors can be rotated by being premultiplied by a quaternion, assuming the quaternion is normalized. On Vec4s, the w element is preserved, so you can safely rotate points and directions.

use std::f32::consts::PI;

let v = Vec4::unit_x();

let q = Quaternion::<f32>::identity();
assert_relative_eq!(q * v, v);

let q = Quaternion::rotation_z(PI);
assert_relative_eq!(q * v, -v);

let q = Quaternion::rotation_z(PI * 0.5);
assert_relative_eq!(q * v, Vec4::unit_y());

let q = Quaternion::rotation_z(PI * 1.5);
assert_relative_eq!(q * v, -Vec4::unit_y());

let angles = 32;
for i in 0..angles {
    let theta = PI * 2. * (i as f32) / (angles as f32);

    // See what rotating unit vectors do for most angles between 0 and 2*PI.
    // It's helpful to picture this as a right-handed coordinate system.

    let v = Vec4::unit_y();
    let q = Quaternion::rotation_x(theta);
    assert_relative_eq!(q * v, Vec4::new(0., theta.cos(), theta.sin(), 0.));

    let v = Vec4::unit_z();
    let q = Quaternion::rotation_y(theta);
    assert_relative_eq!(q * v, Vec4::new(theta.sin(), 0., theta.cos(), 0.));

    let v = Vec4::unit_x();
    let q = Quaternion::rotation_z(theta);
    assert_relative_eq!(q * v, Vec4::new(theta.cos(), theta.sin(), 0., 0.));
}

type Output = SimdVec4<T>

The resulting type after applying the * operator.

fn mul(self, rhs: SimdVec4<T>) -> Self::Output

Performs the * operation.

impl<T> Neg for Quaternion<T> where
    T: Neg<Output = T>, 

type Output = Self

The resulting type after applying the - operator.

fn neg(self) -> Self::Output

Performs the unary - operation.

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

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

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

fn ne(&self, other: &Quaternion<T>) -> bool

This method tests for !=.

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

fn default_max_relative() -> T::Epsilon

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

fn relative_eq(
    &self,
    other: &Self,
    epsilon: T::Epsilon,
    max_relative: T::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<T> Serialize for Quaternion<T> where
    T: 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, Factor> Slerp<Factor> for Quaternion<T> where
    T: Lerp<T, Output = T> + Add<T, Output = T> + Real,
    Factor: Into<T>, 

type Output = Self

The resulting type after performing the SLERP operation.

fn slerp_unclamped(from: Self, to: Self, factor: Factor) -> Self

Performs spherical linear interpolation without implictly constraining factor to be between 0 and 1.

fn slerp(from: Self, to: Self, factor: Factor) -> Self::Output where
    Factor: Clamp + Zero + One, 

Performs spherical linear interpolation, constraining factor to be between 0 and 1.

impl<'a, T, Factor> Slerp<Factor> for &'a Quaternion<T> where
    T: Lerp<T, Output = T> + Add<T, Output = T> + Real,
    Factor: Into<T>, 

type Output = Quaternion<T>

The resulting type after performing the SLERP operation.

fn slerp_unclamped(from: Self, to: Self, factor: Factor) -> Quaternion<T>

Performs spherical linear interpolation without implictly constraining factor to be between 0 and 1.

fn slerp(from: Self, to: Self, factor: Factor) -> Self::Output where
    Factor: Clamp + Zero + One, 

Performs spherical linear interpolation, constraining factor to be between 0 and 1.

impl<T> Sub<Quaternion<T>> for Quaternion<T> where
    T: Sub<Output = T>, 

type Output = Self

The resulting type after applying the - operator.

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

Performs the - operation.

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

fn default_max_ulps() -> u32

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

fn ulps_eq(&self, other: &Self, epsilon: T::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: Copy> Copy for Quaternion<T>

impl<T: Eq> Eq for Quaternion<T>

impl<T> StructuralEq for Quaternion<T>

impl<T> StructuralPartialEq for Quaternion<T>