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

Methods

impl<T> Quaternion<T>

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.

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.

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().

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.

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);

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());

fn inverse(self) -> Self where
    T: Neg<Output = T> + Copy + Sum + 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());

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

Gets the dot product between two quaternions.

fn normalized(self) -> Self where
    T: Float + Sum, 

Gets a normalized copy of this quaternion.

fn magnitude_squared(self) -> T where
    T: Float + Sum, 

Gets this quaternion's magnitude, squared.

fn magnitude(self) -> T where
    T: Float + Sum, 

Gets this quaternion's magnitude.

fn rotation_from_to_3d<V: Into<Vec3<T>>>(from: V, to: V) -> Self where
    T: Float + Sum, 

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

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

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);

fn rotation_3d<V: Into<Vec3<T>>>(angle_radians: T, axis: V) -> Self where
    T: Float + Sum, 

Creates a Quaternion from an angle and axis.

fn rotation_x(angle_radians: T) -> Self where
    T: Float + Sum, 

fn rotation_y(angle_radians: T) -> Self where
    T: Float + Sum, 

fn rotation_z(angle_radians: T) -> Self where
    T: Float + Sum, 

fn rotated_3d<V: Into<Vec3<T>>>(self, angle_radians: T, axis: V) -> Self where
    T: Float + Sum, 

fn rotated_x(self, angle_radians: T) -> Self where
    T: Float + Sum, 

fn rotated_y(self, angle_radians: T) -> Self where
    T: Float + Sum, 

fn rotated_z(self, angle_radians: T) -> Self where
    T: Float + Sum, 

fn rotate_3d<V: Into<Vec3<T>>>(&mut self, angle_radians: T, axis: V) where
    T: Float + Sum, 

fn rotate_x(&mut self, angle_radians: T) where
    T: Float + Sum, 

fn rotate_y(&mut self, angle_radians: T) where
    T: Float + Sum, 

fn rotate_z(&mut self, angle_radians: T) where
    T: Float + Sum, 

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

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

fn into_vec4(self) -> Vec4<T>

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

fn into_vec3(self) -> Vec3<T>

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

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.

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>, 

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.

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> + Sum + Float, 

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);
}

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: Debug> Debug for Quaternion<T>

fn fmt(&self, __arg_0: &mut Formatter) -> Result

Formats the value using the given formatter.

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: Copy> Copy for Quaternion<T>

impl<T: Hash> Hash for Quaternion<T>

fn hash<__HT: Hasher>(&self, __arg_0: &mut __HT)

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: Eq> Eq for Quaternion<T>

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

fn eq(&self, __arg_0: &Quaternion<T>) -> bool

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

fn ne(&self, __arg_0: &Quaternion<T>) -> bool

This method tests for !=.

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<T, Factor> Lerp<Factor> for Quaternion<T> where
    T: Lerp<Factor, Output = T> + Sum + Float,
    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(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_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).

impl<'a, T, Factor> Lerp<Factor> for &'a Quaternion<T> where
    T: Lerp<Factor, Output = T> + Sum + Float,
    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(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_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).

impl<T, Factor> Slerp<Factor> for Quaternion<T> where
    T: Lerp<T, Output = T> + Sum + Float,
    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> + Sum + Float,
    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> 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> 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> Add 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> Sub 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> Mul for Quaternion<T> where
    T: Copy + Mul<Output = T> + Sub<Output = T> + Zero + Sum, 

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: ApproxEq> ApproxEq 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 default_max_relative() -> T::Epsilon

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

fn default_max_ulps() -> u32

The default ULPs to tolerate when 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 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 relative_ne(
    &self,
    other: &Self,
    epsilon: Self::Epsilon,
    max_relative: Self::Epsilon
) -> bool

The inverse of ApproxEq::relative_eq.

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

The inverse of ApproxEq::ulps_eq.

impl<T: Float + Sum> 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: Float + Sum> 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> 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.