Struct num_quaternion::UnitQuaternion

pub struct UnitQuaternion<T>(/* private fields */);

A quaternion with norm $1$.

Unit quaternions form a non-commutative group that can be conveniently used for rotating 3D vectors. A 3D vector can be interpreted as a pure quaternion (a quaternion with real part zero). Such a pure quaternion $v$ can be rotated in 3D space by computing $q^{-1}\cdot v\cdot q$ for a unit quaternion $q$. The resulting product is again a pure quaternion which is $v$ rotated around the axis given by the imaginary part of $q$. The angle of rotation is double the angle between $1$ and $q$ interpreted as 4D vectors.

Multiplying two unit quaternions yields again unit quaternion in theory. However, due to limited machine precision, rounding errors accumulate in practice and the resulting norm may deviate from $1$ more and more. Thus, when you multiply a unit quaternions many times, then you need to adjust the norm. This can be done by calling the function adjust_norm().

See also Quaternion.

Implementations

impl<T> UnitQuaternion<T>

where T: Float,

pub fn from_euler_angles(roll: T, pitch: T, yaw: T) -> Self

Creates a new Quaternion from roll, pitch and yaw angles.

pub fn from_rotation_vector(v: &[T; 3]) -> Self

Returns a quaternion from a vector which is parallel to the rotation axis and whose norm is the rotation angle.

impl<T> UnitQuaternion<T>

where T: ConstZero + ConstOne,

pub const ONE: Self = _

A constant UnitQuaternion of value $1$.

See also UnitQuaternion::one(), Quaternion::ONE.

pub const I: Self = _

A constant UnitQuaternion of value $i$.

See also UnitQuaternion::i(), Quaternion::I.

pub const J: Self = _

A constant UnitQuaternion of value $j$.

See also UnitQuaternion::j(), Quaternion::J.

pub const K: Self = _

A constant UnitQuaternion of value $k$.

See also UnitQuaternion::k(), Quaternion::K.

impl<T> UnitQuaternion<T>

where T: Zero + One,

pub fn i() -> Self

Returns the imaginary unit $i$.

See also UnitQuaternion::I, Quaternion::i().

pub fn j() -> Self

Returns the imaginary unit $j$.

See also UnitQuaternion::J, Quaternion::j().

pub fn k() -> Self

Returns the imaginary unit $k$.

See also UnitQuaternion::K, Quaternion::k().

impl<T> UnitQuaternion<T>

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

Returns the inner quaternion.

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

Returns a reference to the inner quaternion.

impl<T> UnitQuaternion<T>

where T: Clone + Neg<Output = T>,

pub fn conj(&self) -> Self

Returns the conjugate quaternion. i.e. the imaginary part is negated.

impl<T> UnitQuaternion<T>

where T: Clone + Neg<Output = T>,

pub fn inv(&self) -> Self

Returns the multiplicative inverse 1/self.

This is the same as the conjugate of self.

impl<T> UnitQuaternion<T>

where T: Float,

pub fn adjust_norm(self) -> Self

Renormalizes self.

By many multiplications of unit quaternions, round off errors can lead to norms which are deviating from $1$. This function fix that inaccuracy.

impl<T> UnitQuaternion<T>

where T: Add<Output = T> + Sub<Output = T> + Mul<Output = T> + Clone,

pub fn rotate_vector(self, vector: [T; 3]) -> [T; 3]

Rotates a vector using a quaternion.

Given a unit quaternion $q$ and a pure quaternion $v$ (i. e. a quaternion with real part zero), the mapping $v \mapsto q^*vq$ is a 3D rotation in the space of pure quaternions. This function performs this 3D rotation efficiently.

Trait Implementations

impl<T> Add<Quaternion<T>> for UnitQuaternion<T>

where Quaternion<T>: Add<Output = Quaternion<T>>,

type Output = Quaternion<T>

The resulting type after applying the + operator.

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

Performs the + operation.

impl<T> Add<T> for UnitQuaternion<T>

where Quaternion<T>: Add<T, Output = Quaternion<T>>,

type Output = Quaternion<T>

The resulting type after applying the + operator.

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

Performs the + operation.

impl<T> Add<UnitQuaternion<T>> for Quaternion<T>

where T: Add<T, Output = T>,

type Output = Quaternion<T>

The resulting type after applying the + operator.

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

Performs the + operation.

impl Add<UnitQuaternion<f32>> for f32

type Output = Quaternion<f32>

The resulting type after applying the + operator.

fn add(self, rhs: UQ32) -> Self::Output

Performs the + operation.

impl Add<UnitQuaternion<f64>> for f64

type Output = Quaternion<f64>

The resulting type after applying the + operator.

fn add(self, rhs: UQ64) -> Self::Output

Performs the + operation.

impl<T> Add for UnitQuaternion<T>

where Quaternion<T>: Add<Output = Quaternion<T>>,

type Output = Quaternion<T>

The resulting type after applying the + operator.

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

Performs the + operation.

impl<T> Borrow<Quaternion<T>> for UnitQuaternion<T>

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

Immutably borrows from an owned value.

impl<T: Clone> Clone for UnitQuaternion<T>

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

Returns a copy of the value.

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

Performs copy-assignment from source.

impl<T> ConstOne for UnitQuaternion<T>

where T: ConstZero + ConstOne + Num + Clone,

const ONE: Self = Self::ONE

The multiplicative identity element of Self, 1.

impl<T: Debug> Debug for UnitQuaternion<T>

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

Formats the value using the given formatter.

impl<T> Default for UnitQuaternion<T>

where T: Num + Clone,

fn default() -> Self

Returns the “default value” for a type.

impl<T> Div<Quaternion<T>> for UnitQuaternion<T>

where Quaternion<T>: Div<Output = Quaternion<T>>,

type Output = Quaternion<T>

The resulting type after applying the / operator.

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

Performs the / operation.

impl<T> Div<T> for UnitQuaternion<T>

where Quaternion<T>: Div<T, Output = Quaternion<T>>,

type Output = Quaternion<T>

The resulting type after applying the / operator.

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

Performs the / operation.

impl<T> Div<UnitQuaternion<T>> for Quaternion<T>

where Quaternion<T>: Mul<Output = Quaternion<T>>, T: Neg<Output = T> + Clone,

type Output = Quaternion<T>

The resulting type after applying the / operator.

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

Performs the / operation.

impl Div<UnitQuaternion<f32>> for f32

type Output = Quaternion<f32>

The resulting type after applying the / operator.

fn div(self, rhs: UQ32) -> Self::Output

Performs the / operation.

impl Div<UnitQuaternion<f64>> for f64

type Output = Quaternion<f64>

The resulting type after applying the / operator.

fn div(self, rhs: UQ64) -> Self::Output

Performs the / operation.

impl<T> Div for UnitQuaternion<T>

where Quaternion<T>: Mul<Output = Quaternion<T>>, T: Clone + Neg<Output = T>,

type Output = UnitQuaternion<T>

The resulting type after applying the / operator.

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

Performs the / operation.

impl<'a, T> From<&'a UnitQuaternion<T>> for &'a Quaternion<T>

fn from(q: &'a UnitQuaternion<T>) -> Self

Converts to this type from the input type.

impl<T> From<UnitQuaternion<T>> for Quaternion<T>

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

Converts to this type from the input type.

impl<T: Hash> Hash for UnitQuaternion<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> Inv for &UnitQuaternion<T>

where T: Clone + Neg<Output = T>,

type Output = UnitQuaternion<T>

The result after applying the operator.

fn inv(self) -> Self::Output

Returns the multiplicative inverse of self.

impl<T> Inv for UnitQuaternion<T>

where T: Clone + Neg<Output = T>,

type Output = UnitQuaternion<T>

The result after applying the operator.

fn inv(self) -> Self::Output

Returns the multiplicative inverse of self.

impl<T> Mul<Quaternion<T>> for UnitQuaternion<T>

where Quaternion<T>: Mul<Output = Quaternion<T>>,

type Output = Quaternion<T>

The resulting type after applying the * operator.

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

Performs the * operation.

impl<T> Mul<T> for UnitQuaternion<T>

where Quaternion<T>: Mul<T, Output = Quaternion<T>>,

type Output = Quaternion<T>

The resulting type after applying the * operator.

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

Performs the * operation.

impl<T> Mul<UnitQuaternion<T>> for Quaternion<T>

where Quaternion<T>: Mul<Output = Quaternion<T>>,

type Output = Quaternion<T>

The resulting type after applying the * operator.

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

Performs the * operation.

impl Mul<UnitQuaternion<f32>> for f32

type Output = Quaternion<f32>

The resulting type after applying the * operator.

fn mul(self, rhs: UQ32) -> Self::Output

Performs the * operation.

impl Mul<UnitQuaternion<f64>> for f64

type Output = Quaternion<f64>

The resulting type after applying the * operator.

fn mul(self, rhs: UQ64) -> Self::Output

Performs the * operation.

impl<T> Mul for UnitQuaternion<T>

where Quaternion<T>: Mul<Output = Quaternion<T>>,

type Output = UnitQuaternion<T>

The resulting type after applying the * operator.

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

Performs the * operation.

impl<T> Neg for UnitQuaternion<T>

where T: Neg<Output = T>,

type Output = UnitQuaternion<T>

The resulting type after applying the - operator.

fn neg(self) -> Self::Output

Performs the unary - operation.

impl<T> One for UnitQuaternion<T>

where T: Num + Clone,

fn one() -> Self

Returns the multiplicative identity element of Self, 1.

fn is_one(&self) -> bool

Returns true if self is equal to the multiplicative identity.

fn set_one(&mut self)

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

impl<T: PartialEq> PartialEq for UnitQuaternion<T>

fn eq(&self, other: &UnitQuaternion<T>) -> 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> Sub<Quaternion<T>> for UnitQuaternion<T>

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

type Output = Quaternion<T>

The resulting type after applying the - operator.

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

Performs the - operation.

impl<T> Sub<T> for UnitQuaternion<T>

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

type Output = Quaternion<T>

The resulting type after applying the - operator.

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

Performs the - operation.

impl<T> Sub<UnitQuaternion<T>> for Quaternion<T>

where T: Sub<T, Output = T>,

type Output = Quaternion<T>

The resulting type after applying the - operator.

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

Performs the - operation.

impl Sub<UnitQuaternion<f32>> for f32

type Output = Quaternion<f32>

The resulting type after applying the - operator.

fn sub(self, rhs: UQ32) -> Self::Output

Performs the - operation.

impl Sub<UnitQuaternion<f64>> for f64

type Output = Quaternion<f64>

The resulting type after applying the - operator.

fn sub(self, rhs: UQ64) -> Self::Output

Performs the - operation.

impl<T> Sub for UnitQuaternion<T>

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

type Output = Quaternion<T>

The resulting type after applying the - operator.

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

Performs the - operation.

impl<T: Copy> Copy for UnitQuaternion<T>

impl<T: Eq> Eq for UnitQuaternion<T>

impl<T> StructuralPartialEq for UnitQuaternion<T>