Type Alias Rotation3

pub type Rotation3<T> = Rotation<T, 3>;

A 3-dimensional rotation matrix.

Because this is an alias, not all its methods are listed here. See the Rotation type too.

Aliased Type

struct Rotation3<T> { /* private fields */ }

Implementations

impl<T: SimdRealField> Rotation3<T>

pub fn slerp(&self, other: &Self, t: T) -> Self

where T: RealField,

Spherical linear interpolation between two rotation matrices.

Panics if the angle between both rotations is 180 degrees (in which case the interpolation is not well-defined). Use .try_slerp instead to avoid the panic.

Examples:

let q1 = Rotation3::from_euler_angles(std::f32::consts::FRAC_PI_4, 0.0, 0.0);
let q2 = Rotation3::from_euler_angles(-std::f32::consts::PI, 0.0, 0.0);

let q = q1.slerp(&q2, 1.0 / 3.0);

assert_eq!(q.euler_angles(), (std::f32::consts::FRAC_PI_2, 0.0, 0.0));

pub fn try_slerp(&self, other: &Self, t: T, epsilon: T) -> Option<Self>

where T: RealField,

Computes the spherical linear interpolation between two rotation matrices or returns None if both rotations are approximately 180 degrees apart (in which case the interpolation is not well-defined).

Arguments

• self: the first rotation to interpolate from.
• other: the second rotation to interpolate toward.
• t: the interpolation parameter. Should be between 0 and 1.
• epsilon: the value below which the sinus of the angle separating both rotations must be to return None.

impl<T: SimdRealField> Rotation3<T>

where T::Element: SimdRealField,

Construction from a 3D axis and/or angles

pub fn new<SB: Storage<T, U3>>(axisangle: Vector<T, U3, SB>) -> Self

Builds a 3 dimensional rotation matrix from an axis and an angle.

Arguments

• axisangle - A vector representing the rotation. Its magnitude is the amount of rotation in radian. Its direction is the axis of rotation.

Example

let axisangle = Vector3::y() * f32::consts::FRAC_PI_2;
// Point and vector being transformed in the tests.
let pt = Point3::new(4.0, 5.0, 6.0);
let vec = Vector3::new(4.0, 5.0, 6.0);
let rot = Rotation3::new(axisangle);

assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);

// A zero vector yields an identity.
assert_eq!(Rotation3::new(Vector3::<f32>::zeros()), Rotation3::identity());

pub fn from_scaled_axis<SB: Storage<T, U3>>( axisangle: Vector<T, U3, SB>, ) -> Self

Builds a 3D rotation matrix from an axis scaled by the rotation angle.

This is the same as Self::new(axisangle).

Example

let axisangle = Vector3::y() * f32::consts::FRAC_PI_2;
// Point and vector being transformed in the tests.
let pt = Point3::new(4.0, 5.0, 6.0);
let vec = Vector3::new(4.0, 5.0, 6.0);
let rot = Rotation3::new(axisangle);

assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);

// A zero vector yields an identity.
assert_eq!(Rotation3::from_scaled_axis(Vector3::<f32>::zeros()), Rotation3::identity());

pub fn from_axis_angle<SB>(axis: &Unit<Vector<T, U3, SB>>, angle: T) -> Self

where SB: Storage<T, U3>,

Builds a 3D rotation matrix from an axis and a rotation angle.

Example

let axis = Vector3::y_axis();
let angle = f32::consts::FRAC_PI_2;
// Point and vector being transformed in the tests.
let pt = Point3::new(4.0, 5.0, 6.0);
let vec = Vector3::new(4.0, 5.0, 6.0);
let rot = Rotation3::from_axis_angle(&axis, angle);

assert_eq!(rot.axis().unwrap(), axis);
assert_eq!(rot.angle(), angle);
assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);

// A zero vector yields an identity.
assert_eq!(Rotation3::from_scaled_axis(Vector3::<f32>::zeros()), Rotation3::identity());

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

Creates a new rotation from Euler angles.

The primitive rotations are applied in order: 1 roll − 2 pitch − 3 yaw.

Example

let rot = Rotation3::from_euler_angles(0.1, 0.2, 0.3);
let euler = rot.euler_angles();
assert_relative_eq!(euler.0, 0.1, epsilon = 1.0e-6);
assert_relative_eq!(euler.1, 0.2, epsilon = 1.0e-6);
assert_relative_eq!(euler.2, 0.3, epsilon = 1.0e-6);

impl<T: SimdRealField> Rotation3<T>

where T::Element: SimdRealField,

Construction from a 3D eye position and target point

pub fn face_towards<SB, SC>( dir: &Vector<T, U3, SB>, up: &Vector<T, U3, SC>, ) -> Self

where SB: Storage<T, U3>, SC: Storage<T, U3>,

Creates a rotation that corresponds to the local frame of an observer standing at the origin and looking toward dir.

It maps the z axis to the direction dir.

Arguments

• dir - The look direction, that is, direction the matrix z axis will be aligned with.
• up - The vertical direction. The only requirement of this parameter is to not be collinear to dir. Non-collinearity is not checked.

Example

let dir = Vector3::new(1.0, 2.0, 3.0);
let up = Vector3::y();

let rot = Rotation3::face_towards(&dir, &up);
assert_relative_eq!(rot * Vector3::z(), dir.normalize());

pub fn new_observer_frames<SB, SC>( dir: &Vector<T, U3, SB>, up: &Vector<T, U3, SC>, ) -> Self

where SB: Storage<T, U3>, SC: Storage<T, U3>,

👎Deprecated: renamed to face_towards

Deprecated: Use Rotation3::face_towards instead.

pub fn look_at_rh<SB, SC>( dir: &Vector<T, U3, SB>, up: &Vector<T, U3, SC>, ) -> Self

where SB: Storage<T, U3>, SC: Storage<T, U3>,

Builds a right-handed look-at view matrix without translation.

It maps the view direction dir to the negative z axis. This conforms to the common notion of right handed look-at matrix from the computer graphics community.

Arguments

• dir - The direction toward which the camera looks.
• up - A vector approximately aligned with required the vertical axis. The only requirement of this parameter is to not be collinear to dir.

Example

let dir = Vector3::new(1.0, 2.0, 3.0);
let up = Vector3::y();

let rot = Rotation3::look_at_rh(&dir, &up);
assert_relative_eq!(rot * dir.normalize(), -Vector3::z());

pub fn look_at_lh<SB, SC>( dir: &Vector<T, U3, SB>, up: &Vector<T, U3, SC>, ) -> Self

where SB: Storage<T, U3>, SC: Storage<T, U3>,

Builds a left-handed look-at view matrix without translation.

It maps the view direction dir to the positive z axis. This conforms to the common notion of left handed look-at matrix from the computer graphics community.

Arguments

• dir - The direction toward which the camera looks.
• up - A vector approximately aligned with required the vertical axis. The only requirement of this parameter is to not be collinear to dir.

Example

let dir = Vector3::new(1.0, 2.0, 3.0);
let up = Vector3::y();

let rot = Rotation3::look_at_lh(&dir, &up);
assert_relative_eq!(rot * dir.normalize(), Vector3::z());

impl<T: SimdRealField> Rotation3<T>

where T::Element: SimdRealField,

Construction from an existing 3D matrix or rotations

pub fn rotation_between<SB, SC>( a: &Vector<T, U3, SB>, b: &Vector<T, U3, SC>, ) -> Option<Self>

where T: RealField, SB: Storage<T, U3>, SC: Storage<T, U3>,

The rotation matrix required to align a and b but with its angle.

This is the rotation R such that (R * a).angle(b) == 0 && (R * a).dot(b).is_positive().

Example

let a = Vector3::new(1.0, 2.0, 3.0);
let b = Vector3::new(3.0, 1.0, 2.0);
let rot = Rotation3::rotation_between(&a, &b).unwrap();
assert_relative_eq!(rot * a, b, epsilon = 1.0e-6);
assert_relative_eq!(rot.inverse() * b, a, epsilon = 1.0e-6);

pub fn scaled_rotation_between<SB, SC>( a: &Vector<T, U3, SB>, b: &Vector<T, U3, SC>, n: T, ) -> Option<Self>

where T: RealField, SB: Storage<T, U3>, SC: Storage<T, U3>,

The smallest rotation needed to make a and b collinear and point toward the same direction, raised to the power s.

Example

let a = Vector3::new(1.0, 2.0, 3.0);
let b = Vector3::new(3.0, 1.0, 2.0);
let rot2 = Rotation3::scaled_rotation_between(&a, &b, 0.2).unwrap();
let rot5 = Rotation3::scaled_rotation_between(&a, &b, 0.5).unwrap();
assert_relative_eq!(rot2 * rot2 * rot2 * rot2 * rot2 * a, b, epsilon = 1.0e-6);
assert_relative_eq!(rot5 * rot5 * a, b, epsilon = 1.0e-6);

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

The rotation matrix needed to make self and other coincide.

The result is such that: self.rotation_to(other) * self == other.

Example

let rot1 = Rotation3::from_axis_angle(&Vector3::y_axis(), 1.0);
let rot2 = Rotation3::from_axis_angle(&Vector3::x_axis(), 0.1);
let rot_to = rot1.rotation_to(&rot2);
assert_relative_eq!(rot_to * rot1, rot2, epsilon = 1.0e-6);

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

where T: RealField,

Raise the rotation to a given floating power, i.e., returns the rotation with the same axis as self and an angle equal to self.angle() multiplied by n.

Example

let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
let angle = 1.2;
let rot = Rotation3::from_axis_angle(&axis, angle);
let pow = rot.powf(2.0);
assert_relative_eq!(pow.axis().unwrap(), axis, epsilon = 1.0e-6);
assert_eq!(pow.angle(), 2.4);

pub fn from_basis_unchecked(basis: &[Vector3<T>; 3]) -> Self

Builds a rotation from a basis assumed to be orthonormal.

In order to get a valid rotation matrix, the input must be an orthonormal basis, i.e., all vectors are normalized, and the are all orthogonal to each other. These invariants are not checked by this method.

pub fn from_matrix(m: &Matrix3<T>) -> Self

where T: RealField,

Builds a rotation matrix by extracting the rotation part of the given transformation m.

This is an iterative method. See .from_matrix_eps to provide mover convergence parameters and starting solution. This implements “A Robust Method to Extract the Rotational Part of Deformations” by Müller et al.

pub fn from_matrix_eps( m: &Matrix3<T>, eps: T, max_iter: usize, guess: Self, ) -> Self

where T: RealField,

Builds a rotation matrix by extracting the rotation part of the given transformation m.

This implements “A Robust Method to Extract the Rotational Part of Deformations” by Müller et al.

Parameters

• m: the matrix from which the rotational part is to be extracted.
• eps: the angular errors tolerated between the current rotation and the optimal one.
• max_iter: the maximum number of iterations. Loops indefinitely until convergence if set to 0.
• guess: a guess of the solution. Convergence will be significantly faster if an initial solution close to the actual solution is provided. Can be set to Rotation3::identity() if no other guesses come to mind.

pub fn renormalize(&mut self)

where T: RealField,

Ensure this rotation is an orthonormal rotation matrix. This is useful when repeated computations might cause the matrix from progressively not being orthonormal anymore.

impl<T: SimdRealField> Rotation3<T>

3D axis and angle extraction

pub fn angle(&self) -> T

The rotation angle in [0; pi].

Example

let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
let rot = Rotation3::from_axis_angle(&axis, 1.78);
assert_relative_eq!(rot.angle(), 1.78);

pub fn axis(&self) -> Option<Unit<Vector3<T>>>

where T: RealField,

The rotation axis. Returns None if the rotation angle is zero or PI.

Example

let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
let angle = 1.2;
let rot = Rotation3::from_axis_angle(&axis, angle);
assert_relative_eq!(rot.axis().unwrap(), axis);

// Case with a zero angle.
let rot = Rotation3::from_axis_angle(&axis, 0.0);
assert!(rot.axis().is_none());

pub fn scaled_axis(&self) -> Vector3<T>

where T: RealField,

The rotation axis multiplied by the rotation angle.

Example

let axisangle = Vector3::new(0.1, 0.2, 0.3);
let rot = Rotation3::new(axisangle);
assert_relative_eq!(rot.scaled_axis(), axisangle, epsilon = 1.0e-6);

pub fn axis_angle(&self) -> Option<(Unit<Vector3<T>>, T)>

where T: RealField,

The rotation axis and angle in (0, pi] of this rotation matrix.

Returns None if the angle is zero.

Example

let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
let angle = 1.2;
let rot = Rotation3::from_axis_angle(&axis, angle);
let axis_angle = rot.axis_angle().unwrap();
assert_relative_eq!(axis_angle.0, axis);
assert_relative_eq!(axis_angle.1, angle);

// Case with a zero angle.
let rot = Rotation3::from_axis_angle(&axis, 0.0);
assert!(rot.axis_angle().is_none());

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

where T::Element: SimdRealField,

The rotation angle needed to make self and other coincide.

Example

let rot1 = Rotation3::from_axis_angle(&Vector3::y_axis(), 1.0);
let rot2 = Rotation3::from_axis_angle(&Vector3::x_axis(), 0.1);
assert_relative_eq!(rot1.angle_to(&rot2), 1.0045657, epsilon = 1.0e-6);

pub fn to_euler_angles(self) -> (T, T, T)

where T: RealField,

👎Deprecated: This is renamed to use .euler_angles().

Creates Euler angles from a rotation.

The angles are produced in the form (roll, pitch, yaw).

pub fn euler_angles(&self) -> (T, T, T)

where T: RealField,

Euler angles corresponding to this rotation from a rotation.

The angles are produced in the form (roll, pitch, yaw).

Example

let rot = Rotation3::from_euler_angles(0.1, 0.2, 0.3);
let euler = rot.euler_angles();
assert_relative_eq!(euler.0, 0.1, epsilon = 1.0e-6);
assert_relative_eq!(euler.1, 0.2, epsilon = 1.0e-6);
assert_relative_eq!(euler.2, 0.3, epsilon = 1.0e-6);

pub fn euler_angles_ordered( &self, seq: [Unit<Vector3<T>>; 3], extrinsic: bool, ) -> ([T; 3], bool)

where T: RealField + Copy,

Represent this rotation as Euler angles.

Returns the angles produced in the order provided by seq parameter, along with the observability flag. The Euler axes passed to seq must form an orthonormal basis. If the rotation is gimbal locked, then the observability flag is false.

Panics

Panics if the Euler axes in seq are not orthonormal.

Example 1:

use std::f64::consts::PI;
use approx::assert_relative_eq;
use nalgebra::{Matrix3, Rotation3, Unit, Vector3};

// 3-1-2
let n = [
    Unit::new_unchecked(Vector3::new(0.0, 0.0, 1.0)),
    Unit::new_unchecked(Vector3::new(1.0, 0.0, 0.0)),
    Unit::new_unchecked(Vector3::new(0.0, 1.0, 0.0)),
];

let r1 = Rotation3::from_axis_angle(&n[2], 20.0 * PI / 180.0);
let r2 = Rotation3::from_axis_angle(&n[1], 30.0 * PI / 180.0);
let r3 = Rotation3::from_axis_angle(&n[0], 45.0 * PI / 180.0);

let d = r3 * r2 * r1;

let (angles, observable) = d.euler_angles_ordered(n, false);
assert!(observable);
assert_relative_eq!(angles[0] * 180.0 / PI, 45.0, epsilon = 1e-12);
assert_relative_eq!(angles[1] * 180.0 / PI, 30.0, epsilon = 1e-12);
assert_relative_eq!(angles[2] * 180.0 / PI, 20.0, epsilon = 1e-12);

Example 2:

use std::f64::consts::PI;
use approx::assert_relative_eq;
use nalgebra::{Matrix3, Rotation3, Unit, Vector3};

let sqrt_2 = 2.0_f64.sqrt();
let n = [
    Unit::new_unchecked(Vector3::new(1.0 / sqrt_2, 1.0 / sqrt_2, 0.0)),
    Unit::new_unchecked(Vector3::new(1.0 / sqrt_2, -1.0 / sqrt_2, 0.0)),
    Unit::new_unchecked(Vector3::new(0.0, 0.0, 1.0)),
];

let r1 = Rotation3::from_axis_angle(&n[2], 20.0 * PI / 180.0);
let r2 = Rotation3::from_axis_angle(&n[1], 30.0 * PI / 180.0);
let r3 = Rotation3::from_axis_angle(&n[0], 45.0 * PI / 180.0);

let d = r3 * r2 * r1;

let (angles, observable) = d.euler_angles_ordered(n, false);
assert!(observable);
assert_relative_eq!(angles[0] * 180.0 / PI, 45.0, epsilon = 1e-12);
assert_relative_eq!(angles[1] * 180.0 / PI, 30.0, epsilon = 1e-12);
assert_relative_eq!(angles[2] * 180.0 / PI, 20.0, epsilon = 1e-12);

Algorithm based on: Malcolm D. Shuster, F. Landis Markley, “General formula for extraction the Euler angles”, Journal of guidance, control, and dynamics, vol. 29.1, pp. 215-221. 2006, and modified to be able to produce extrinsic rotations.

Trait Implementations

impl From<DQuat> for Rotation3<f64>

fn from(e: DQuat) -> Rotation3<f64>

Converts to this type from the input type.

impl From<Quat> for Rotation3<f32>

fn from(e: Quat) -> Rotation3<f32>

Converts to this type from the input type.

impl<T: SimdRealField> From<Unit<Quaternion<T>>> for Rotation3<T>

where T::Element: SimdRealField,

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

Converts to this type from the input type.

impl<T1, T2> SubsetOf<Unit<DualQuaternion<T2>>> for Rotation3<T1>

where T1: RealField, T2: RealField + SupersetOf<T1>,

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

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

fn is_in_subset(dq: &UnitDualQuaternion<T2>) -> bool

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

fn from_superset_unchecked(dq: &UnitDualQuaternion<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<T1, T2> SubsetOf<Unit<Quaternion<T2>>> for Rotation3<T1>

where T1: RealField, T2: RealField + SupersetOf<T1>,

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

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

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

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

fn from_superset_unchecked(q: &UnitQuaternion<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.