cgmath/src/euler.rs
2019-11-05 07:16:38 +01:00

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// Copyright 2016 The CGMath Developers. For a full listing of the authors,
// refer to the Cargo.toml file at the top-level directory of this distribution.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
use num_traits::cast;
#[cfg(feature = "rand")]
use rand::{
distributions::{Distribution, Standard},
Rng,
};
use structure::*;
use angle::Rad;
use approx;
#[cfg(feature = "mint")]
use mint;
use num::BaseFloat;
use quaternion::Quaternion;
/// A set of [Euler angles] representing a rotation in three-dimensional space.
///
/// This type is marked as `#[repr(C)]`.
///
/// The axis rotation sequence is XYZ. That is, the rotation is first around
/// the X axis, then the Y axis, and lastly the Z axis (using intrinsic
/// rotations). Since all three rotation axes are used, the angles are
/// TaitBryan angles rather than proper Euler angles.
///
/// # Ranges
///
/// - x: [-pi, pi]
/// - y: [-pi/2, pi/2]
/// - z: [-pi, pi]
///
/// # Defining rotations using Euler angles
///
/// Note that while [Euler angles] are intuitive to define, they are prone to
/// [gimbal lock] and are challenging to interpolate between. Instead we
/// recommend that you convert them to a more robust representation, such as a
/// quaternion or a rotation matrix. To this end, `From<Euler<A>>` conversions
/// are provided for the following types:
///
/// - [`Basis3`](struct.Basis3.html)
/// - [`Matrix3`](struct.Matrix3.html)
/// - [`Matrix4`](struct.Matrix4.html)
/// - [`Quaternion`](struct.Quaternion.html)
///
/// For example, to define a quaternion that applies the following:
///
/// 1. a 90° rotation around the _x_ axis
/// 2. a 45° rotation around the _y_ axis
/// 3. a 15° rotation around the _z_ axis
///
/// you can use the following code:
///
/// ```
/// use cgmath::{Deg, Euler, Quaternion};
///
/// let rotation = Quaternion::from(Euler {
/// x: Deg(90.0),
/// y: Deg(45.0),
/// z: Deg(15.0),
/// });
/// ```
///
/// [Euler angles]: https://en.wikipedia.org/wiki/Euler_angles
/// [gimbal lock]: https://en.wikipedia.org/wiki/Gimbal_lock#Gimbal_lock_in_applied_mathematics
/// [convert]: #defining-rotations-using-euler-angles
#[repr(C)]
#[derive(Copy, Clone, Debug, PartialEq, Eq)]
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
pub struct Euler<A> {
/// The angle to apply around the _x_ axis. Also known at the _pitch_.
pub x: A,
/// The angle to apply around the _y_ axis. Also known at the _yaw_.
pub y: A,
/// The angle to apply around the _z_ axis. Also known at the _roll_.
pub z: A,
}
impl<A> Euler<A> {
/// Construct a set of euler angles.
///
/// # Arguments
///
/// * `x` - The angle to apply around the _x_ axis. Also known at the _pitch_.
/// * `y` - The angle to apply around the _y_ axis. Also known at the _yaw_.
/// * `z` - The angle to apply around the _z_ axis. Also known at the _roll_.
pub const fn new(x: A, y: A, z: A) -> Euler<A> {
Euler { x: x, y: y, z: z }
}
}
impl<S: BaseFloat> From<Quaternion<S>> for Euler<Rad<S>> {
fn from(src: Quaternion<S>) -> Euler<Rad<S>> {
let sig: S = cast(0.499).unwrap();
let two: S = cast(2).unwrap();
let one: S = cast(1).unwrap();
let (qw, qx, qy, qz) = (src.s, src.v.x, src.v.y, src.v.z);
let (sqw, sqx, sqy, sqz) = (qw * qw, qx * qx, qy * qy, qz * qz);
let unit = sqx + sqz + sqy + sqw;
let test = qx * qz + qy * qw;
// We set x to zero and z to the value, but the other way would work too.
if test > sig * unit {
// x + z = 2 * atan(x / w)
Euler {
x: Rad::zero(),
y: Rad::turn_div_4(),
z: Rad::atan2(qx, qw) * two,
}
} else if test < -sig * unit {
// x - z = 2 * atan(x / w)
Euler {
x: Rad::zero(),
y: -Rad::turn_div_4(),
z: -Rad::atan2(qx, qw) * two,
}
} else {
// Using the quat-to-matrix equation from either
// http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToMatrix/index.htm
// or equation 15 on page 7 of
// http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770024290.pdf
// to fill in the equations on page A-2 of the NASA document gives the below.
Euler {
x: Rad::atan2(two * (-qy * qz + qx * qw), one - two * (sqx + sqy)),
y: Rad::asin(two * (qx * qz + qy * qw)),
z: Rad::atan2(two * (-qx * qy + qz * qw), one - two * (sqy + sqz)),
}
}
}
}
impl<A: Angle> approx::AbsDiffEq for Euler<A> {
type Epsilon = A::Epsilon;
#[inline]
fn default_epsilon() -> A::Epsilon {
A::default_epsilon()
}
#[inline]
fn abs_diff_eq(&self, other: &Self, epsilon: A::Epsilon) -> bool {
A::abs_diff_eq(&self.x, &other.x, epsilon)
&& A::abs_diff_eq(&self.y, &other.y, epsilon)
&& A::abs_diff_eq(&self.z, &other.z, epsilon)
}
}
impl<A: Angle> approx::RelativeEq for Euler<A> {
#[inline]
fn default_max_relative() -> A::Epsilon {
A::default_max_relative()
}
#[inline]
fn relative_eq(&self, other: &Self, epsilon: A::Epsilon, max_relative: A::Epsilon) -> bool {
A::relative_eq(&self.x, &other.x, epsilon, max_relative)
&& A::relative_eq(&self.y, &other.y, epsilon, max_relative)
&& A::relative_eq(&self.z, &other.z, epsilon, max_relative)
}
}
impl<A: Angle> approx::UlpsEq for Euler<A> {
#[inline]
fn default_max_ulps() -> u32 {
A::default_max_ulps()
}
#[inline]
fn ulps_eq(&self, other: &Self, epsilon: A::Epsilon, max_ulps: u32) -> bool {
A::ulps_eq(&self.x, &other.x, epsilon, max_ulps)
&& A::ulps_eq(&self.y, &other.y, epsilon, max_ulps)
&& A::ulps_eq(&self.z, &other.z, epsilon, max_ulps)
}
}
#[cfg(feature = "rand")]
impl<A> Distribution<Euler<A>> for Standard
where
Standard: Distribution<A>,
A: Angle,
{
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Euler<A> {
Euler {
x: rng.gen(),
y: rng.gen(),
z: rng.gen(),
}
}
}
#[cfg(feature = "mint")]
type MintEuler<S> = mint::EulerAngles<S, mint::IntraXYZ>;
#[cfg(feature = "mint")]
impl<S, A: Angle + From<S>> From<MintEuler<S>> for Euler<A> {
fn from(mint: MintEuler<S>) -> Self {
Euler {
x: mint.a.into(),
y: mint.b.into(),
z: mint.c.into(),
}
}
}
#[cfg(feature = "mint")]
impl<S: Clone, A: Angle + Into<S>> Into<MintEuler<S>> for Euler<A> {
fn into(self) -> MintEuler<S> {
MintEuler::from([self.x.into(), self.y.into(), self.z.into()])
}
}