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quaternion trigonometry

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This commit is contained in:
Adam Nemecek 2019-02-26 18:12:30 -08:00
parent aeff67ecbd
commit edb08cd900
3 changed files with 266 additions and 10 deletions
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3
Cargo.toml
View File

@ -47,6 +47,9 @@ quickcheck = { version = "0.8", optional = true }
pest = { version = "2.0", optional = true }
pest_derive = { version = "2.0", optional = true }
[patch.crates-io]
alga = { git = "https://github.com/rustsim/alga", branch = "dev" }
[dev-dependencies]
serde_json = "1.0"
rand_xorshift = "0.1"

252
src/geometry/quaternion.rs
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@ -506,6 +506,255 @@ impl<N: Real> Quaternion<N> {
pub fn normalize_mut(&mut self) -> N {
self.coords.normalize_mut()
}
/// Calculates square of a quaternion.
#[inline]
pub fn squared(&self) -> Self {
self * self
}
/// Divides quaternion into two.
#[inline]
pub fn half(&self) -> Self {
self / ::convert(2.0f64)
}
/// Calculates square root.
#[inline]
pub fn sqrt(&self) -> Self {
self.powf(::convert(0.5))
}
/// Check if the quaternion is pure.
#[inline]
pub fn is_pure(&self) -> bool {
self.w == N::zero()
}
/// Convert quaternion to pure quaternion.
#[inline]
pub fn pure(&self) -> Self {
Self::from_imag(self.imag())
}
/// Calculates the quaternionic cosinus.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let expected = Quaternion::new(58.93364616794395, -34.086183690465596, -51.1292755356984, -68.17236738093119);
/// let result = input.cos();
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn cos(&self) -> Self {
let z = self.imag().magnitude();
let w = -self.w.sin() * z.sinhc();
Self::from_parts(self.w.cos() * z.cosh(), self.imag() * w)
}
/// Calculates the quaternionic arccosinus.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let result = input.cos().acos();
/// assert_relative_eq!(input, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn acos(&self) -> Self {
let u = Self::from_imag(self.imag().normalize());
let identity = Self::identity();
let z = (self + (self.squared() - identity).sqrt()).ln();
-(u * z)
}
/// Calculates the quaternionic sinus.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let expected = Quaternion::new(91.78371578403467, 21.886486853029176, 32.82973027954377, 43.77297370605835);
/// let result = input.sin();
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn sin(&self) -> Self {
let z = self.imag().magnitude();
let w = self.w.cos() * z.sinhc();
Self::from_parts(self.w.sin() * z.cosh(), self.imag() * w)
}
/// Calculates the quaternionic arcsinus.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let result = input.sin().asin();
/// assert_relative_eq!(input, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn asin(&self) -> Self {
let u = Self::from_imag(self.imag().normalize());
let identity = Self::identity();
let z = ((u * self) + (identity - self.squared()).sqrt()).ln();
-(u * z)
}
/// Calculates the quaternionic tangent.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let expected = Quaternion::new(0.00003821631725009489, 0.3713971716439371, 0.5570957574659058, 0.7427943432878743);
/// let result = input.tan();
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn tan(&self) -> Self {
let s = self.sin();
let c = self.cos();
let ci = c.try_inverse().unwrap();
s * ci
}
/// Calculates the quaternionic arctangent.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let result = input.tan().atan();
/// assert_relative_eq!(input, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn atan(&self) -> Self {
let u = Self::from_imag(self.imag().normalize());
let num = u + self;
let den = u - self;
let fr = num * den.try_inverse().unwrap();
let ln = fr.ln();
(u.half()) * ln
}
/// Calculates the hyperbolic quaternionic sinus.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let expected = Quaternion::new(0.7323376060463428, -0.4482074499805421, -0.6723111749708133, -0.8964148999610843);
/// let result = input.sinh();
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn sinh(&self) -> Self {
(self.exp() - (-self).exp()).half()
}
/// Calculates the hyperbolic quaternionic arcsinus.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let expected = Quaternion::new(2.385889902585242, 0.514052600662788, 0.7710789009941821, 1.028105201325576);
/// let result = input.asinh();
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn asinh(&self) -> Self {
let identity = Self::identity();
(self + (identity + self.squared()).sqrt()).ln()
}
/// Calculates the hyperbolic quaternionic cosinus.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let expected = Quaternion::new(0.9615851176369566, -0.3413521745610167, -0.5120282618415251, -0.6827043491220334);
/// let result = input.cosh();
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn cosh(&self) -> Self {
(self.exp() + (-self).exp()).half()
}
/// Calculates the hyperbolic quaternionic arccosinus.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let expected = Quaternion::new(2.4014472020074007, 0.5162761016176176, 0.7744141524264264, 1.0325522032352352);
/// let result = input.acosh();
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn acosh(&self) -> Self {
let identity = Self::identity();
(self + (self + identity).sqrt() * (self - identity).sqrt()).ln()
}
/// Calculates the hyperbolic quaternionic tangent.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let expected = Quaternion::new(1.0248695360556623, -0.10229568178876419, -0.1534435226831464, -0.20459136357752844);
/// let result = input.tanh();
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn tanh(&self) -> Self {
let s = self.sinh();
let c = self.cosh();
let ci = c.try_inverse().unwrap();
s * ci
}
/// Calculates the hyperbolic quaternionic arctangent.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Quaternion;
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
/// let expected = Quaternion::new(0.03230293287000163, 0.5173453683196951, 0.7760180524795426, 1.0346907366393903);
/// let result = input.atanh();
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
/// ```
#[inline]
pub fn atanh(&self) -> Self {
let identity = Self::identity();
((identity + self).ln() - (identity - self).ln()).half()
}
}
impl<N: Real + AbsDiffEq<Epsilon = N>> AbsDiffEq for Quaternion<N> {
@ -879,7 +1128,7 @@ impl<N: Real> UnitQuaternion<N> {
#[inline]
pub fn ln(&self) -> Quaternion<N> {
if let Some(v) = self.axis() {
Quaternion::from_parts(N::zero(), v.into_inner() * self.angle())
Quaternion::from_imag(v.into_inner() * self.angle())
} else {
Quaternion::zero()
}
@ -1073,3 +1322,4 @@ impl<N: Real + UlpsEq<Epsilon = N>> UlpsEq for UnitQuaternion<N> {
self.as_ref().ulps_eq(other.as_ref(), epsilon, max_ulps)
}
}

21
src/geometry/quaternion_construction.rs
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@ -13,9 +13,7 @@ use alga::general::Real;
use base::dimension::U3;
use base::storage::Storage;
#[cfg(feature = "arbitrary")]
use base::Vector3;
use base::{Unit, Vector, Vector4, Matrix3};
use base::{Unit, Vector, Vector3, Vector4, Matrix3};
use geometry::{Quaternion, Rotation3, UnitQuaternion};
@ -43,8 +41,13 @@ impl<N: Real> Quaternion<N> {
/// ```
#[inline]
pub fn new(w: N, i: N, j: N, k: N) -> Self {
let v = Vector4::<N>::new(i, j, k, w);
Self::from(v)
Self::from(Vector4::new(i, j, k, w))
}
/// Constructs a pure quaternion.
#[inline]
pub fn from_imag(vector: Vector3<N>) -> Self {
Self::from_parts(N::zero(), vector)
}
/// Creates a new quaternion from its scalar and vector parts. Note that the arguments order does
@ -92,7 +95,7 @@ impl<N: Real> Quaternion<N> {
/// ```
#[inline]
pub fn identity() -> Self {
Self::new(N::one(), N::zero(), N::zero(), N::zero())
Self::from_parts(N::one(), Vector3::zero())
}
}
@ -106,7 +109,7 @@ impl<N: Real> One for Quaternion<N> {
impl<N: Real> Zero for Quaternion<N> {
#[inline]
fn zero() -> Self {
Self::new(N::zero(), N::zero(), N::zero(), N::zero())
Self::from(Vector4::zero())
}
#[inline]
@ -579,7 +582,7 @@ impl<N: Real> UnitQuaternion<N> {
pub fn new<SB>(axisangle: Vector<N, U3, SB>) -> Self
where SB: Storage<N, U3> {
let two: N = ::convert(2.0f64);
let q = Quaternion::<N>::from_parts(N::zero(), axisangle / two).exp();
let q = Quaternion::<N>::from_imag(axisangle / two).exp();
Self::new_unchecked(q)
}
@ -608,7 +611,7 @@ impl<N: Real> UnitQuaternion<N> {
pub fn new_eps<SB>(axisangle: Vector<N, U3, SB>, eps: N) -> Self
where SB: Storage<N, U3> {
let two: N = ::convert(2.0f64);
let q = Quaternion::<N>::from_parts(N::zero(), axisangle / two).exp_eps(eps);
let q = Quaternion::<N>::from_imag(axisangle / two).exp_eps(eps);
Self::new_unchecked(q)
}