Introduce DualQuaternion type

This commit introduces the `DualQuaternion` type, in line with the plan
laid out in [#487].

[#487]: https://github.com/dimforge/nalgebra/issues/487

src/geometry/dual_quaternion.rs

@@ -0,0 +1,116 @@
+use crate::{Quaternion, SimdRealField};
+
+/// A dual quaternion.
+///
+/// # Indexing
+///
+/// DualQuaternions are stored as \[..real, ..dual\].
+/// Both of the quaternion components are laid out in `w, i, j, k` order.
+///
+/// ```
+/// # use nalgebra::{DualQuaternion, Quaternion};
+///
+/// let real = Quaternion::new(1.0, 2.0, 3.0, 4.0);
+/// let dual = Quaternion::new(5.0, 6.0, 7.0, 8.0);
+///
+/// let dq = DualQuaternion::from_real_and_dual(real, dual);
+/// assert_eq!(dq[0], 1.0);
+/// assert_eq!(dq[4], 5.0);
+/// assert_eq!(dq[6], 7.0);
+/// ```
+///
+/// NOTE:
+///  As of December 2020, dual quaternion support is a work in progress.
+///  If a feature that you need is missing, feel free to open an issue or a PR.
+///  See https://github.com/dimforge/nalgebra/issues/487
+#[repr(C)]
+#[derive(Debug, Default, Eq, PartialEq, Copy, Clone)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct DualQuaternion<N: SimdRealField> {
+    // [real(w, i, j, k), dual(w, i, j, k)]
+    pub(crate) dq: [N; 8],
+}
+
+impl<N: SimdRealField> DualQuaternion<N> {
+    /// Get the first quaternion component.
+    ///
+    /// # Example
+    /// ```
+    /// # #[macro_use] extern crate approx;
+    /// # use nalgebra::{DualQuaternion, Quaternion};
+    ///
+    /// let real = Quaternion::new(1.0, 2.0, 3.0, 4.0);
+    /// let dual = Quaternion::new(5.0, 6.0, 7.0, 8.0);
+    ///
+    /// let dq = DualQuaternion::from_real_and_dual(real, dual);
+    /// relative_eq!(dq.real(), real);
+    /// ```
+    #[inline]
+    pub fn real(&self) -> Quaternion<N> {
+        Quaternion::new(self[0], self[1], self[2], self[3])
+    }
+
+    /// Get the second quaternion component.
+    ///
+    /// # Example
+    /// ```
+    /// # #[macro_use] extern crate approx;
+    /// # use nalgebra::{DualQuaternion, Quaternion};
+    ///
+    /// let real = Quaternion::new(1.0, 2.0, 3.0, 4.0);
+    /// let dual = Quaternion::new(5.0, 6.0, 7.0, 8.0);
+    ///
+    /// let dq = DualQuaternion::from_real_and_dual(real, dual);
+    /// relative_eq!(dq.dual(), dual);
+    /// ```
+    #[inline]
+    pub fn dual(&self) -> Quaternion<N> {
+        Quaternion::new(self[4], self[5], self[6], self[7])
+    }
+}
+
+impl<N: SimdRealField> DualQuaternion<N>
+where
+    N::Element: SimdRealField,
+{
+    /// Normalizes this quaternion.
+    ///
+    /// # Example
+    /// ```
+    /// # #[macro_use] extern crate approx;
+    /// # use nalgebra::{DualQuaternion, Quaternion};
+    /// let real = Quaternion::new(1.0, 2.0, 3.0, 4.0);
+    /// let dual = Quaternion::new(5.0, 6.0, 7.0, 8.0);
+    /// let dq = DualQuaternion::from_real_and_dual(real, dual);
+    ///
+    /// let dq_normalized = dq.normalize();
+    ///
+    /// relative_eq!(dq_normalized.real().norm(), 1.0);
+    /// ```
+    #[inline]
+    #[must_use = "Did you mean to use normalize_mut()?"]
+    pub fn normalize(&self) -> Self {
+        let real_norm = self.real().norm();
+
+        Self::from_real_and_dual(self.real() / real_norm, self.dual() / real_norm)
+    }
+
+    /// Normalizes this quaternion.
+    ///
+    /// # Example
+    /// ```
+    /// # #[macro_use] extern crate approx;
+    /// # use nalgebra::{DualQuaternion, Quaternion};
+    /// let real = Quaternion::new(1.0, 2.0, 3.0, 4.0);
+    /// let dual = Quaternion::new(5.0, 6.0, 7.0, 8.0);
+    /// let mut dq = DualQuaternion::from_real_and_dual(real, dual);
+    ///
+    /// dq.normalize_mut();
+    ///
+    /// relative_eq!(dq.real().norm(), 1.0);
+    /// ```
+    #[inline]
+    pub fn normalize_mut(&mut self) {
+        *self = self.normalize();
+    }
+}

src/geometry/dual_quaternion_construction.rs

@@ -0,0 +1,49 @@
+use crate::{DualQuaternion, Quaternion, SimdRealField};
+
+impl<N: SimdRealField> DualQuaternion<N> {
+    /// Creates a dual quaternion from its rotation and translation components.
+    ///
+    /// # Example
+    /// ```
+    /// # use nalgebra::{DualQuaternion, Quaternion};
+    /// let rot = Quaternion::new(1.0, 2.0, 3.0, 4.0);
+    /// let trans = Quaternion::new(5.0, 6.0, 7.0, 8.0);
+    ///
+    /// let dq = DualQuaternion::from_real_and_dual(rot, trans);
+    /// assert_eq!(dq.real().w, 1.0);
+    /// ```
+    #[inline]
+    pub fn from_real_and_dual(real: Quaternion<N>, dual: Quaternion<N>) -> Self {
+        Self {
+            dq: [
+                real.w, real.i, real.j, real.k, dual.w, dual.i, dual.j, dual.k,
+            ],
+        }
+    }
+}
+
+impl<N: SimdRealField> DualQuaternion<N> {
+    /// The dual quaternion multiplicative identity
+    ///
+    /// # Example
+    ///
+    /// ```
+    /// # use nalgebra::{DualQuaternion, Quaternion};
+    ///
+    /// let dq1 = DualQuaternion::identity();
+    /// let dq2 = DualQuaternion::from_real_and_dual(
+    ///     Quaternion::new(1.,2.,3.,4.),
+    ///     Quaternion::new(5.,6.,7.,8.)
+    /// );
+    ///
+    /// assert_eq!(dq1 * dq2, dq2);
+    /// assert_eq!(dq2 * dq1, dq2);
+    /// ```
+    #[inline]
+    pub fn identity() -> Self {
+        Self::from_real_and_dual(
+            Quaternion::from_real(N::one()),
+            Quaternion::from_real(N::zero()),
+        )
+    }
+}

src/geometry/dual_quaternion_ops.rs

@@ -0,0 +1,95 @@
+/*
+ * This file provides:
+ *
+ * NOTE: Work in progress https://github.com/dimforge/nalgebra/issues/487
+ *
+ * (Dual Quaternion)
+ *
+ * Index<usize>
+ * IndexMut<usize>
+ *
+ * (Assignment Operators)
+ *
+ * DualQuaternion × Scalar
+ * DualQuaternion × DualQuaternion
+ * DualQuaternion + DualQuaternion
+ * DualQuaternion - DualQuaternion
+ *
+ * ---
+ *
+ * References:
+ *   Multiplication:
+ *   - https://cs.gmu.edu/~jmlien/teaching/cs451/uploads/Main/dual-quaternion.pdf
+ */
+
+use crate::base::allocator::Allocator;
+use crate::{DefaultAllocator, DualQuaternion, SimdRealField, U1, U4};
+use simba::simd::SimdValue;
+use std::ops::{Add, Index, IndexMut, Mul, Sub};
+
+impl<N: SimdRealField> Index<usize> for DualQuaternion<N> {
+    type Output = N;
+
+    #[inline]
+    fn index(&self, i: usize) -> &Self::Output {
+        &self.dq[i]
+    }
+}
+
+impl<N: SimdRealField> IndexMut<usize> for DualQuaternion<N> {
+    #[inline]
+    fn index_mut(&mut self, i: usize) -> &mut N {
+        &mut self.dq[i]
+    }
+}
+
+impl<N: SimdRealField> Mul<DualQuaternion<N>> for DualQuaternion<N>
+where
+    N::Element: SimdRealField,
+    DefaultAllocator: Allocator<N, U4, U1> + Allocator<N, U4, U1>,
+{
+    type Output = DualQuaternion<N>;
+
+    fn mul(self, rhs: Self) -> Self::Output {
+        Self::from_real_and_dual(
+            self.real() * rhs.real(),
+            self.real() * rhs.dual() + self.dual() * rhs.real(),
+        )
+    }
+}
+
+impl<N: SimdRealField> Mul<N> for DualQuaternion<N>
+where
+    N::Element: SimdRealField + SimdValue,
+    DefaultAllocator: Allocator<N, U4, U1> + Allocator<N, U4, U1>,
+{
+    type Output = DualQuaternion<N>;
+
+    fn mul(self, rhs: N) -> Self::Output {
+        Self::from_real_and_dual(self.real() * rhs, self.dual() * rhs)
+    }
+}
+
+impl<N: SimdRealField> Add<DualQuaternion<N>> for DualQuaternion<N>
+where
+    N::Element: SimdRealField,
+    DefaultAllocator: Allocator<N, U4, U1> + Allocator<N, U4, U1>,
+{
+    type Output = DualQuaternion<N>;
+
+    fn add(self, rhs: DualQuaternion<N>) -> Self::Output {
+        Self::from_real_and_dual(self.real() + rhs.real(), self.dual() + rhs.dual())
+    }
+}
+
+impl<N: SimdRealField> Sub<DualQuaternion<N>> for DualQuaternion<N>
+where
+    N::Element: SimdRealField,
+    DefaultAllocator: Allocator<N, U4, U1> + Allocator<N, U4, U1>,
+{
+    type Output = DualQuaternion<N>;
+
+    fn sub(self, rhs: DualQuaternion<N>) -> Self::Output {
+        Self::from_real_and_dual(self.real() - rhs.real(), self.dual() - rhs.dual())
+    }
+}

src/geometry/mod.rs

@@ -35,6 +35,10 @@ mod quaternion_coordinates;
 mod quaternion_ops;
 mod quaternion_simba;
 
+mod dual_quaternion;
+mod dual_quaternion_construction;
+mod dual_quaternion_ops;
+
 mod unit_complex;
 #[cfg(feature = "alga")]
 mod unit_complex_alga;
@@ -98,6 +102,8 @@ pub use self::rotation_alias::*;
 
 pub use self::quaternion::*;
 
+pub use self::dual_quaternion::*;
+
 pub use self::unit_complex::*;
 
 pub use self::translation::*;

src/geometry/quaternion.rs

@@ -1,7 +1,6 @@
 use approx::{AbsDiffEq, RelativeEq, UlpsEq};
 use num::Zero;
 use std::fmt;
-use std::hash;
 #[cfg(feature = "abomonation-serialize")]
 use std::io::{Result as IOResult, Write};
 
@@ -14,7 +13,7 @@ use serde::{Deserialize, Deserializer, Serialize, Serializer};
 use abomonation::Abomonation;
 
 use simba::scalar::{ClosedNeg, RealField};
-use simba::simd::{SimdBool, SimdOption, SimdRealField, SimdValue};
+use simba::simd::{SimdBool, SimdOption, SimdRealField};
 
 use crate::base::dimension::{U1, U3, U4};
 use crate::base::storage::{CStride, RStride};
@@ -23,7 +22,6 @@ use crate::base::{
 };
 
 use crate::geometry::{Point3, Rotation};
-use std::ops::Neg;
 
 /// A quaternion. See the type alias `UnitQuaternion = Unit<Quaternion>` for a quaternion
 /// that may be used as a rotation.

src/geometry/quaternion_construction.rs

@@ -10,7 +10,7 @@ use rand::distributions::{Distribution, OpenClosed01, Standard};
 use rand::Rng;
 
 use simba::scalar::RealField;
-use simba::simd::{SimdBool, SimdValue};
+use simba::simd::SimdBool;
 
 use crate::base::dimension::U3;
 use crate::base::storage::Storage;