nalgebra/geometry/unit_complex.rs
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use approx::{AbsDiffEq, RelativeEq, UlpsEq};
use num_complex::Complex;
use std::fmt;
use crate::base::{Matrix2, Matrix3, Normed, Unit, Vector1, Vector2};
use crate::geometry::{Point2, Rotation2};
use crate::Scalar;
use simba::scalar::RealField;
use simba::simd::SimdRealField;
use std::cmp::{Eq, PartialEq};
/// A 2D rotation represented as a complex number with magnitude 1.
///
/// All the methods specific [`UnitComplex`](crate::UnitComplex) are listed here. You may also
/// read the documentation of the [`Complex`](crate::Complex) type which
/// is used internally and accessible with `unit_complex.complex()`.
///
/// # Construction
/// * [Identity <span style="float:right;">`identity`</span>](#identity)
/// * [From a 2D rotation angle <span style="float:right;">`new`, `from_cos_sin_unchecked`…</span>](#construction-from-a-2d-rotation-angle)
/// * [From an existing 2D matrix or complex number <span style="float:right;">`from_matrix`, `rotation_to`, `powf`…</span>](#construction-from-an-existing-2d-matrix-or-complex-number)
/// * [From two vectors <span style="float:right;">`rotation_between`, `scaled_rotation_between_axis`…</span>](#construction-from-two-vectors)
///
/// # Transformation and composition
/// * [Angle extraction <span style="float:right;">`angle`, `angle_to`…</span>](#angle-extraction)
/// * [Transformation of a vector or a point <span style="float:right;">`transform_vector`, `inverse_transform_point`…</span>](#transformation-of-a-vector-or-a-point)
/// * [Conjugation and inversion <span style="float:right;">`conjugate`, `inverse_mut`…</span>](#conjugation-and-inversion)
/// * [Interpolation <span style="float:right;">`slerp`…</span>](#interpolation)
///
/// # Conversion
/// * [Conversion to a matrix <span style="float:right;">`to_rotation_matrix`, `to_homogeneous`…</span>](#conversion-to-a-matrix)
pub type UnitComplex<T> = Unit<Complex<T>>;
impl<T: Scalar + PartialEq> PartialEq for UnitComplex<T> {
#[inline]
fn eq(&self, rhs: &Self) -> bool {
(**self).eq(&**rhs)
}
}
impl<T: Scalar + Eq> Eq for UnitComplex<T> {}
impl<T: SimdRealField> Normed for Complex<T> {
type Norm = T::SimdRealField;
#[inline]
fn norm(&self) -> T::SimdRealField {
// We don't use `.norm_sqr()` because it requires
// some very strong Num trait requirements.
(self.re.clone() * self.re.clone() + self.im.clone() * self.im.clone()).simd_sqrt()
}
#[inline]
fn norm_squared(&self) -> T::SimdRealField {
// We don't use `.norm_sqr()` because it requires
// some very strong Num trait requirements.
self.re.clone() * self.re.clone() + self.im.clone() * self.im.clone()
}
#[inline]
fn scale_mut(&mut self, n: Self::Norm) {
self.re *= n.clone();
self.im *= n;
}
#[inline]
fn unscale_mut(&mut self, n: Self::Norm) {
self.re /= n.clone();
self.im /= n;
}
}
/// # Angle extraction
impl<T: SimdRealField> UnitComplex<T>
where
T::Element: SimdRealField,
{
/// The rotation angle in `]-pi; pi]` of this unit complex number.
///
/// # Example
/// ```
/// # use nalgebra::UnitComplex;
/// let rot = UnitComplex::new(1.78);
/// assert_eq!(rot.angle(), 1.78);
/// ```
#[inline]
#[must_use]
pub fn angle(&self) -> T {
self.im.clone().simd_atan2(self.re.clone())
}
/// The sine of the rotation angle.
///
/// # Example
/// ```
/// # use nalgebra::UnitComplex;
/// let angle = 1.78f32;
/// let rot = UnitComplex::new(angle);
/// assert_eq!(rot.sin_angle(), angle.sin());
/// ```
#[inline]
#[must_use]
pub fn sin_angle(&self) -> T {
self.im.clone()
}
/// The cosine of the rotation angle.
///
/// # Example
/// ```
/// # use nalgebra::UnitComplex;
/// let angle = 1.78f32;
/// let rot = UnitComplex::new(angle);
/// assert_eq!(rot.cos_angle(),angle.cos());
/// ```
#[inline]
#[must_use]
pub fn cos_angle(&self) -> T {
self.re.clone()
}
/// The rotation angle returned as a 1-dimensional vector.
///
/// This is generally used in the context of generic programming. Using
/// the `.angle()` method instead is more common.
#[inline]
#[must_use]
pub fn scaled_axis(&self) -> Vector1<T> {
Vector1::new(self.angle())
}
/// The rotation axis and angle in (0, pi] of this complex number.
///
/// This is generally used in the context of generic programming. Using
/// the `.angle()` method instead is more common.
/// Returns `None` if the angle is zero.
#[inline]
#[must_use]
pub fn axis_angle(&self) -> Option<(Unit<Vector1<T>>, T)>
where
T: RealField,
{
let ang = self.angle();
if ang.is_zero() {
None
} else if ang.is_sign_positive() {
Some((Unit::new_unchecked(Vector1::x()), ang))
} else {
Some((Unit::new_unchecked(-Vector1::<T>::x()), -ang))
}
}
/// The rotation angle needed to make `self` and `other` coincide.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::UnitComplex;
/// let rot1 = UnitComplex::new(0.1);
/// let rot2 = UnitComplex::new(1.7);
/// assert_relative_eq!(rot1.angle_to(&rot2), 1.6);
/// ```
#[inline]
#[must_use]
pub fn angle_to(&self, other: &Self) -> T {
let delta = self.rotation_to(other);
delta.angle()
}
}
/// # Conjugation and inversion
impl<T: SimdRealField> UnitComplex<T>
where
T::Element: SimdRealField,
{
/// Compute the conjugate of this unit complex number.
///
/// # Example
/// ```
/// # use nalgebra::UnitComplex;
/// let rot = UnitComplex::new(1.78);
/// let conj = rot.conjugate();
/// assert_eq!(rot.complex().im, -conj.complex().im);
/// assert_eq!(rot.complex().re, conj.complex().re);
/// ```
#[inline]
#[must_use = "Did you mean to use conjugate_mut()?"]
pub fn conjugate(&self) -> Self {
Self::new_unchecked(self.conj())
}
/// Inverts this complex number if it is not zero.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::UnitComplex;
/// let rot = UnitComplex::new(1.2);
/// let inv = rot.inverse();
/// assert_relative_eq!(rot * inv, UnitComplex::identity(), epsilon = 1.0e-6);
/// assert_relative_eq!(inv * rot, UnitComplex::identity(), epsilon = 1.0e-6);
/// ```
#[inline]
#[must_use = "Did you mean to use inverse_mut()?"]
pub fn inverse(&self) -> Self {
self.conjugate()
}
/// Compute in-place the conjugate of this unit complex number.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::UnitComplex;
/// let angle = 1.7;
/// let rot = UnitComplex::new(angle);
/// let mut conj = UnitComplex::new(angle);
/// conj.conjugate_mut();
/// assert_eq!(rot.complex().im, -conj.complex().im);
/// assert_eq!(rot.complex().re, conj.complex().re);
/// ```
#[inline]
pub fn conjugate_mut(&mut self) {
let me = self.as_mut_unchecked();
me.im = -me.im.clone();
}
/// Inverts in-place this unit complex number.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::UnitComplex;
/// let angle = 1.7;
/// let mut rot = UnitComplex::new(angle);
/// rot.inverse_mut();
/// assert_relative_eq!(rot * UnitComplex::new(angle), UnitComplex::identity());
/// assert_relative_eq!(UnitComplex::new(angle) * rot, UnitComplex::identity());
/// ```
#[inline]
pub fn inverse_mut(&mut self) {
self.conjugate_mut()
}
}
/// # Conversion to a matrix
impl<T: SimdRealField> UnitComplex<T>
where
T::Element: SimdRealField,
{
/// Builds the rotation matrix corresponding to this unit complex number.
///
/// # Example
/// ```
/// # use nalgebra::{UnitComplex, Rotation2};
/// # use std::f32;
/// let rot = UnitComplex::new(f32::consts::FRAC_PI_6);
/// let expected = Rotation2::new(f32::consts::FRAC_PI_6);
/// assert_eq!(rot.to_rotation_matrix(), expected);
/// ```
#[inline]
#[must_use]
pub fn to_rotation_matrix(self) -> Rotation2<T> {
let r = self.re.clone();
let i = self.im.clone();
Rotation2::from_matrix_unchecked(Matrix2::new(r.clone(), -i.clone(), i, r))
}
/// Converts this unit complex number into its equivalent homogeneous transformation matrix.
///
/// # Example
/// ```
/// # use nalgebra::{UnitComplex, Matrix3};
/// # use std::f32;
/// let rot = UnitComplex::new(f32::consts::FRAC_PI_6);
/// let expected = Matrix3::new(0.8660254, -0.5, 0.0,
/// 0.5, 0.8660254, 0.0,
/// 0.0, 0.0, 1.0);
/// assert_eq!(rot.to_homogeneous(), expected);
/// ```
#[inline]
#[must_use]
pub fn to_homogeneous(self) -> Matrix3<T> {
self.to_rotation_matrix().to_homogeneous()
}
}
/// # Transformation of a vector or a point
impl<T: SimdRealField> UnitComplex<T>
where
T::Element: SimdRealField,
{
/// Rotate the given point by this unit complex number.
///
/// This is the same as the multiplication `self * pt`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{UnitComplex, Point2};
/// # use std::f32;
/// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
/// let transformed_point = rot.transform_point(&Point2::new(1.0, 2.0));
/// assert_relative_eq!(transformed_point, Point2::new(-2.0, 1.0), epsilon = 1.0e-6);
/// ```
#[inline]
#[must_use]
pub fn transform_point(&self, pt: &Point2<T>) -> Point2<T> {
self * pt
}
/// Rotate the given vector by this unit complex number.
///
/// This is the same as the multiplication `self * v`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{UnitComplex, Vector2};
/// # use std::f32;
/// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
/// let transformed_vector = rot.transform_vector(&Vector2::new(1.0, 2.0));
/// assert_relative_eq!(transformed_vector, Vector2::new(-2.0, 1.0), epsilon = 1.0e-6);
/// ```
#[inline]
#[must_use]
pub fn transform_vector(&self, v: &Vector2<T>) -> Vector2<T> {
self * v
}
/// Rotate the given point by the inverse of this unit complex number.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{UnitComplex, Point2};
/// # use std::f32;
/// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
/// let transformed_point = rot.inverse_transform_point(&Point2::new(1.0, 2.0));
/// assert_relative_eq!(transformed_point, Point2::new(2.0, -1.0), epsilon = 1.0e-6);
/// ```
#[inline]
#[must_use]
pub fn inverse_transform_point(&self, pt: &Point2<T>) -> Point2<T> {
// TODO: would it be useful performance-wise not to call inverse explicitly (i-e. implement
// the inverse transformation explicitly here) ?
self.inverse() * pt
}
/// Rotate the given vector by the inverse of this unit complex number.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{UnitComplex, Vector2};
/// # use std::f32;
/// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
/// let transformed_vector = rot.inverse_transform_vector(&Vector2::new(1.0, 2.0));
/// assert_relative_eq!(transformed_vector, Vector2::new(2.0, -1.0), epsilon = 1.0e-6);
/// ```
#[inline]
#[must_use]
pub fn inverse_transform_vector(&self, v: &Vector2<T>) -> Vector2<T> {
self.inverse() * v
}
/// Rotate the given vector by the inverse of this unit complex number.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{UnitComplex, Vector2};
/// # use std::f32;
/// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
/// let transformed_vector = rot.inverse_transform_unit_vector(&Vector2::x_axis());
/// assert_relative_eq!(transformed_vector, -Vector2::y_axis(), epsilon = 1.0e-6);
/// ```
#[inline]
#[must_use]
pub fn inverse_transform_unit_vector(&self, v: &Unit<Vector2<T>>) -> Unit<Vector2<T>> {
self.inverse() * v
}
}
/// # Interpolation
impl<T: SimdRealField> UnitComplex<T>
where
T::Element: SimdRealField,
{
/// Spherical linear interpolation between two rotations represented as unit complex numbers.
///
/// # Examples:
///
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::geometry::UnitComplex;
///
/// let rot1 = UnitComplex::new(std::f32::consts::FRAC_PI_4);
/// let rot2 = UnitComplex::new(-std::f32::consts::PI);
///
/// let rot = rot1.slerp(&rot2, 1.0 / 3.0);
///
/// assert_relative_eq!(rot.angle(), std::f32::consts::FRAC_PI_2);
/// ```
#[inline]
#[must_use]
pub fn slerp(&self, other: &Self, t: T) -> Self {
let delta = other / self;
self * Self::new(delta.angle() * t)
}
}
impl<T: RealField + fmt::Display> fmt::Display for UnitComplex<T> {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
write!(f, "UnitComplex angle: {}", self.angle())
}
}
impl<T: RealField> AbsDiffEq for UnitComplex<T> {
type Epsilon = T;
#[inline]
fn default_epsilon() -> Self::Epsilon {
T::default_epsilon()
}
#[inline]
fn abs_diff_eq(&self, other: &Self, epsilon: Self::Epsilon) -> bool {
self.re.abs_diff_eq(&other.re, epsilon.clone()) && self.im.abs_diff_eq(&other.im, epsilon)
}
}
impl<T: RealField> RelativeEq for UnitComplex<T> {
#[inline]
fn default_max_relative() -> Self::Epsilon {
T::default_max_relative()
}
#[inline]
fn relative_eq(
&self,
other: &Self,
epsilon: Self::Epsilon,
max_relative: Self::Epsilon,
) -> bool {
self.re
.relative_eq(&other.re, epsilon.clone(), max_relative.clone())
&& self.im.relative_eq(&other.im, epsilon, max_relative)
}
}
impl<T: RealField> UlpsEq for UnitComplex<T> {
#[inline]
fn default_max_ulps() -> u32 {
T::default_max_ulps()
}
#[inline]
fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
self.re.ulps_eq(&other.re, epsilon.clone(), max_ulps)
&& self.im.ulps_eq(&other.im, epsilon, max_ulps)
}
}