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// https://pomax.github.io/bezierinfo
use num_traits::Float;
use core::ops::*;
use core::iter::Sum;
use vec::repr_c::{
Vec3 as CVec3,
Vec4 as CVec4,
};
// WISH: into_iter, iter_mut, etc (for concisely applying the same xform to all points)
// WISH: AABBs from beziers
// WISH: OOBBs from beziers
// WISH: "Tracing a curve at fixed distance intervals"
// WISH: project a point on a curve using e.g binary search after a coarse linear search
macro_rules! bezier_impl_any {
($Bezier:ident $Point:ident) => {
impl<T> $Bezier<T> {
/// Evaluates the normalized tangent at interpolation factor `t`.
pub fn normalized_tangent(self, t: T) -> $Point<T> where T: Float + Sum {
self.evaluate_derivative(t).normalized()
}
// WISH: better length approximation estimations (e.g see https://math.stackexchange.com/a/61796)
/// Approximates the curve's length by subdividing it into step_count+1 segments.
pub fn length_by_discretization(self, step_count: u32) -> T
where T: Float + AddAssign + Sum
{
let mut length = T::zero();
let mut prev_point = self.evaluate(T::zero());
for i in 1..step_count+2 {
let t = T::from(i).unwrap()/(T::from(step_count).unwrap()+T::one());
let next_point = self.evaluate(t);
length += (next_point - prev_point).magnitude();
prev_point = next_point;
}
length
}
}
}
}
macro_rules! bezier_impl_quadratic {
($QuadraticBezier:ident $Point:ident $LineSegment:ident) => {
#[derive(Debug, Default, Copy, Clone, Hash, PartialEq, Eq, /*PartialOrd, Ord*/)]
#[cfg_attr(feature="serde", derive(Serialize, Deserialize))]
pub struct $QuadraticBezier<T>(pub $Point<T>, pub $Point<T>, pub $Point<T>);
impl<T: Float> $QuadraticBezier<T> {
/// Evaluates the position of the point lying on the curve at interpolation factor `t`.
///
/// This is one of the most important Bézier curve operations,
/// because, in one way or another, it is used to render a curve
/// to the screen.
/// The common use case is to successively evaluate a curve at a set of values
/// that range from 0 to 1, to approximate the curve as an array of
/// line segments which are then rendered.
pub fn evaluate(self, t: T) -> $Point<T> {
let l = T::one();
let two = l+l;
self.0*(l-t)*(l-t) + self.1*two*(l-t)*t + self.2*t*t
}
/// Evaluates the derivative tangent at interpolation factor `t`, which happens to give
/// a non-normalized tangent vector.
///
/// See also `normalized_tangent()`.
pub fn evaluate_derivative(self, t: T) -> $Point<T> {
let l = T::one();
let n = l+l;
(self.1-self.0)*(l-t)*n + (self.2-self.1)*t*n
}
/// Creates a quadratic Bézier curve from a single segment.
pub fn from_line_segment(line: $LineSegment<T>) -> Self {
$QuadraticBezier(line.a, line.a, line.b)
}
/// Returns the constant matrix M such that,
/// given `T = [1, t*t, t*t*t]` and `P` the vector of control points,
/// `dot(T * M, P)` evalutes the Bezier curve at 't'.
///
/// This function name is arguably dubious.
pub fn matrix() -> Mat3<T> {
let zero = T::zero();
let one = T::one();
let two = one+one;
Mat3 {
rows: CVec3::new(
Vec3::new( one, zero, zero),
Vec3::new(-two, two, zero),
Vec3::new( one, -two, one),
)
}
}
/// Splits this quadratic Bézier curve into two curves, at interpolation factor `t`.
// NOTE that some computations may be reused, but the compiler can
// reason about these. Clarity wins here IMO.
pub fn split(self, t: T) -> (Self, Self) {
let l = T::one();
let two = l+l;
let first = $QuadraticBezier(
self.0,
self.1*t - self.0*(t-l),
self.2*t*t - self.1*two*t*(t-l) + self.0*(t-l)*(t-l),
);
let second = $QuadraticBezier(
self.2*t*t - self.1*two*t*(t-l) + self.0*(t-l)*(t-l),
self.2*t - self.1*(t-l),
self.2,
);
(first, second)
}
}
impl<T> From<Vec3<$Point<T>>> for $QuadraticBezier<T> {
fn from(v: Vec3<$Point<T>>) -> Self {
$QuadraticBezier(v.x, v.y, v.z)
}
}
impl<T> From<$QuadraticBezier<T>> for Vec3<$Point<T>> {
fn from(v: $QuadraticBezier<T>) -> Self {
Vec3::new(v.0, v.1, v.2)
}
}
bezier_impl_any!($QuadraticBezier $Point);
}
}
macro_rules! bezier_impl_cubic {
($CubicBezier:ident $Point:ident $LineSegment:ident) => {
#[derive(Debug, Default, Copy, Clone, Hash, PartialEq, Eq, /*PartialOrd, Ord*/)]
#[cfg_attr(feature="serde", derive(Serialize, Deserialize))]
pub struct $CubicBezier<T>(pub $Point<T>, pub $Point<T>, pub $Point<T>, pub $Point<T>);
impl<T: Float> $CubicBezier<T> {
/// Evaluates the position of the point lying on the curve at interpolation factor `t`.
///
/// This is one of the most important Bézier curve operations,
/// because, in one way or another, it is used to render a curve
/// to the screen.
/// The common use case is to successively evaluate a curve at a set of values
/// that range from 0 to 1, to approximate the curve as an array of
/// line segments which are then rendered.
pub fn evaluate(self, t: T) -> $Point<T> {
let l = T::one();
let three = l+l+l;
self.0*(l-t)*(l-t)*(l-t) + self.1*three*(l-t)*(l-t)*t + self.2*three*(l-t)*t*t + self.3*t*t*t
}
/// Evaluates the derivative tangent at interpolation factor `t`, which happens to give
/// a non-normalized tangent vector.
///
/// See also `normalized_tangent()`.
pub fn evaluate_derivative(self, t: T) -> $Point<T> {
let l = T::one();
let n = l+l+l;
let two = l+l;
(self.1-self.0)*(l-t)*(l-t)*n + (self.2-self.1)*two*(l-t)*t*n + (self.3-self.2)*t*t*n
}
/// Creates a cubic Bézier curve from a single segment.
pub fn from_line_segment(line: $LineSegment<T>) -> Self {
$CubicBezier(line.a, line.a, line.b, line.b)
}
/// Returns the constant matrix M such that,
/// given `T = [1, t*t, t*t*t, t*t*t*t]` and `P` the vector of control points,
/// `dot(T * M, P)` evalutes the Bezier curve at 't'.
///
/// This function name is arguably dubious.
pub fn matrix() -> Mat4<T> {
let zero = T::zero();
let one = T::one();
let three = one+one+one;
let six = three + three;
Mat4 {
rows: CVec4::new(
Vec4::new( one, zero, zero, zero),
Vec4::new(-three, three, zero, zero),
Vec4::new( three, -six, three, zero),
Vec4::new(-one, three, -three, one),
)
}
}
/// Splits this cubic Bézier curve into two curves, at interpolation factor `t`.
// NOTE that some computations may be reused, but the compiler can
// reason about these. Clarity wins here IMO.
pub fn split(self, t: T) -> (Self, Self) {
let l = T::one();
let two = l+l;
let three = l+l+l;
let first = $CubicBezier(
self.0,
self.1*t - self.0*(t-l),
self.2*t*t - self.1*two*t*(t-l) + self.0*(t-l)*(t-l),
self.3*t*t*t - self.2*three*t*t*(t-l) + self.1*three*t*(t-l)*(t-l) - self.0*(t-l)*(t-l)*(t-l),
);
let second = $CubicBezier(
self.3*t*t*t - self.2*three*t*t*(t-l) + self.1*three*t*(t-l)*(t-l) - self.0*(t-l)*(t-l)*(t-l),
self.3*t*t - self.2*two*t*(t-l) + self.1*(t-l)*(t-l),
self.3*t - self.2*(t-l),
self.3,
);
(first, second)
}
// WISH: CubicBezier::circle(radius)
// pub fn circle(radius: T, curve_count: u32) ->
}
impl<T> From<Vec4<$Point<T>>> for $CubicBezier<T> {
fn from(v: Vec4<$Point<T>>) -> Self {
$CubicBezier(v.x, v.y, v.z, v.w)
}
}
impl<T> From<$CubicBezier<T>> for Vec4<$Point<T>> {
fn from(v: $CubicBezier<T>) -> Self {
Vec4::new(v.0, v.1, v.2, v.3)
}
}
bezier_impl_any!($CubicBezier $Point);
}
}
#[cfg(all(nightly, feature="repr_simd"))]
pub mod repr_simd {
use super::*;
use vec::repr_simd::{Vec3, Vec4, Vec2};
use mat::repr_simd::row_major::{Mat3, Mat4};
use geom::repr_simd::{LineSegment2, LineSegment3};
bezier_impl_quadratic!(QuadraticBezier2 Vec2 LineSegment2);
bezier_impl_quadratic!(QuadraticBezier3 Vec3 LineSegment3);
bezier_impl_cubic!(CubicBezier2 Vec2 LineSegment2);
bezier_impl_cubic!(CubicBezier3 Vec3 LineSegment3);
}
pub mod repr_c {
use super::*;
use vec::repr_c::{Vec3, Vec4, Vec2};
use mat::repr_c::row_major::{Mat3, Mat4};
use geom::repr_c::{LineSegment2, LineSegment3};
bezier_impl_quadratic!(QuadraticBezier2 Vec2 LineSegment2);
bezier_impl_quadratic!(QuadraticBezier3 Vec3 LineSegment3);
bezier_impl_cubic!(CubicBezier2 Vec2 LineSegment2);
bezier_impl_cubic!(CubicBezier3 Vec3 LineSegment3);
}
pub use self::repr_c::*;