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//! # Spline interpolation made easy. //! //! This crate exposes splines for which each sections can be interpolated independently of each //! other – i.e. it’s possible to interpolate with a linear interpolator on one section and then //! switch to a cubic Hermite interpolator for the next section. //! //! Most of the crate consists of three types: //! //! - [`Key`], which represents the control points by which the spline must pass. //! - [`Interpolation`], the type of possible interpolation for each segment. //! - [`Spline`], a spline from which you can *sample* points by interpolation. //! //! When adding control points, you add new sections. Two control points define a section – i.e. //! it’s not possible to define a spline without at least two control points. Every time you add a //! new control point, a new section is created. Each section is assigned an interpolation mode that //! is picked from its lower control point. //! //! # Quickly create splines //! //! ``` //! use splines::{Interpolation, Key, Spline}; //! //! let start = Key::new(0., 0., Interpolation::Linear); //! let end = Key::new(1., 10., Interpolation::default()); //! let spline = Spline::from_vec(vec![start, end]); //! ``` //! //! You will notice that we used `Interpolation::Linear` for the first key. The first key `start`’s //! interpolation will be used for the whole segment defined by those two keys. The `end`’s //! interpolation won’t be used. You can in theory use any [`Interpolation`] you want for the last //! key. We use the default one because we don’t care. //! //! # Interpolate values //! //! The whole purpose of splines is to interpolate discrete values to yield continuous ones. This is //! usually done with the `Spline::sample` method. This method expects the interpolation parameter //! (often, this will be the time of your simulation) as argument and will yield an interpolated //! value. //! //! If you try to sample in out-of-bounds interpolation parameter, you’ll get no value. //! //! ``` //! # use splines::{Interpolation, Key, Spline}; //! # let start = Key::new(0., 0., Interpolation::Linear); //! # let end = Key::new(1., 10., Interpolation::Linear); //! # let spline = Spline::from_vec(vec![start, end]); //! assert_eq!(spline.sample(0.), Some(0.)); //! assert_eq!(spline.clamped_sample(1.), 10.); //! assert_eq!(spline.sample(1.1), None); //! ``` //! //! It’s possible that you want to get a value even if you’re out-of-bounds. This is especially //! important for simulations / animations. Feel free to use the `Spline::clamped_interpolation` for //! that purpose. //! //! ``` //! # use splines::{Interpolation, Key, Spline}; //! # let start = Key::new(0., 0., Interpolation::Linear); //! # let end = Key::new(1., 10., Interpolation::Linear); //! # let spline = Spline::from_vec(vec![start, end]); //! assert_eq!(spline.clamped_sample(-0.9), 0.); // clamped to the first key //! assert_eq!(spline.clamped_sample(1.1), 10.); // clamped to the last key //! ``` //! # Features and customization //! //! This crate was written with features baked in and hidden behind feature-gates. The idea is that //! the default configuration (i.e. you just add `"spline = …"` to your `Cargo.toml`) will always //! give you the minimal, core and raw concepts of what splines, keys / knots and interpolation //! modes are. However, you might want more. Instead of letting other people do the extra work to //! add implementations for very famous and useful traits – and do it in less efficient way, because //! they wouldn’t have access to the internals of this crate, it’s possible to enable features in an //! ad hoc way. //! //! This mechanism is not final and this is currently an experiment to see how people like it or //! not. It’s especially important to see how it copes with the documentation. //! //! So here’s a list of currently supported features and how to enable them: //! //! - **Serialization / deserialization.** //! + This feature implements both the `Serialize` and `Deserialize` traits from `serde` for all //! types exported by this crate. //! + Enable with the `"serialization"` feature. //! - **[cgmath](https://crates.io/crates/cgmath) implementors.** //! + Adds some usefull implementations of `Interpolate` for some cgmath types. //! + Enable with the `"impl-cgmath"` feature. //! - **Standard library / no standard library.** //! + It’s possible to compile against the standard library or go on your own without it. //! + Compiling with the standard library is enabled by default. //! + Use `default-features = []` in your `Cargo.toml` to disable. //! + Enable explicitly with the `"std"` feature. #![cfg_attr(not(feature = "std"), no_std)] #![cfg_attr(not(feature = "std"), feature(alloc))] #![cfg_attr(not(feature = "std"), feature(core_intrinsics))] // on no_std, we also need the alloc crate for Vec #[cfg(not(feature = "std"))] extern crate alloc; #[cfg(feature = "impl-cgmath")] extern crate cgmath; #[cfg(feature = "serialization")] extern crate serde; #[cfg(feature = "serialization")] #[macro_use] extern crate serde_derive; #[cfg(feature = "impl-cgmath")] use cgmath::{InnerSpace, Quaternion, Vector2, Vector3, Vector4}; #[cfg(feature = "std")] use std::cmp::Ordering; #[cfg(feature = "std")] use std::f32::consts; #[cfg(feature = "std")] use std::ops::{Add, Div, Mul, Sub}; #[cfg(not(feature = "std"))] use alloc::vec::Vec; #[cfg(not(feature = "std"))] use core::cmp::Ordering; #[cfg(not(feature = "std"))] use core::f32::consts; #[cfg(not(feature = "std"))] use core::ops::{Add, Div, Mul, Sub}; /// A spline control point. /// /// This type associates a value at a given interpolation parameter value. It also contains an /// interpolation hint used to determine how to interpolate values on the segment defined by this /// key and the next one – if existing. #[derive(Copy, Clone, Debug)] #[cfg_attr(feature = "serialization", derive(Deserialize, Serialize))] #[cfg_attr(feature = "serialization", serde(rename_all = "snake_case"))] pub struct Key<T> { /// Interpolation parameter at which the [`Key`] should be reached. pub t: f32, /// Held value. pub value: T, /// Interpolation mode. pub interpolation: Interpolation } impl<T> Key<T> { /// Create a new key. pub fn new(t: f32, value: T, interpolation: Interpolation) -> Self { Key { t: t, value: value, interpolation: interpolation } } } /// Interpolation mode. #[derive(Copy, Clone, Debug)] #[cfg_attr(feature = "serialization", derive(Deserialize, Serialize))] #[cfg_attr(feature = "serialization", serde(rename_all = "snake_case"))] pub enum Interpolation { /// Hold a [`Key`] until the time passes the normalized step threshold, in which case the next /// key is used. /// /// *Note: if you set the threshold to `0.5`, the first key will be used until the time is half /// between the two keys; the second key will be in used afterwards. If you set it to `1.0`, the /// first key will be kept until the next key. Set it to `0.` and the first key will never be /// used.* Step(f32), /// Linear interpolation between a key and the next one. Linear, /// Cosine interpolation between a key and the next one. Cosine, /// Catmull-Rom interpolation. CatmullRom } impl Default for Interpolation { /// `Interpolation::Linear` is the default. fn default() -> Self { Interpolation::Linear } } /// Spline curve used to provide interpolation between control points (keys). #[derive(Debug, Clone)] #[cfg_attr(feature = "serialization", derive(Deserialize, Serialize))] pub struct Spline<T>(Vec<Key<T>>); impl<T> Spline<T> { /// Create a new spline out of keys. The keys don’t have to be sorted even though it’s recommended /// to provide ascending sorted ones (for performance purposes). pub fn from_vec(mut keys: Vec<Key<T>>) -> Self { keys.sort_by(|k0, k1| k0.t.partial_cmp(&k1.t).unwrap_or(Ordering::Less)); Spline(keys) } /// Create a new spline by consuming an `Iterater<Item = Key<T>>`. They keys don’t have to be /// sorted. /// /// # Note on iterators /// /// It’s valid to use any iterator that implements `Iterator<Item = Key<T>>`. However, you should /// use `Spline::from_vec` if you are passing a `Vec<_>`. This will remove dynamic allocations. pub fn from_iter<I>(iter: I) -> Self where I: Iterator<Item = Key<T>> { Self::from_vec(iter.collect()) } /// Retrieve the keys of a spline. pub fn keys(&self) -> &[Key<T>] { &self.0 } /// Sample a spline at a given time. /// /// The current implementation, based on immutability, cannot perform in constant time. This means /// that sampling’s processing complexity is currently *O(log n)*. It’s possible to achieve *O(1)* /// performance by using a slightly different spline type. If you are interested by this feature, /// an implementation for a dedicated type is foreseen yet not started yet. /// /// # Return /// /// `None` if you try to sample a value at a time that has no key associated with. That can also /// happen if you try to sample between two keys with a specific interpolation mode that make the /// sampling impossible. For instance, `Interpolate::CatmullRom` requires *four* keys. If you’re /// near the beginning of the spline or its end, ensure you have enough keys around to make the /// sampling. pub fn sample(&self, t: f32) -> Option<T> where T: Interpolate { let keys = &self.0; let i = search_lower_cp(keys, t)?; let cp0 = &keys[i]; match cp0.interpolation { Interpolation::Step(threshold) => { let cp1 = &keys[i+1]; let nt = normalize_time(t, cp0, cp1); Some(if nt < threshold { cp0.value } else { cp1.value }) }, Interpolation::Linear => { let cp1 = &keys[i+1]; let nt = normalize_time(t, cp0, cp1); Some(Interpolate::lerp(cp0.value, cp1.value, nt)) }, Interpolation::Cosine => { let cp1 = &keys[i+1]; let nt = normalize_time(t, cp0, cp1); let cos_nt = { #[cfg(feature = "std")] { (1. - f32::cos(nt * consts::PI)) * 0.5 } #[cfg(not(feature = "std"))] { use core::intrinsics::cosf32; unsafe { (1. - cosf32(nt * consts::PI)) * 0.5 } } }; Some(Interpolate::lerp(cp0.value, cp1.value, cos_nt)) }, Interpolation::CatmullRom => { // We need at least four points for Catmull Rom; ensure we have them, otherwise, return // None. if i == 0 || i >= keys.len() - 2 { None } else { let cp1 = &keys[i+1]; let cpm0 = &keys[i-1]; let cpm1 = &keys[i+2]; let nt = normalize_time(t, cp0, cp1); Some(Interpolate::cubic_hermite((cpm0.value, cpm0.t), (cp0.value, cp0.t), (cp1.value, cp1.t), (cpm1.value, cpm1.t), nt)) } } } } /// Sample a spline at a given time with clamping. /// /// # Return /// /// If you sample before the first key or after the last one, return the first key or the last /// one, respectively. Otherwise, behave the same way as `Spline::sample`. /// /// # Panic /// /// This function panics if you have no key. pub fn clamped_sample(&self, t: f32) -> T where T: Interpolate { let first = self.0.first().unwrap(); let last = self.0.last().unwrap(); if t <= first.t { return first.value; } else if t >= last.t { return last.value; } self.sample(t).unwrap() } } /// Iterator over spline keys. /// /// This iterator type assures you to iterate over sorted keys. pub struct Iter<'a, T> where T: 'a { anim_param: &'a Spline<T>, i: usize } impl<'a, T> Iterator for Iter<'a, T> { type Item = &'a Key<T>; fn next(&mut self) -> Option<Self::Item> { let r = self.anim_param.0.get(self.i); if let Some(_) = r { self.i += 1; } r } } impl<'a, T> IntoIterator for &'a Spline<T> { type Item = &'a Key<T>; type IntoIter = Iter<'a, T>; fn into_iter(self) -> Self::IntoIter { Iter { anim_param: self, i: 0 } } } /// Keys that can be interpolated in between. Implementing this trait is required to perform /// sampling on splines. pub trait Interpolate: Copy { /// Linear interpolation. fn lerp(a: Self, b: Self, t: f32) -> Self; /// Cubic hermite interpolation. /// /// Default to `Self::lerp`. fn cubic_hermite(_: (Self, f32), a: (Self, f32), b: (Self, f32), _: (Self, f32), t: f32) -> Self { Self::lerp(a.0, b.0, t) } } impl Interpolate for f32 { fn lerp(a: Self, b: Self, t: f32) -> Self { a * (1. - t) + b * t } fn cubic_hermite(x: (Self, f32), a: (Self, f32), b: (Self, f32), y: (Self, f32), t: f32) -> Self { cubic_hermite(x, a, b, y, t) } } #[cfg(feature = "impl-cgmath")] impl Interpolate for Vector2<f32> { fn lerp(a: Self, b: Self, t: f32) -> Self { a.lerp(b, t) } fn cubic_hermite(x: (Self, f32), a: (Self, f32), b: (Self, f32), y: (Self, f32), t: f32) -> Self { cubic_hermite(x, a, b, y, t) } } #[cfg(feature = "impl-cgmath")] impl Interpolate for Vector3<f32> { fn lerp(a: Self, b: Self, t: f32) -> Self { a.lerp(b, t) } fn cubic_hermite(x: (Self, f32), a: (Self, f32), b: (Self, f32), y: (Self, f32), t: f32) -> Self { cubic_hermite(x, a, b, y, t) } } #[cfg(feature = "impl-cgmath")] impl Interpolate for Vector4<f32> { fn lerp(a: Self, b: Self, t: f32) -> Self { a.lerp(b, t) } fn cubic_hermite(x: (Self, f32), a: (Self, f32), b: (Self, f32), y: (Self, f32), t: f32) -> Self { cubic_hermite(x, a, b, y, t) } } #[cfg(feature = "impl-cgmath")] impl Interpolate for Quaternion<f32> { fn lerp(a: Self, b: Self, t: f32) -> Self { a.nlerp(b, t) } } // Default implementation of Interpolate::cubic_hermit. pub(crate) fn cubic_hermite<T>(x: (T, f32), a: (T, f32), b: (T, f32), y: (T, f32), t: f32) -> T where T: Copy + Add<Output = T> + Sub<Output = T> + Mul<f32, Output = T> + Div<f32, Output = T> { // time stuff let t2 = t * t; let t3 = t2 * t; let two_t3 = 2. * t3; let three_t2 = 3. * t2; // tangents let m0 = (b.0 - x.0) / (b.1 - x.1); let m1 = (y.0 - a.0) / (y.1 - a.1); a.0 * (two_t3 - three_t2 + 1.) + m0 * (t3 - 2. * t2 + t) + b.0 * (-two_t3 + three_t2) + m1 * (t3 - t2) } // Normalize a time ([0;1]) given two control points. #[inline(always)] pub(crate) fn normalize_time<T>(t: f32, cp: &Key<T>, cp1: &Key<T>) -> f32 { assert!(cp1.t != cp.t, "overlapping keys"); (t - cp.t) / (cp1.t - cp.t) } // Find the lower control point corresponding to a given time. fn search_lower_cp<T>(cps: &[Key<T>], t: f32) -> Option<usize> { let mut i = 0; let len = cps.len(); if len < 2 { return None; } loop { let cp = &cps[i]; let cp1 = &cps[i+1]; if t >= cp1.t { if i >= len - 2 { return None; } i += 1; } else if t < cp.t { if i == 0 { return None; } i -= 1; } else { break; // found } } Some(i) }