Merge pull request #31 from phaazon/fix/bezier-interpolation

Fix/bezier interpolation

.github/workflows/ci.yaml

@@ -0,0 +1,46 @@
+name: CI
+on: [push]
+
+jobs:
+  build-linux:
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v1
+      - name: Build
+        run: |
+          cargo build --verbose --all-features
+      - name: Test
+        run: |
+          cargo test --verbose --all-features
+
+
+  build-windows:
+    runs-on: windows-latest
+    steps:
+      - uses: actions/checkout@v1
+      - name: Build
+        run: |
+          cargo build --verbose --all-features
+      - name: Test
+        run: |
+          cargo test --verbose --all-features
+
+  build-macosx:
+    runs-on: macosx-latest
+    steps:
+      - uses: actions/checkout@v1
+      - name: Build
+        run: |
+          cargo build --verbose --all-features
+      - name: Test
+        run: |
+          cargo test --verbose --all-features
+
+  check-readme:
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v1
+      - name: Install cargo-sync-readme
+        run: cargo install --force cargo-sync-readme
+      - name: Check
+        run: cargo sync-readme -c

.travis.yml

@@ -1,23 +0,0 @@
-language: rust
-
-rust:
-  - stable
-  - beta
-  - nightly
-
-os:
-  - linux
-  - osx
-
-script:
-  - rustc --version
-  - cargo --version
-  - echo "Testing default crate configuration"
-  - cargo build --verbose
-  - cargo test --verbose
-  - cd examples && cargo check --verbose
-  - echo "Testing feature serialization"
-  - cargo build --verbose --features serialization
-  - cargo test --verbose --features serialization
-  - echo "Building without std"
-  - cargo build --verbose --no-default-features

CHANGELOG.md

@@ -1,43 +1,86 @@
-## 0.2.3
+# 2.0.1
+
+> Tue Sep 24th 2019
+
+- Fix the cubic Bézier curve interpolation. The “output” tangent is now taken by mirroring the
+  next key’s tangent around its control point.
+
+# 2.0.0
+
+> Mon Sep 23rd 2019
+
+## Major changes
+
+- Add support for [Bézier curves](https://en.wikipedia.org/wiki/B%C3%A9zier_curve).
+- Because of Bézier curves, the `Interpolation` type now has one more type variable to know how we
+  should interpolate with Bézier.
+
+## Minor changes
+
+- Add `Spline::get`, `Spline::get_mut` and `Spline::replace`.
+
+# 1.0
+
+> Sun Sep 22nd 2019
+
+## Major changes
+
+- Make `Spline::clamped_sample` failible via `Option` instead of panicking.
+- Add support for polymorphic sampling type.
+
+## Minor changes
+
+- Add the `std` feature (and hence support for `no_std`).
+- Add `impl-nalgebra` feature.
+- Add `impl-cgmath` feature.
+- Add support for adding keys to splines.
+- Add support for removing keys from splines.
+
+## Patch changes
+
+- Migrate to Rust 2018.
+- Documentation typo fixes.
+
+# 0.2.3
 
 > Sat 13th October 2018
 
-  - Add the `"impl-nalgebra"` feature gate. It gives access to some implementors for the `nalgebra`
-    crate.
-  - Enhance the documentation.
+- Add the `"impl-nalgebra"` feature gate. It gives access to some implementors for the `nalgebra`
+  crate.
+- Enhance the documentation.
 
-## 0.2.2
+# 0.2.2
 
 > Sun 30th September 2018
 
-  - Bump version numbers (`splines-0.2`) in examples.
-  - Fix several typos in the documentation.
+- Bump version numbers (`splines-0.2`) in examples.
+- Fix several typos in the documentation.
 
-## 0.2.1
+# 0.2.1
 
 > Thu 20th September 2018
 
-  - Enhance the features documentation.
+- Enhance the features documentation.
 
 # 0.2
 
 > Thu 6th September 2018
 
-  - Add the `"std"` feature gate, that can be used to compile with the standard library.
-  - Add the `"impl-cgmath"` feature gate in order to make optional, if wanted, the `cgmath`
-    dependency.
-  - Enhance the documentation.
+- Add the `"std"` feature gate, that can be used to compile with the standard library.
+- Add the `"impl-cgmath"` feature gate in order to make optional, if wanted, the `cgmath`
+  dependency.
+- Enhance the documentation.
 
-## 0.1.1
+# 0.1.1
 
 > Wed 8th August 2018
 
-  - Add a feature gate, `"serialization"`, that can be used to automatically derive `Serialize` and
-    `Deserialize` from the [serde](https://crates.io/crates/serde) crate.
-  - Enhance the documentation.
+- Add a feature gate, `"serialization"`, that can be used to automatically derive `Serialize` and
+  `Deserialize` from the [serde](https://crates.io/crates/serde) crate.
+- Enhance the documentation.
 
 # 0.1
 
 > Sunday 5th August 2018
 
-  - Initial revision.
+- Initial revision.

Cargo.toml

@@ -1,6 +1,6 @@
 [package]
 name = "splines"
-version = "1.0.0-rc.2"
+version = "2.0.1"
 license = "BSD-3-Clause"
 authors = ["Dimitri Sabadie <dimitri.sabadie@gmail.com>"]
 description = "Spline interpolation made easy"
@@ -33,3 +33,6 @@ nalgebra = { version = ">=0.14, <0.19", optional = true }
 num-traits = { version = "0.2", optional = true }
 serde =  { version = "1", optional = true }
 serde_derive = { version = "1", optional = true }
+
+[package.metadata.docs.rs]
+all-features = true

README.md

@@ -13,9 +13,9 @@ 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.
+  - [`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
@@ -40,17 +40,13 @@ 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
+usually done with the [`Spline::sample`] method. This method expects the sampling 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.
+If you try to sample in out-of-bounds sampling 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.), Some(10.));
 assert_eq!(spline.sample(1.1), None);
@@ -61,14 +57,17 @@ important for simulations / animations. Feel free to use the `Spline::clamped_in
 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), Some(0.)); // clamped to the first key
 assert_eq!(spline.clamped_sample(1.1), Some(10.)); // clamped to the last key
 ```
 
+# Polymorphic sampling types
+
+[`Spline`] curves are parametered both by the carried value (being interpolated) but also the
+sampling type. It’s very typical to use `f32` or `f64` but really, you can in theory use any
+kind of type; that type must, however, implement a contract defined by a set of traits to
+implement. See [the documentation of this module](crate::interpolate) for further details.
+
 # Features and customization
 
 This crate was written with features baked in and hidden behind feature-gates. The idea is that
@@ -84,20 +83,22 @@ 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 useful implementations of `Interpolate` for some cgmath types.
-+ Enable with the `"impl-cgmath"` feature.
-- **[nalgebra](https://crates.io/crates/nalgebra) implementors.**
-+ Adds some useful implementations of `Interpolate` for some nalgebra types.
-+ Enable with the `"impl-nalgebra"` 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.
+  - **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 useful implementations of `Interpolate` for some cgmath types.
+    - Enable with the `"impl-cgmath"` feature.
+  - **[nalgebra](https://crates.io/crates/nalgebra) implementors.**
+    - Adds some useful implementations of `Interpolate` for some nalgebra types.
+    - Enable with the `"impl-nalgebra"` 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.
+
+[`Interpolation`]: crate::interpolation::Interpolation
 
 <!-- cargo-sync-readme end -->

src/cgmath.rs

@@ -2,7 +2,9 @@ use cgmath::{
   BaseFloat, BaseNum, InnerSpace, Quaternion, Vector1, Vector2, Vector3, Vector4, VectorSpace
 };
 
-use crate::interpolate::{Additive, Interpolate, Linear, One, cubic_hermite_def};
+use crate::interpolate::{
+  Additive, Interpolate, Linear, One, cubic_bezier_def, cubic_hermite_def, quadratic_bezier_def
+};
 
 macro_rules! impl_interpolate_vec {
   ($($t:tt)*) => {
@@ -29,6 +31,16 @@ macro_rules! impl_interpolate_vec {
       fn cubic_hermite(x: (Self, T), a: (Self, T), b: (Self, T), y: (Self, T), t: T) -> Self {
         cubic_hermite_def(x, a, b, y, t)
       }
+
+      #[inline(always)]
+      fn quadratic_bezier(a: Self, u: Self, b: Self, t: T) -> Self {
+        quadratic_bezier_def(a, u, b, t)
+      }
+
+      #[inline(always)]
+      fn cubic_bezier(a: Self, u: Self, v: Self, b: Self, t: T) -> Self {
+        cubic_bezier_def(a, u, v, b, t)
+      }
     }
   }
 }
@@ -61,4 +73,14 @@ where Self: InnerSpace<Scalar = T>, T: Additive + BaseFloat + One {
   fn cubic_hermite(x: (Self, T), a: (Self, T), b: (Self, T), y: (Self, T), t: T) -> Self {
     cubic_hermite_def(x, a, b, y, t)
   }
+
+  #[inline(always)]
+  fn quadratic_bezier(a: Self, u: Self, b: Self, t: T) -> Self {
+    quadratic_bezier_def(a, u, b, t)
+  }
+
+  #[inline(always)]
+  fn cubic_bezier(a: Self, u: Self, v: Self, b: Self, t: T) -> Self {
+    cubic_bezier_def(a, u, v, b, t)
+  }
 }

src/interpolate.rs

@@ -57,6 +57,12 @@ pub trait Interpolate<T>: Sized + Copy {
   fn cubic_hermite(_: (Self, T), a: (Self, T), b: (Self, T), _: (Self, T), t: T) -> Self {
     Self::lerp(a.0, b.0, t)
   }
+
+  /// Quadratic Bézier interpolation.
+  fn quadratic_bezier(a: Self, u: Self, b: Self, t: T) -> Self;
+
+  /// Cubic Bézier interpolation.
+  fn cubic_bezier(a: Self, u: Self, v: Self, b: Self, t: T) -> Self;
 }
 
 /// Set of types that support additions and subtraction.
@@ -212,6 +218,34 @@ where V: Linear<T>,
   a.0.outer_mul(two_t3 - three_t2 + one_t) + m0.outer_mul(t3 - t2 * two_t + t) + b.0.outer_mul(three_t2 - two_t3) + m1.outer_mul(t3 - t2)
 }
 
+/// Default implementation of [`Interpolate::quadratic_bezier`].
+///
+/// `V` is the value being interpolated. `T` is the sampling value (also sometimes called time).
+pub fn quadratic_bezier_def<V, T>(a: V, u: V, b: V, t: T) -> V
+where V: Linear<T>,
+      T: Additive + Mul<T, Output = T> + One {
+  let one_t = T::one() - t;
+  let one_t_2 = one_t * one_t;
+  u + (a - u).outer_mul(one_t_2) + (b - u).outer_mul(t * t)
+}
+
+/// Default implementation of [`Interpolate::cubic_bezier`].
+///
+/// `V` is the value being interpolated. `T` is the sampling value (also sometimes called time).
+pub fn cubic_bezier_def<V, T>(a: V, u: V, v: V, b: V, t: T) -> V
+where V: Linear<T>,
+      T: Additive + Mul<T, Output = T> + One {
+  let one_t = T::one() - t;
+  let one_t_2 = one_t * one_t;
+  let one_t_3 = one_t_2 * one_t;
+  let three = T::one() + T::one() + T::one();
+
+  // mirror the “output” tangent based on the next key “input” tangent
+  let v_ = b + b - v;
+
+  a.outer_mul(one_t_3) + u.outer_mul(three * one_t_2 * t) + v_.outer_mul(three * one_t * t * t) + b.outer_mul(t * t * t)
+}
+
 macro_rules! impl_interpolate_simple {
   ($t:ty) => {
     impl Interpolate<$t> for $t {
@@ -222,6 +256,14 @@ macro_rules! impl_interpolate_simple {
       fn cubic_hermite(x: (Self, $t), a: (Self, $t), b: (Self, $t), y: (Self, $t), t: $t) -> Self {
         cubic_hermite_def(x, a, b, y, t)
       }
+
+      fn quadratic_bezier(a: Self, u: Self, b: Self, t: $t) -> Self {
+        quadratic_bezier_def(a, u, b, t)
+      }
+
+      fn cubic_bezier(a: Self, u: Self, v: Self, b: Self, t: $t) -> Self {
+        cubic_bezier_def(a, u, v, b, t)
+      }
     }
   }
 }
@@ -239,6 +281,14 @@ macro_rules! impl_interpolate_via {
       fn cubic_hermite((x, xt): (Self, $t), (a, at): (Self, $t), (b, bt): (Self, $t), (y, yt): (Self, $t), t: $t) -> Self {
         cubic_hermite_def((x, xt as $v), (a, at as $v), (b, bt as $v), (y, yt as $v), t as $v)
       }
+
+      fn quadratic_bezier(a: Self, u: Self, b: Self, t: $t) -> Self {
+        quadratic_bezier_def(a, u, b, t as $v)
+      }
+
+      fn cubic_bezier(a: Self, u: Self, v: Self, b: Self, t: $t) -> Self {
+        cubic_bezier_def(a, u, v, b, t as $v)
+      }
     }
   }
 }

src/interpolation.rs

@@ -5,11 +5,11 @@
 /// Available kind of interpolations.
 ///
 /// Feel free to visit each variant for more documentation.
-#[derive(Copy, Clone, Debug)]
+#[derive(Copy, Clone, Debug, Eq, PartialEq)]
 #[cfg_attr(feature = "serialization", derive(Deserialize, Serialize))]
 #[cfg_attr(feature = "serialization", serde(rename_all = "snake_case"))]
-pub enum Interpolation<T> {
-  /// Hold a [`Key<T, _>`] until the sampling value passes the normalized step threshold, in which
+pub enum Interpolation<T, V> {
+  /// Hold a [`Key`] until the sampling value 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 half the time
@@ -17,20 +17,36 @@ pub enum Interpolation<T> {
   /// > first key will be kept until the next key. Set it to `0.` and the first key will never be
   /// > used.
   ///
-  /// [`Key<T, _>`]: crate::key::Key
+  /// [`Key`]: crate::key::Key
   Step(T),
   /// Linear interpolation between a key and the next one.
   Linear,
   /// Cosine interpolation between a key and the next one.
   Cosine,
   /// Catmull-Rom interpolation, performing a cubic Hermite interpolation using four keys.
-  CatmullRom
+  CatmullRom,
+  /// Bézier interpolation.
+  ///
+  /// A control point that uses such an interpolation is associated with an extra point. The segmant
+  /// connecting both is called the _tangent_ of this point. The part of the spline defined between
+  /// this control point and the next one will be interpolated across with Bézier interpolation. Two
+  /// cases are possible:
+  ///
+  /// - The next control point also has a Bézier interpolation mode. In this case, its tangent is
+  ///   used for the interpolation process. This is called _cubic Bézier interpolation_ and it
+  ///   kicks ass.
+  /// - The next control point doesn’t have a Bézier interpolation mode set. In this case, the
+  ///   tangent used for the next control point is defined as the segment connecting that control
+  ///   point and the current control point’s associated point. This is called _quadratic Bézer
+  ///   interpolation_ and it kicks ass too, but a bit less than cubic.
+  Bezier(V),
+  #[doc(hidden)]
+  __NonExhaustive
 }
 
-impl<T> Default for Interpolation<T> {
+impl<T, V> Default for Interpolation<T, V> {
   /// [`Interpolation::Linear`] is the default.
   fn default() -> Self {
     Interpolation::Linear
   }
 }
-

src/key.rs

@@ -17,7 +17,7 @@ use crate::interpolation::Interpolation;
 /// key and the next one – if existing. Have a look at [`Interpolation`] for further details.
 ///
 /// [`Interpolation`]: crate::interpolation::Interpolation
-#[derive(Copy, Clone, Debug)]
+#[derive(Copy, Clone, Debug, Eq, PartialEq)]
 #[cfg_attr(feature = "serialization", derive(Deserialize, Serialize))]
 #[cfg_attr(feature = "serialization", serde(rename_all = "snake_case"))]
 pub struct Key<T, V> {
@@ -26,12 +26,12 @@ pub struct Key<T, V> {
   /// Carried value.
   pub value: V,
   /// Interpolation mode.
-  pub interpolation: Interpolation<T>
+  pub interpolation: Interpolation<T, V>
 }
 
 impl<T, V> Key<T, V> {
   /// Create a new key.
-  pub fn new(t: T, value: V, interpolation: Interpolation<T>) -> Self {
+  pub fn new(t: T, value: V, interpolation: Interpolation<T, V>) -> Self {
     Key { t, value, interpolation }
   }
 }

src/lib.rs

@@ -85,20 +85,22 @@
 //! 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
+//!     - This feature implements both the `Serialize` and `Deserialize` traits from `serde` for all
 //!       types exported by this crate.
-//!     + Enable with the `"serialization"` feature.
+//!     - Enable with the `"serialization"` feature.
 //!   - **[cgmath](https://crates.io/crates/cgmath) implementors.**
-//!     + Adds some useful implementations of `Interpolate` for some cgmath types.
-//!     + Enable with the `"impl-cgmath"` feature.
+//!     - Adds some useful implementations of `Interpolate` for some cgmath types.
+//!     - Enable with the `"impl-cgmath"` feature.
 //!   - **[nalgebra](https://crates.io/crates/nalgebra) implementors.**
-//!     + Adds some useful implementations of `Interpolate` for some nalgebra types.
-//!     + Enable with the `"impl-nalgebra"` feature.
+//!     - Adds some useful implementations of `Interpolate` for some nalgebra types.
+//!     - Enable with the `"impl-nalgebra"` 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.
+//!     - 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.
+//!
+//! [`Interpolation`]: crate::interpolation::Interpolation
 
 #![cfg_attr(not(feature = "std"), no_std)]
 #![cfg_attr(not(feature = "std"), feature(alloc))]

src/nalgebra.rs

@@ -1,18 +1,16 @@
 use alga::general::{ClosedAdd, ClosedDiv, ClosedMul, ClosedSub};
-use nalgebra::{
-  DefaultAllocator, DimName, Point, Scalar, Vector, Vector1, Vector2, Vector3, Vector4, Vector5,
-  Vector6
-};
-use nalgebra::allocator::Allocator;
+use nalgebra::{Scalar, Vector, Vector1, Vector2, Vector3, Vector4, Vector5, Vector6};
 use num_traits as nt;
 use std::ops::Mul;
 
-use crate::interpolate::{Interpolate, Linear, Additive, One, cubic_hermite_def};
+use crate::interpolate::{
+  Interpolate, Linear, Additive, One, cubic_bezier_def, cubic_hermite_def, quadratic_bezier_def
+};
 
 macro_rules! impl_interpolate_vector {
   ($($t:tt)*) => {
     // implement Linear
-    impl<T> Linear<T> for $($t)*<T> where T: Scalar + ClosedMul + ClosedDiv {
+    impl<T> Linear<T> for $($t)*<T> where T: Scalar + ClosedAdd + ClosedSub + ClosedMul + ClosedDiv {
       #[inline(always)]
       fn outer_mul(self, t: T) -> Self {
         self * t
@@ -44,6 +42,16 @@ macro_rules! impl_interpolate_vector {
       fn cubic_hermite(x: (Self, T), a: (Self, T), b: (Self, T), y: (Self, T), t: T) -> Self {
         cubic_hermite_def(x, a, b, y, t)
       }
+
+      #[inline(always)]
+      fn quadratic_bezier(a: Self, u: Self, b: Self, t: T) -> Self {
+        quadratic_bezier_def(a, u, b, t)
+      }
+
+      #[inline(always)]
+      fn cubic_bezier(a: Self, u: Self, v: Self, b: Self, t: T) -> Self {
+        cubic_bezier_def(a, u, v, b, t)
+      }
     }
   }
 }
@@ -54,19 +62,3 @@ impl_interpolate_vector!(Vector3);
 impl_interpolate_vector!(Vector4);
 impl_interpolate_vector!(Vector5);
 impl_interpolate_vector!(Vector6);
-
-impl<T, D> Linear<T> for Point<T, D>
-where D: DimName,
-      DefaultAllocator: Allocator<T, D>,
-      <DefaultAllocator as Allocator<T, D>>::Buffer: Copy,
-      T: Scalar + ClosedDiv + ClosedMul {
-  #[inline(always)]
-  fn outer_mul(self, t: T) -> Self {
-    self * t
-  }
-
-  #[inline(always)]
-  fn outer_div(self, t: T) -> Self {
-    self / t
-  }
-}

src/spline.rs

@@ -28,12 +28,17 @@ use crate::key::Key;
 pub struct Spline<T, V>(pub(crate) Vec<Key<T, V>>);
 
 impl<T, V> Spline<T, V> {
+  /// Internal sort to ensure invariant of sorting keys is valid.
+  fn internal_sort(&mut self) where T: PartialOrd {
+    self.0.sort_by(|k0, k1| k0.t.partial_cmp(&k1.t).unwrap_or(Ordering::Less));
+  }
+
   /// 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, V>>) -> Self where T: PartialOrd {
-    keys.sort_by(|k0, k1| k0.t.partial_cmp(&k1.t).unwrap_or(Ordering::Less));
-
-    Spline(keys)
+  pub fn from_vec(keys: Vec<Key<T, V>>) -> Self where T: PartialOrd {
+    let mut spline = Spline(keys);
+    spline.internal_sort();
+    spline
   }
 
   /// Create a new spline by consuming an `Iterater<Item = Key<T>>`. They keys don’t have to be
@@ -42,7 +47,7 @@ impl<T, V> Spline<T, V> {
   /// # 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.
+  /// use [`Spline::from_vec`] if you are passing a [`Vec`].
   pub fn from_iter<I>(iter: I) -> Self where I: Iterator<Item = Key<T, V>>, T: PartialOrd {
     Self::from_vec(iter.collect())
   }
@@ -52,6 +57,18 @@ impl<T, V> Spline<T, V> {
     &self.0
   }
 
+  /// Number of keys.
+  #[inline(always)]
+  pub fn len(&self) -> usize {
+    self.0.len()
+  }
+
+  /// Check whether the spline has no key.
+  #[inline(always)]
+  pub fn is_empty(&self) -> bool {
+    self.0.is_empty()
+  }
+
   /// Sample a spline at a given time.
   ///
   /// The current implementation, based on immutability, cannot perform in constant time. This means
@@ -76,13 +93,13 @@ impl<T, V> Spline<T, V> {
 
     match cp0.interpolation {
       Interpolation::Step(threshold) => {
-        let cp1 = &keys[i+1];
+        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 cp1 = &keys[i + 1];
         let nt = normalize_time(t, cp0, cp1);
 
         Some(Interpolate::lerp(cp0.value, cp1.value, nt))
@@ -90,7 +107,7 @@ impl<T, V> Spline<T, V> {
 
       Interpolation::Cosine => {
         let two_t = T::one() + T::one();
-        let cp1 = &keys[i+1];
+        let cp1 = &keys[i + 1];
         let nt = normalize_time(t, cp0, cp1);
         let cos_nt = (T::one() - (nt * T::pi()).cos()) / two_t;
 
@@ -103,14 +120,28 @@ impl<T, V> Spline<T, V> {
         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 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))
         }
       }
+
+      Interpolation::Bezier(u) => {
+        // We need to check the next control point to see whether we want quadratic or cubic Bezier.
+        let cp1 = &keys[i + 1];
+        let nt = normalize_time(t, cp0, cp1);
+
+        if let Interpolation::Bezier(v) = cp1.interpolation {
+          Some(Interpolate::cubic_bezier(cp0.value, u, v, cp1.value, nt))
+        } else {
+          Some(Interpolate::quadratic_bezier(cp0.value, u, cp1.value, nt))
+        }
+      }
+
+      Interpolation::__NonExhaustive => unreachable!(),
     }
   }
 
@@ -146,6 +177,69 @@ impl<T, V> Spline<T, V> {
       }
     })
   }
+
+  /// Add a key into the spline.
+  pub fn add(&mut self, key: Key<T, V>) where T: PartialOrd {
+    self.0.push(key);
+    self.internal_sort();
+  }
+
+  /// Remove a key from the spline.
+  pub fn remove(&mut self, index: usize) -> Option<Key<T, V>> {
+    if index >= self.0.len() {
+      None
+    } else {
+      Some(self.0.remove(index))
+    }
+  }
+
+  /// Update a key and return the key already present.
+  ///
+  /// The key is updated — if present — with the provided function.
+  ///
+  /// # Notes
+  ///
+  /// That function makes sense only if you want to change the interpolator (i.e. [`Key::t`]) of
+  /// your key. If you just want to change the interpolation mode or the carried value, consider
+  /// using the [`Spline::get_mut`] method instead as it will be way faster.
+  pub fn replace<F>(
+    &mut self,
+    index: usize,
+    f: F
+  ) -> Option<Key<T, V>>
+  where
+    F: FnOnce(&Key<T, V>) -> Key<T, V>,
+    T: PartialOrd
+  {
+    let key = self.remove(index)?;
+    self.add(f(&key));
+    Some(key)
+  }
+
+  /// Get a key at a given index.
+  pub fn get(&self, index: usize) -> Option<&Key<T, V>> {
+    self.0.get(index)
+  }
+
+  /// Mutably get a key at a given index.
+  pub fn get_mut(&mut self, index: usize) -> Option<KeyMut<T, V>> {
+    self.0.get_mut(index).map(|key| KeyMut {
+      value: &mut key.value,
+      interpolation: &mut key.interpolation
+    })
+  }
+}
+
+/// A mutable [`Key`].
+///
+/// Mutable keys allow to edit the carried values and the interpolation mode but not the actual
+/// interpolator value as it would invalidate the internal structure of the [`Spline`]. If you
+/// want to achieve this, you’re advised to use [`Spline::replace`].
+pub struct KeyMut<'a, T, V> {
+  /// Carried value.
+  pub value: &'a mut V,
+  /// Interpolation mode to use for that key.
+  pub interpolation: &'a mut Interpolation<T, V>,
 }
 
 // Normalize a time ([0;1]) given two control points.

tests/mod.rs

@@ -172,3 +172,51 @@ fn nalgebra_vector_interpolation() {
   assert_eq!(Interpolate::lerp(start, end, 1.0), end);
   assert_eq!(Interpolate::lerp(start, end, 0.5), mid);
 }
+
+#[test]
+fn add_key_empty() {
+  let mut spline: Spline<f32, f32> = Spline::from_vec(vec![]);
+  spline.add(Key::new(0., 0., Interpolation::Linear));
+
+  assert_eq!(spline.keys(), &[Key::new(0., 0., Interpolation::Linear)]);
+}
+
+#[test]
+fn add_key() {
+  let start = Key::new(0., 0., Interpolation::Step(0.5));
+  let k1 = Key::new(1., 5., Interpolation::Linear);
+  let k2 = Key::new(2., 0., Interpolation::Step(0.1));
+  let k3 = Key::new(3., 1., Interpolation::Linear);
+  let k4 = Key::new(10., 2., Interpolation::Linear);
+  let end = Key::new(11., 4., Interpolation::default());
+  let new = Key::new(2.4, 40., Interpolation::Linear);
+  let mut spline = Spline::from_vec(vec![start, k1, k2.clone(), k3, k4, end]);
+
+  assert_eq!(spline.keys(), &[start, k1, k2, k3, k4, end]);
+  spline.add(new);
+  assert_eq!(spline.keys(), &[start, k1, k2, new, k3, k4, end]);
+}
+
+#[test]
+fn remove_element_empty() {
+  let mut spline: Spline<f32, f32> = Spline::from_vec(vec![]);
+  let removed = spline.remove(0);
+
+  assert_eq!(removed, None);
+  assert!(spline.is_empty());
+}
+
+#[test]
+fn remove_element() {
+  let start = Key::new(0., 0., Interpolation::Step(0.5));
+  let k1 = Key::new(1., 5., Interpolation::Linear);
+  let k2 = Key::new(2., 0., Interpolation::Step(0.1));
+  let k3 = Key::new(3., 1., Interpolation::Linear);
+  let k4 = Key::new(10., 2., Interpolation::Linear);
+  let end = Key::new(11., 4., Interpolation::default());
+  let mut spline = Spline::from_vec(vec![start, k1, k2.clone(), k3, k4, end]);
+  let removed = spline.remove(2);
+
+  assert_eq!(removed, Some(k2));
+  assert_eq!(spline.len(), 5);
+}