diff --git a/src/interpolate.rs b/src/interpolate.rs
index 2ef09ba..6234258 100644
--- a/src/interpolate.rs
+++ b/src/interpolate.rs
@@ -42,277 +42,126 @@ use core::ops::{Add, Mul, Sub};
 use std::f32;
 #[cfg(feature = "std")]
 use std::f64;
-#[cfg(feature = "std")]
-use std::ops::{Add, Mul, Sub};
 
-/// Keys that can be interpolated in between. Implementing this trait is required to perform
-/// sampling on splines.
+/// Types that can be used as interpolator in splines.
 ///
-/// `T` is the variable used to sample with. Typical implementations use [`f32`] or [`f64`], but
-/// you’re free to use the ones you like. Feel free to have a look at [`Spline::sample`] for
-/// instance to know which trait your type must implement to be usable.
+/// An interpolator value is like the fabric on which control keys (and sampled values) live on.
+pub trait Interpolator: Sized + Copy + PartialOrd {
+  /// Normalize the interpolator.
+  fn normalize(self, start: Self, end: Self) -> Self;
+}
+
+macro_rules! impl_Interpolator {
+  ($t:ty) => {
+    impl Interpolator for $t {
+      fn normalize(self, start: Self, end: Self) -> Self {
+        (self - start) / (end - start)
+      }
+    }
+  };
+}
+
+impl_Interpolator!(f32);
+impl_Interpolator!(f64);
+
+/// Values that can be interpolated. Implementing this trait is required to perform sampling on splines.
 ///
-/// [`Spline::sample`]: crate::spline::Spline::sample
-pub trait Interpolate<T>: Sized + Copy + Linear<T> {
+/// `T` is the interpolator used to sample with. Typical implementations use [`f32`] or [`f64`], but
+/// you’re free to use the ones you like.
+pub trait Interpolate<T>: Sized + Copy {
+  /// Step interpolation.
+  fn step(t: T, threshold: T, a: Self, b: Self) -> Self;
+
   /// Linear interpolation.
-  fn lerp(a: Self, b: Self, t: T) -> Self;
+  fn lerp(t: T, a: Self, b: Self) -> Self;
+
+  /// Cosine interpolation.
+  fn cosine(t: T, a: Self, b: Self) -> Self;
 
   /// Cubic hermite interpolation.
-  ///
-  /// Default to [`lerp`].
-  ///
-  /// [`lerp`]: Interpolate::lerp
-  fn cubic_hermite(_: (Self, T), a: (Self, T), b: (Self, T), _: (Self, T), t: T) -> Self {
-    Self::lerp(a.0, b.0, t)
-  }
+  fn cubic_hermite(t: T, x: (T, Self), a: (T, Self), b: (T, Self), y: (T, Self)) -> Self;
 
   /// Quadratic Bézier interpolation.
-  fn quadratic_bezier(a: Self, u: Self, b: Self, t: T) -> Self;
+  ///
+  /// `a` is the first point; `b` is the second point and `u` is the tangent of `a` to the curve.
+  fn quadratic_bezier(t: T, a: Self, u: Self, b: Self) -> Self;
 
   /// Cubic Bézier interpolation.
-  fn cubic_bezier(a: Self, u: Self, v: Self, b: Self, t: T) -> Self;
+  ///
+  /// `a` is the first point; `b` is the second point; `u` is the output tangent of `a` to the curve and `v` is the
+  /// input tangent of `b` to the curve.
+  fn cubic_bezier(t: T, a: Self, u: Self, v: Self, b: Self) -> Self;
+
+  /// Cubic Bézier interpolation – special case for non-explicit second tangent.
+  ///
+  /// This version does the same computation as [`Interpolate::cubic_bezier`] but computes the second tangent by
+  /// inversing it (typical when the next point uses a Bézier interpolation, where input and output tangents are
+  /// mirrored for the same key).
+  fn cubic_bezier_mirrored(t: T, a: Self, u: Self, v: Self, b: Self) -> Self;
 }
 
-/// Set of types that support additions and subtraction.
-///
-/// The [`Copy`] trait is also a supertrait as it’s likely to be used everywhere.
-pub trait Additive: Copy + Add<Self, Output = Self> + Sub<Self, Output = Self> {}
-
-impl<T> Additive for T where T: Copy + Add<Self, Output = Self> + Sub<Self, Output = Self> {}
-
-/// Set of additive types that support outer multiplication and division, making them linear.
-pub trait Linear<T>: Additive {
-  /// Apply an outer multiplication law.
-  fn outer_mul(self, t: T) -> Self;
-
-  /// Apply an outer division law.
-  fn outer_div(self, t: T) -> Self;
-}
-
-macro_rules! impl_linear_simple {
-  ($t:ty) => {
-    impl Linear<$t> for $t {
-      fn outer_mul(self, t: $t) -> Self {
-        self * t
+#[macro_export]
+macro_rules! impl_Interpolate {
+  ($t:ty, $v:ty, $pi:expr) => {
+    impl $crate::interpolate::Interpolate<$t> for $v {
+      fn step(t: $t, threshold: $t, a: Self, b: Self) -> Self {
+        if t < threshold {
+          a
+        } else {
+          b
+        }
       }
 
-      /// Apply an outer division law.
-      fn outer_div(self, t: $t) -> Self {
-        self / t
-      }
-    }
-  };
-}
-
-impl_linear_simple!(f32);
-impl_linear_simple!(f64);
-
-macro_rules! impl_linear_cast {
-  ($t:ty, $q:ty) => {
-    impl Linear<$t> for $q {
-      fn outer_mul(self, t: $t) -> Self {
-        self * t as $q
+      fn cosine(t: $t, a: Self, b: Self) -> Self {
+        let cos_nt = (1. - (t * $pi).cos()) * 0.5;
+        <Self as $crate::interpolate::Interpolate<$t>>::lerp(cos_nt, a, b)
       }
 
-      /// Apply an outer division law.
-      fn outer_div(self, t: $t) -> Self {
-        self / t as $q
-      }
-    }
-  };
-}
-
-impl_linear_cast!(f32, f64);
-impl_linear_cast!(f64, f32);
-
-/// Types with a neutral element for multiplication.
-pub trait One {
-  /// The neutral element for the multiplicative monoid — typically called `1`.
-  fn one() -> Self;
-}
-
-macro_rules! impl_one_float {
-  ($t:ty) => {
-    impl One for $t {
-      #[inline(always)]
-      fn one() -> Self {
-        1.
-      }
-    }
-  };
-}
-
-impl_one_float!(f32);
-impl_one_float!(f64);
-
-/// Types with a sane definition of π and cosine.
-pub trait Trigo {
-  /// π.
-  fn pi() -> Self;
-
-  /// Cosine of the argument.
-  fn cos(self) -> Self;
-}
-
-impl Trigo for f32 {
-  #[inline(always)]
-  fn pi() -> Self {
-    f32::consts::PI
-  }
-
-  #[inline(always)]
-  fn cos(self) -> Self {
-    #[cfg(feature = "std")]
-    {
-      self.cos()
-    }
-
-    #[cfg(not(feature = "std"))]
-    {
-      unsafe { cosf32(self) }
-    }
-  }
-}
-
-impl Trigo for f64 {
-  #[inline(always)]
-  fn pi() -> Self {
-    f64::consts::PI
-  }
-
-  #[inline(always)]
-  fn cos(self) -> Self {
-    #[cfg(feature = "std")]
-    {
-      self.cos()
-    }
-
-    #[cfg(not(feature = "std"))]
-    {
-      unsafe { cosf64(self) }
-    }
-  }
-}
-
-/// Default implementation of [`Interpolate::cubic_hermite`].
-///
-/// `V` is the value being interpolated. `T` is the sampling value (also sometimes called time).
-pub fn cubic_hermite_def<V, T>(x: (V, T), a: (V, T), b: (V, T), y: (V, T), t: T) -> V
-where
-  V: Linear<T>,
-  T: Additive + Mul<T, Output = T> + One,
-{
-  // some stupid generic constants, because Rust doesn’t have polymorphic literals…
-  let one_t = T::one();
-  let two_t = one_t + one_t; // lolololol
-  let three_t = two_t + one_t; // megalol
-
-  // sampler stuff
-  let t2 = t * t;
-  let t3 = t2 * t;
-  let two_t3 = t3 * two_t;
-  let three_t2 = t2 * three_t;
-
-  // tangents
-  let m0 = (b.0 - x.0).outer_div(b.1 - x.1);
-  let m1 = (y.0 - a.0).outer_div(y.1 - a.1);
-
-  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();
-
-  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 {
-      fn lerp(a: Self, b: Self, t: $t) -> Self {
+      fn lerp(t: $t, a: Self, b: Self) -> Self {
         a * (1. - t) + b * t
       }
 
-      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 cubic_hermite(t: $t, x: ($t, Self), a: ($t, Self), b: ($t, Self), y: ($t, Self)) -> Self {
+        // sampler stuff
+        let two_t = t * 2.;
+        let three_t = t * 3.;
+        let t2 = t * t;
+        let t3 = t2 * t;
+        let two_t3 = t3 * two_t;
+        let three_t2 = t2 * three_t;
+
+        // tangents
+        let m0 = (b.1 - x.1) / (b.0 - x.0);
+        let m1 = (y.1 - a.1) / (y.0 - a.0);
+
+        a.1 * (two_t3 - three_t2 + 1.)
+          + m0 * (t3 - t2 * two_t + t)
+          + b.1 * (three_t2 - two_t3)
+          + m1 * (t3 - t2)
       }
 
-      fn quadratic_bezier(a: Self, u: Self, b: Self, t: $t) -> Self {
-        quadratic_bezier_def(a, u, b, t)
+      fn quadratic_bezier(t: $t, a: Self, u: Self, b: Self) -> Self {
+        let one_t = 1. - t;
+        let one_t2 = one_t * one_t;
+
+        u + (a - u) * one_t2 + (b - u) * t * t
       }
 
-      fn cubic_bezier(a: Self, u: Self, v: Self, b: Self, t: $t) -> Self {
-        cubic_bezier_def(a, u, v, b, t)
+      fn cubic_bezier(t: $t, a: Self, u: Self, v: Self, b: Self) -> Self {
+        let one_t = 1. - t;
+        let one_t2 = one_t * one_t;
+        let one_t3 = one_t2 * one_t;
+        let t2 = t * t;
+
+        a * one_t3 + (u * one_t2 * t + v * one_t * t2) * 3. + b * t2 * t
+      }
+
+      fn cubic_bezier_mirrored(t: $t, a: Self, u: Self, v: Self, b: Self) -> Self {
+        <Self as $crate::interpolate::Interpolate<$t>>::cubic_bezier(t, a, u, b + b - v, b)
       }
     }
   };
 }
 
-impl_interpolate_simple!(f32);
-impl_interpolate_simple!(f64);
-
-macro_rules! impl_interpolate_via {
-  ($t:ty, $v:ty) => {
-    impl Interpolate<$t> for $v {
-      fn lerp(a: Self, b: Self, t: $t) -> Self {
-        a * (1. - t as $v) + b * t as $v
-      }
-
-      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)
-      }
-    }
-  };
-}
-
-impl_interpolate_via!(f32, f64);
-impl_interpolate_via!(f64, f32);
+impl_Interpolate!(f32, f32, std::f32::consts::PI);
+impl_Interpolate!(f64, f64, std::f64::consts::PI);
diff --git a/src/spline.rs b/src/spline.rs
index ba260b7..8090008 100644
--- a/src/spline.rs
+++ b/src/spline.rs
@@ -1,5 +1,9 @@
 //! Spline curves and operations.
 
+#[cfg(feature = "std")]
+use crate::interpolate::{Interpolate, Interpolator};
+use crate::interpolation::Interpolation;
+use crate::key::Key;
 #[cfg(not(feature = "std"))]
 use alloc::vec::Vec;
 #[cfg(not(feature = "std"))]
@@ -10,12 +14,6 @@ use core::ops::{Div, Mul};
 use serde_derive::{Deserialize, Serialize};
 #[cfg(feature = "std")]
 use std::cmp::Ordering;
-#[cfg(feature = "std")]
-use std::ops::{Div, Mul};
-
-use crate::interpolate::{Additive, Interpolate, One, Trigo};
-use crate::interpolation::Interpolation;
-use crate::key::Key;
 
 /// Spline curve used to provide interpolation between control points (keys).
 ///
@@ -104,8 +102,8 @@ impl<T, V> Spline<T, V> {
   /// the sampling.
   pub fn sample_with_key(&self, t: T) -> Option<SampledWithKey<V>>
   where
-    T: Additive + One + Trigo + Mul<T, Output = T> + Div<T, Output = T> + PartialOrd,
-    V: Additive + Interpolate<T>,
+    T: Interpolator,
+    V: Interpolate<T>,
   {
     let keys = &self.0;
     let i = search_lower_cp(keys, t)?;
@@ -114,26 +112,24 @@ impl<T, V> Spline<T, V> {
     let value = match cp0.interpolation {
       Interpolation::Step(threshold) => {
         let cp1 = &keys[i + 1];
-        let nt = normalize_time(t, cp0, cp1);
-        let value = if nt < threshold { cp0.value } else { cp1.value };
+        let nt = t.normalize(cp0.t, cp1.t);
+        let value = V::step(nt, threshold, cp0.value, cp1.value);
 
         Some(value)
       }
 
       Interpolation::Linear => {
         let cp1 = &keys[i + 1];
-        let nt = normalize_time(t, cp0, cp1);
-        let value = Interpolate::lerp(cp0.value, cp1.value, nt);
+        let nt = t.normalize(cp0.t, cp1.t);
+        let value = V::lerp(nt, cp0.value, cp1.value);
 
         Some(value)
       }
 
       Interpolation::Cosine => {
-        let two_t = T::one() + T::one();
         let cp1 = &keys[i + 1];
-        let nt = normalize_time(t, cp0, cp1);
-        let cos_nt = (T::one() - (nt * T::pi()).cos()) / two_t;
-        let value = Interpolate::lerp(cp0.value, cp1.value, cos_nt);
+        let nt = t.normalize(cp0.t, cp1.t);
+        let value = V::cosine(nt, cp0.value, cp1.value);
 
         Some(value)
       }
@@ -147,13 +143,13 @@ impl<T, V> Spline<T, V> {
           let cp1 = &keys[i + 1];
           let cpm0 = &keys[i - 1];
           let cpm1 = &keys[i + 2];
-          let nt = normalize_time(t, cp0, cp1);
-          let value = Interpolate::cubic_hermite(
-            (cpm0.value, cpm0.t),
-            (cp0.value, cp0.t),
-            (cp1.value, cp1.t),
-            (cpm1.value, cpm1.t),
+          let nt = t.normalize(cp0.t, cp1.t);
+          let value = V::cubic_hermite(
             nt,
+            (cpm0.t, cpm0.value),
+            (cp0.t, cp0.value),
+            (cp1.t, cp1.value),
+            (cpm1.t, cpm1.value),
           );
 
           Some(value)
@@ -163,18 +159,14 @@ impl<T, V> Spline<T, V> {
       Interpolation::Bezier(u) | Interpolation::StrokeBezier(_, 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);
+        let nt = t.normalize(cp0.t, cp1.t);
 
         let value = match cp1.interpolation {
-          Interpolation::Bezier(v) => {
-            Interpolate::cubic_bezier(cp0.value, u, cp1.value + cp1.value - v, cp1.value, nt)
-          }
+          Interpolation::Bezier(v) => V::cubic_bezier_mirrored(nt, cp0.value, u, v, cp1.value),
 
-          Interpolation::StrokeBezier(v, _) => {
-            Interpolate::cubic_bezier(cp0.value, u, v, cp1.value, nt)
-          }
+          Interpolation::StrokeBezier(v, _) => V::cubic_bezier(nt, cp0.value, u, v, cp1.value),
 
-          _ => Interpolate::quadratic_bezier(cp0.value, u, cp1.value, nt),
+          _ => V::quadratic_bezier(nt, cp0.value, u, cp1.value),
         };
 
         Some(value)
@@ -188,8 +180,8 @@ impl<T, V> Spline<T, V> {
   ///
   pub fn sample(&self, t: T) -> Option<V>
   where
-    T: Additive + One + Trigo + Mul<T, Output = T> + Div<T, Output = T> + PartialOrd,
-    V: Additive + Interpolate<T>,
+    T: Interpolator,
+    V: Interpolate<T>,
   {
     self.sample_with_key(t).map(|sampled| sampled.value)
   }
@@ -207,8 +199,8 @@ impl<T, V> Spline<T, V> {
   /// This function returns [`None`] if you have no key.
   pub fn clamped_sample_with_key(&self, t: T) -> Option<SampledWithKey<V>>
   where
-    T: Additive + One + Trigo + Mul<T, Output = T> + Div<T, Output = T> + PartialOrd,
-    V: Additive + Interpolate<T>,
+    T: Interpolator,
+    V: Interpolate<T>,
   {
     if self.0.is_empty() {
       return None;
@@ -242,8 +234,8 @@ impl<T, V> Spline<T, V> {
   /// Sample a spline at a given time with clamping.
   pub fn clamped_sample(&self, t: T) -> Option<V>
   where
-    T: Additive + One + Trigo + Mul<T, Output = T> + Div<T, Output = T> + PartialOrd,
-    V: Additive + Interpolate<T>,
+    T: Interpolator,
+    V: Interpolate<T>,
   {
     self.clamped_sample_with_key(t).map(|sampled| sampled.value)
   }
@@ -322,16 +314,6 @@ pub struct KeyMut<'a, T, V> {
   pub interpolation: &'a mut Interpolation<T, V>,
 }
 
-// Normalize a time ([0;1]) given two control points.
-#[inline(always)]
-pub(crate) fn normalize_time<T, V>(t: T, cp: &Key<T, V>, cp1: &Key<T, V>) -> T
-where
-  T: Additive + Div<T, Output = T> + PartialEq,
-{
-  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, V>(cps: &[Key<T, V>], t: T) -> Option<usize>
 where