# Objective
Partially address #13408
Rework of #13613
Unify the very nice forms of interpolation specifically present in
`bevy_math` under a shared trait upon which further behavior can be
based.
The ideas in this PR were prompted by [Lerp smoothing is broken by Freya
Holmer](https://www.youtube.com/watch?v=LSNQuFEDOyQ).
## Solution
There is a new trait `StableInterpolate` in `bevy_math::common_traits`
which enshrines a quite-specific notion of interpolation with a lot of
guarantees:
```rust
/// A type with a natural interpolation that provides strong subdivision guarantees.
///
/// Although the only required method is `interpolate_stable`, many things are expected of it:
///
/// 1. The notion of interpolation should follow naturally from the semantics of the type, so
/// that inferring the interpolation mode from the type alone is sensible.
///
/// 2. The interpolation recovers something equivalent to the starting value at `t = 0.0`
/// and likewise with the ending value at `t = 1.0`.
///
/// 3. Importantly, the interpolation must be *subdivision-stable*: for any interpolation curve
/// between two (unnamed) values and any parameter-value pairs `(t0, p)` and `(t1, q)`, the
/// interpolation curve between `p` and `q` must be the *linear* reparametrization of the original
/// interpolation curve restricted to the interval `[t0, t1]`.
///
/// The last of these conditions is very strong and indicates something like constant speed. It
/// is called "subdivision stability" because it guarantees that breaking up the interpolation
/// into segments and joining them back together has no effect.
///
/// Here is a diagram depicting it:
/// ```text
/// top curve = u.interpolate_stable(v, t)
///
/// t0 => p t1 => q
/// |-------------|---------|-------------|
/// 0 => u / \ 1 => v
/// / \
/// / \
/// / linear \
/// / reparametrization \
/// / t = t0 * (1 - s) + t1 * s \
/// / \
/// |-------------------------------------|
/// 0 => p 1 => q
///
/// bottom curve = p.interpolate_stable(q, s)
/// ```
///
/// Note that some common forms of interpolation do not satisfy this criterion. For example,
/// [`Quat::lerp`] and [`Rot2::nlerp`] are not subdivision-stable.
///
/// Furthermore, this is not to be used as a general trait for abstract interpolation.
/// Consumers rely on the strong guarantees in order for behavior based on this trait to be
/// well-behaved.
///
/// [`Quat::lerp`]: crate::Quat::lerp
/// [`Rot2::nlerp`]: crate::Rot2::nlerp
pub trait StableInterpolate: Clone {
/// Interpolate between this value and the `other` given value using the parameter `t`.
/// Note that the parameter `t` is not necessarily clamped to lie between `0` and `1`.
/// When `t = 0.0`, `self` is recovered, while `other` is recovered at `t = 1.0`,
/// with intermediate values lying between the two.
fn interpolate_stable(&self, other: &Self, t: f32) -> Self;
}
```
This trait has a blanket implementation over `NormedVectorSpace`, where
`lerp` is used, along with implementations for `Rot2`, `Quat`, and the
direction types using variants of `slerp`. Other areas may choose to
implement this trait in order to hook into its functionality, but the
stringent requirements must actually be met.
This trait bears no direct relationship with `bevy_animation`'s
`Animatable` trait, although they may choose to use `interpolate_stable`
in their trait implementations if they wish, as both traits involve
type-inferred interpolations of the same kind. `StableInterpolate` is
not a supertrait of `Animatable` for a couple reasons:
1. Notions of interpolation in animation are generally going to be much
more general than those allowed under these constraints.
2. Laying out these generalized interpolation notions is the domain of
`bevy_animation` rather than of `bevy_math`. (Consider also that
inferring interpolation from types is not universally desirable.)
Similarly, this is not implemented on `bevy_color`'s color types,
although their current mixing behavior does meet the conditions of the
trait.
As an aside, the subdivision-stability condition is of interest
specifically for the [Curve
RFC](https://github.com/bevyengine/rfcs/pull/80), where it also ensures
a kind of stability for subsampling.
Importantly, this trait ensures that the "smooth following" behavior
defined in this PR behaves predictably:
```rust
/// Smoothly nudge this value towards the `target` at a given decay rate. The `decay_rate`
/// parameter controls how fast the distance between `self` and `target` decays relative to
/// the units of `delta`; the intended usage is for `decay_rate` to generally remain fixed,
/// while `delta` is something like `delta_time` from an updating system. This produces a
/// smooth following of the target that is independent of framerate.
///
/// More specifically, when this is called repeatedly, the result is that the distance between
/// `self` and a fixed `target` attenuates exponentially, with the rate of this exponential
/// decay given by `decay_rate`.
///
/// For example, at `decay_rate = 0.0`, this has no effect.
/// At `decay_rate = f32::INFINITY`, `self` immediately snaps to `target`.
/// In general, higher rates mean that `self` moves more quickly towards `target`.
///
/// # Example
/// ```
/// # use bevy_math::{Vec3, StableInterpolate};
/// # let delta_time: f32 = 1.0 / 60.0;
/// let mut object_position: Vec3 = Vec3::ZERO;
/// let target_position: Vec3 = Vec3::new(2.0, 3.0, 5.0);
/// // Decay rate of ln(10) => after 1 second, remaining distance is 1/10th
/// let decay_rate = f32::ln(10.0);
/// // Calling this repeatedly will move `object_position` towards `target_position`:
/// object_position.smooth_nudge(&target_position, decay_rate, delta_time);
/// ```
fn smooth_nudge(&mut self, target: &Self, decay_rate: f32, delta: f32) {
self.interpolate_stable_assign(target, 1.0 - f32::exp(-decay_rate * delta));
}
```
As the documentation indicates, the intention is for this to be called
in game update systems, and `delta` would be something like
`Time::delta_seconds` in Bevy, allowing positions, orientations, and so
on to smoothly follow a target. A new example, `smooth_follow`,
demonstrates a basic implementation of this, with a sphere smoothly
following a sharply moving target:
https://github.com/bevyengine/bevy/assets/2975848/7124b28b-6361-47e3-acf7-d1578ebd0347
## Testing
Tested by running the example with various parameters.
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