c60dcea231
# Objective
- For curves that also include derivatives, make accessing derivative
information via the `Curve` API ergonomic: that is, provide access to a
curve that also samples derivative information.
- Implement this functionality for cubic spline curves provided by
`bevy_math`.
Ultimately, this is to serve the purpose of doing more geometric
operations on curves, like reparametrization by arclength and the
construction of moving frames.
## Solution
This has several parts, some of which may seem redundant. However, care
has been put into this to satisfy the following constraints:
- Accessing a `Curve` that samples derivative information should be not
just possible but easy and non-error-prone. For example, given a
differentiable `Curve<Vec2>`, one should be able to access something
like a `Curve<(Vec2, Vec2)>` ergonomically, and not just sample the
derivatives piecemeal from point to point.
- Derivative access should not step on the toes of ordinary curve usage.
In particular, in the above scenario, we want to avoid simply making the
same curve both a `Curve<Vec2>` and a `Curve<(Vec2, Vec2)>` because this
requires manual disambiguation when the API is used.
- Derivative access must work gracefully in both owned and borrowed
contexts.
### `HasTangent`
We introduce a trait `HasTangent` that provides an associated `Tangent`
type for types that have tangent spaces:
```rust
pub trait HasTangent {
/// The tangent type.
type Tangent: VectorSpace;
}
```
(Mathematically speaking, it would be more precise to say that these are
types that represent spaces which are canonically
[parallelized](https://en.wikipedia.org/wiki/Parallelizable_manifold). )
The idea here is that a point moving through a `HasTangent` type may
have a derivative valued in the associated `Tangent` type at each time
in its journey. We reify this with a `WithDerivative<T>` type that uses
`HasTangent` to include derivative information:
```rust
pub struct WithDerivative<T>
where
T: HasTangent,
{
/// The underlying value.
pub value: T,
/// The derivative at `value`.
pub derivative: T::Tangent,
}
```
And we can play the same game with second derivatives as well, since
every `VectorSpace` type is `HasTangent` where `Tangent` is itself (we
may want to be more restrictive with this in practice, but this holds
mathematically).
```rust
pub struct WithTwoDerivatives<T>
where
T: HasTangent,
{
/// The underlying value.
pub value: T,
/// The derivative at `value`.
pub derivative: T::Tangent,
/// The second derivative at `value`.
pub second_derivative: <T::Tangent as HasTangent>::Tangent,
}
```
In this PR, `HasTangent` is only implemented for `VectorSpace` types,
but it would be valuable to have this implementation for types like
`Rot2` and `Quat` as well. We could also do it for the isometry types
and, potentially, transforms as well. (This is in decreasing order of
value in my opinion.)
### `CurveWithDerivative`
This is a trait for a `Curve<T>` which allows the construction of a
`Curve<WithDerivative<T>>` when derivative information is known
intrinsically. It looks like this:
```rust
/// Trait for curves that have a well-defined notion of derivative, allowing for
/// derivatives to be extracted along with values.
pub trait CurveWithDerivative<T>
where
T: HasTangent,
{
/// This curve, but with its first derivative included in sampling.
fn with_derivative(self) -> impl Curve<WithDerivative<T>>;
}
```
The idea here is to provide patterns like this:
```rust
let value_and_derivative = my_curve.with_derivative().sample_clamped(t);
```
One of the main points here is that `Curve<WithDerivative<T>>` is useful
as an output because it can be used durably. For example, in a dynamic
context, something that needs curves with derivatives can store
something like a `Box<dyn Curve<WithDerivative<T>>>`. Note that
`CurveWithDerivative` is not dyn-compatible.
### `SampleDerivative`
Many curves "know" how to sample their derivatives instrinsically, but
implementing `CurveWithDerivative` as given would be onerous or require
an annoying amount of boilerplate. There are also hurdles to overcome
that involve references to curves: for the `Curve` API, the expectation
is that curve transformations like `with_derivative` take things by
value, with the contract that they can still be used by reference
through deref-magic by including `by_ref` in a method chain.
These problems are solved simultaneously by a trait `SampleDerivative`
which, when implemented, automatically derives `CurveWithDerivative` for
a type and all types that dereference to it. It just looks like this:
```rust
pub trait SampleDerivative<T>: Curve<T>
where
T: HasTangent,
{
fn sample_with_derivative_unchecked(&self, t: f32) -> WithDerivative<T>;
// ... other sampling variants as default methods
}
```
The point is that the output of `with_derivative` is a
`Curve<WithDerivative<T>>` that uses the `SampleDerivative`
implementation. On a `SampleDerivative` type, you can also just call
`my_curve.sample_with_derivative(t)` instead of something like
`my_curve.by_ref().with_derivative().sample(t)`, which is more verbose
and less accessible.
In practice, `CurveWithDerivative<T>` is actually a "sealed" extension
trait of `SampleDerivative<T>`.
## Adaptors
`SampleDerivative` has automatic implementations on all curve adaptors
except for `FunctionCurve`, `MapCurve`, and `ReparamCurve` (because we
do not have a notion of differentiable Rust functions).
For example, `CurveReparamCurve` (the reparametrization of a curve by
another curve) can compute derivatives using the chain rule in the case
both its constituents have them.
## Testing
Tests for derivatives on the curve adaptors are included.
---
## Showcase
This development allows derivative information to be included with and
extracted from curves using the `Curve` API.
```rust
let points = [
vec2(-1.0, -20.0),
vec2(3.0, 2.0),
vec2(5.0, 3.0),
vec2(9.0, 8.0),
];
// A cubic spline curve that goes through `points`.
let curve = CubicCardinalSpline::new(0.3, points).to_curve().unwrap();
// Calling `with_derivative` causes derivative output to be included in the output of the curve API.
let curve_with_derivative = curve.with_derivative();
// A `Curve<f32>` that outputs the speed of the original.
let speed_curve = curve_with_derivative.map(|x| x.derivative.norm());
```
---
## Questions
- ~~Maybe we should seal `WithDerivative` or make it require
`SampleDerivative` (i.e. make it unimplementable except through
`SampleDerivative`).~~ I decided this is a good idea.
- ~~Unclear whether `VectorSpace: HasTangent` blanket implementation is
really appropriate. For colors, for example, I'm not sure that the
derivative values can really be interpreted as a color. In any case, it
should still remain the case that `VectorSpace` types are `HasTangent`
and that `HasTangent::Tangent: HasTangent`.~~ I think this is fine.
- Infinity bikeshed on names of traits and things.
## Future
- Faster implementations of `SampleDerivative` for cubic spline curves.
- Improve ergonomics for accessing only derivatives (and other kinds of
transformations on derivative curves).
- Implement `HasTangent` for:
- `Rot2`/`Quat`
- `Isometry` types
- `Transform`, maybe
- Implement derivatives for easing curves.
- Marker traits for continuous/differentiable curves. (It's actually
unclear to me how much value this has in practice, but we have discussed
it in the past.)
---------
Co-authored-by: Alice Cecile <alice.i.cecile@gmail.com>
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What is Bevy?
Bevy is a refreshingly simple data-driven game engine built in Rust. It is free and open-source forever!
WARNING
Bevy is still in the early stages of development. Important features are missing. Documentation is sparse. A new version of Bevy containing breaking changes to the API is released approximately once every 3 months. We provide migration guides, but we can't guarantee migrations will always be easy. Use only if you are willing to work in this environment.
MSRV: Bevy relies heavily on improvements in the Rust language and compiler. As a result, the Minimum Supported Rust Version (MSRV) is generally close to "the latest stable release" of Rust.
Design Goals
- Capable: Offer a complete 2D and 3D feature set
- Simple: Easy for newbies to pick up, but infinitely flexible for power users
- Data Focused: Data-oriented architecture using the Entity Component System paradigm
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About
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Getting Started
We recommend checking out the Quick Start Guide for a brief introduction.
Follow the Setup guide to ensure your development environment is set up correctly. Once set up, you can quickly try out the examples by cloning this repo and running the following commands:
# Switch to the correct version (latest release, default is main development branch)
git checkout latest
# Runs the "breakout" example
cargo run --example breakout
To draw a window with standard functionality enabled, use:
use bevy::prelude::*;
fn main(){
App::new()
.add_plugins(DefaultPlugins)
.run();
}
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Bevy can be built just fine using default configuration on stable Rust. However for really fast iterative compiles, you should enable the "fast compiles" setup by following the instructions here.
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