# Objective
Fill a gap in the functionality of our curve constructions by allowing
users to easily build cyclic curves from control data.
## Solution
Here I opted for something lightweight and discoverable. There is a new
`CyclicCubicGenerator` trait with a method `to_curve_cyclic` which uses
splines' control data to create curves that are cyclic. For now, its
signature is exactly like that of `CubicGenerator` — `to_curve_cyclic`
just yields a `CubicCurve`:
```rust
/// Implement this on cubic splines that can generate a cyclic cubic curve from their spline parameters.
///
/// This makes sense only when the control data can be interpreted cyclically.
pub trait CyclicCubicGenerator<P: VectorSpace> {
/// Build a cyclic [`CubicCurve`] by computing the interpolation coefficients for each curve segment.
fn to_curve_cyclic(&self) -> CubicCurve<P>;
}
```
This trait has been implemented for `CubicHermite`,
`CubicCardinalSpline`, `CubicBSpline`, and `LinearSpline`:
<img width="753" alt="Screenshot 2024-07-01 at 8 58 27 PM"
src="https://github.com/bevyengine/bevy/assets/2975848/69ae0802-3b78-4fb9-b73a-6f842cf3b33c">
<img width="628" alt="Screenshot 2024-07-01 at 9 00 14 PM"
src="https://github.com/bevyengine/bevy/assets/2975848/2992175a-a96c-40fc-b1a1-5206c3572cde">
<img width="606" alt="Screenshot 2024-07-01 at 8 59 36 PM"
src="https://github.com/bevyengine/bevy/assets/2975848/9e99eb3a-dbe6-42da-886c-3d3e00410d03">
<img width="603" alt="Screenshot 2024-07-01 at 8 59 01 PM"
src="https://github.com/bevyengine/bevy/assets/2975848/d037bc0c-396a-43af-ab5c-fad9a29417ef">
(Each type pictured respectively with the control points rendered as
green spheres; tangents not pictured in the case of the Hermite spline.)
These curves are all parametrized so that the output of `to_curve` and
the output of `to_curve_cyclic` are similar. For instance, in
`CubicCardinalSpline`, the first output segment is a curve segment
joining the first and second control points in each, although it is
constructed differently. In the other cases, the segments from
`to_curve` are a subset of those in `to_curve_cyclic`, with the new
segments appearing at the end.
## Testing
I rendered cyclic splines from control data and made sure they looked
reasonable. Existing tests are intact for splines where previous code
was modified. (Note that the coefficient computation for cyclic spline
segments is almost verbatim identical to that of their non-cyclic
counterparts.)
The Bezier benchmarks also look fine.
---
## Changelog
- Added `CyclicCubicGenerator` trait to `bevy_math::cubic_splines` for
creating cyclic curves from control data.
- Implemented `CyclicCubicGenerator` for `CubicHermite`,
`CubicCardinalSpline`, `CubicBSpline`, and `LinearSpline`.
- `bevy_math` now depends on `itertools`.
---
## Discussion
### Design decisions
The biggest thing here is just the approach taken in the first place:
namely, the cyclic constructions use new methods on the same old
structs. This choice was made to reduce friction and increase
discoverability but also because creating new ones just seemed
unnecessary: the underlying data would have been the same, so creating
something like "`CyclicCubicBSpline`" whose internally-held control data
is regarded as cyclic in nature doesn't really accomplish much — the end
result for the user is basically the same either way.
Similarly, I don't presently see a pressing need for `to_curve_cyclic`
to output something other than a `CubicCurve`, although changing this in
the future may be useful. See below.
A notable omission here is that `CyclicCubicGenerator` is not
implemented for `CubicBezier`. This is not a gap waiting to be filled —
`CubicBezier` just doesn't have enough data to join its start with its
end without just making up the requisite control points wholesale. In
all the cases where `CyclicCubicGenerator` has been implemented here,
the fashion in which the ends are connected is quite natural and follows
the semantics of the associated spline construction.
### Future direction
There are two main things here:
1. We should investigate whether we should do something similar for
NURBS. I just don't know that much about NURBS at the moment, so I
regarded this as out of scope for the PR.
2. We may eventually want to change the output type of
`CyclicCubicGenerator::to_curve_cyclic` to a type which reifies the
cyclic nature of the curve output. This wasn't done in this PR because
I'm unsure how much value a type-level guarantee of cyclicity actually
has, but if some useful features make sense only in the case of cyclic
curves, this might be worth pursuing.
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