# Objective
Fix `CubicCurve::iter_samples` iteration count.
## Solution
If I understand the function and the docs correctly, this should iterate
over `0..=subdivisions` instead of `0..subdivisions`.
For example: Now the iteration returns 3 points at `subdivisions = 2`,
as indicated in the documentation.
# Objective
- Make cubic splines more flexible and more performant
- Remove the existing spline implementation that is generic over many degrees
- This is a potential performance footgun and adds type complexity for negligible gain.
- Add implementations of:
- Bezier splines
- Cardinal splines (inc. Catmull-Rom)
- B-Splines
- Hermite splines
https://user-images.githubusercontent.com/2632925/221780519-495d1b20-ab46-45b4-92a3-32c46da66034.mp4https://user-images.githubusercontent.com/2632925/221780524-2b154016-699f-404f-9c18-02092f589b04.mp4https://user-images.githubusercontent.com/2632925/221780525-f934f99d-9ad4-4999-bae2-75d675f5644f.mp4
## Solution
- Implements the concept that splines are curve generators (e.g. https://youtu.be/jvPPXbo87ds?t=3488) via the `CubicGenerator` trait.
- Common splines are bespoke data types that implement this trait. This gives us flexibility to add custom spline-specific methods on these types, while ultimately all generating a `CubicCurve`.
- All splines generate `CubicCurve`s, which are a chain of precomputed polynomial coefficients. This means that all splines have the same evaluation cost, as the calculations for determining position, velocity, and acceleration are all identical. In addition, `CubicCurve`s are simply a list of `CubicSegment`s, which are evaluated from t=0 to t=1. This also means cubic splines of different type can be chained together, as ultimately they all are simply a collection of `CubicSegment`s.
- Because easing is an operation on a singe segment of a Bezier curve, we can simply implement easing on `Beziers` that use the `Vec2` type for points. Higher level crates such as `bevy_ui` can wrap this in a more ergonomic interface as needed.
### Performance
Measured on a desktop i5 8600K (6-year-old CPU):
- easing: 2.7x faster (19ns)
- cubic vec2 position sample: 1.5x faster (1.8ns)
- cubic vec3 position sample: 1.5x faster (2.6ns)
- cubic vec3a position sample: 1.9x faster (1.4ns)
On a laptop i7 11800H:
- easing: 16ns
- cubic vec2 position sample: 1.6ns
- cubic vec3 position sample: 2.3ns
- cubic vec3a position sample: 1.2ns
---
## Changelog
- Added a generic cubic curve trait, and implementation for Cardinal splines (including Catmull-Rom), B-Splines, Beziers, and Hermite Splines. 2D cubic curve segments also implement easing functionality for animation.
