Port the engine prototype to Rust, integrate it into the application prototype, and use it to enforce the constraints.
### Features
To see the engine in action:
1. Add a constraint by shift-clicking to select two spheres in the outline view and then hitting the 🔗 button
2. Click a summary arrow to see the outline item for the new constraint
2. Set the constraint's Lorentz product by entering a value in the text field at the right end of the outline item
* *The display should update as soon as you press* Enter *or focus away from the text field*
The checkbox at the left end of a constraint outline item controls whether the constraint is active. Activating a constraint triggers a solution update. (Deactivating a constraint doesn't, since the remaining active constraints are still satisfied.)
### Precision
The Julia prototype of the engine uses a generic scalar type, so you can pass in any type the linear algebra functions are implemented for. The examples use the [adjustable-precision](https://docs.julialang.org/en/v1/base/numbers/#Base.MPFR.setprecision) `BigFloat` type.
In the Rust port of the engine, the scalar type is currently fixed at `f64`. Switching to generic scalars shouldn't be too hard, but I haven't looked into [which other types](https://www.nalgebra.org/docs/user_guide/generic_programming) the linear algebra functions are implemented for.
### Testing
To confirm quantitatively that the Rust port of the engine is working, you can go to the `app-proto` folder and:
* Run some automated tests by calling `cargo test`.
* Inspect the optimization process in a few examples calling the `run-examples` script. The first example that prints is the same as the Irisawa hexlet example from the engine prototype. If you go into `engine-proto/gram-test`, launch Julia, and then
```
include("irisawa-hexlet.jl")
for (step, scaled_loss) in enumerate(history_alt.scaled_loss)
println(rpad(step-1, 4), " | ", scaled_loss)
end
```
you should see that it prints basically the same loss history until the last few steps, when the lower default precision of the Rust engine really starts to show.
### A small engine revision
The Rust port of the engine improves on the Julia prototype in one part of the constraint-solving routine: projecting the Hessian onto the subspace where the frozen entries stay constant. The Julia prototype does this by removing the rows and columns of the Hessian that correspond to the frozen entries, finding the Newton step from the resulting "compressed" Hessian, and then adding zero entries to the Newton step in the appropriate places. The Rust port instead replaces each frozen row and column with its corresponding standard unit vector, avoiding the finicky compressing and decompressing steps.
To confirm that this version of the constraint-solving routine works the same as the original, I implemented it in Julia as `realize_gram_alt_proj`. The solutions we get from this routine match the ones we get from the original `realize_gram` to very high precision, and in the simplest examples (`sphere-in-tetrahedron.jl` and `tetrahedron-radius-ratio.jl`), the descent paths also match to very high precision. In a more complicated example (`irisawa-hexlet.jl`), the descent paths diverge about a quarter of the way into the search, even though they end up in the same place.
Co-authored-by: Aaron Fenyes <aaron.fenyes@fareycircles.ooo>
Reviewed-on: #15
Co-authored-by: Vectornaut <vectornaut@nobody@nowhere.net>
Co-committed-by: Vectornaut <vectornaut@nobody@nowhere.net>
