# Crate algebraic_gen

source ·## Expand description

Generates performant geometric products for any dimension of space.

The crate provides a macro `generate_geometric_product!`

. Given a function
identifier and a nonnegative integer literal, the macro generates a function
that computes the geometric product of two coefficient representations of
multivectors.

For example, `generate_geometric_product!(my_product, 3)`

generates the
following function:

```
pub fn my_product<A, B, T>(a: &A, b: &B) -> [T; 8]
where
A: ::core::ops::Index<usize, Output = T>,
B: ::core::ops::Index<usize, Output = T>,
T: Copy +
::core::ops::Mul<Output = T> +
::core::ops::Add<Output = T> +
::core::ops::Sub<Output = T>,
{
[
a[0] * b[0] - a[1] * b[1] - a[2] * b[2] - a[3] * b[3] - a[4] * b[4] - a[5] * b[5] - a[6] * b[6] + a[7] * b[7],
a[0] * b[1] + a[1] * b[0] + a[2] * b[3] - a[3] * b[2] + a[4] * b[5] - a[5] * b[4] - a[6] * b[7] - a[7] * b[6],
a[0] * b[2] - a[1] * b[3] + a[2] * b[0] + a[3] * b[1] + a[4] * b[6] + a[5] * b[7] - a[6] * b[4] + a[7] * b[5],
a[0] * b[3] + a[1] * b[2] - a[2] * b[1] + a[3] * b[0] - a[4] * b[7] + a[5] * b[6] - a[6] * b[5] - a[7] * b[4],
a[0] * b[4] - a[1] * b[5] - a[2] * b[6] - a[3] * b[7] + a[4] * b[0] + a[5] * b[1] + a[6] * b[2] - a[7] * b[3],
a[0] * b[5] + a[1] * b[4] + a[2] * b[7] - a[3] * b[6] - a[4] * b[1] + a[5] * b[0] + a[6] * b[3] + a[7] * b[2],
a[0] * b[6] - a[1] * b[7] + a[2] * b[4] + a[3] * b[5] - a[4] * b[2] - a[5] * b[3] + a[6] * b[0] - a[7] * b[1],
a[0] * b[7] + a[1] * b[6] - a[2] * b[5] + a[3] * b[4] + a[4] * b[3] - a[5] * b[2] + a[6] * b[1] + a[7] * b[0],
]
}
```

In this example, the implementation of the function will access `a`

and `b`

at indices `0..8`

. In the general case, it will access its arguments at
indices `0..1 << D`

and return a value of type `[T; 1 << D]`

.

## Example

The functions are straightforward to generate and use. Pass a (small)
nonnegative integer to the macro and it generates the product function.
(Check it out with `cargo expand`

.)

```
use algebraic_gen::generate_geometric_product;
generate_geometric_product!(my_product, 3);
fn main() {
let a: [f64; 8] = [1., 2., 3., 4., 5., 6., 7., 8.];
let b: [f64; 8] = [8., 7., 6., 5., 4., 3., 2., 1.];
let c = my_product(&a, &b);
println!("The geometric product of {:?} and {:?} is {:?}", a, b, c);
assert_eq!(c, [-88.0, -18.0, 60.0, -18.0, 72.0, 102.0, -36.0, 114.0]);
}
```

## Geometric Algebra

A geometric algebra is defined with respect
to a given dimension `D`

of space. In such an algebra, there exist objects
of different *degrees* from zero to `D`

(inclusive). Objects of degree zero
are scalars, objects of degree one are vectors, and objects of degree two,
three, etc. are bivectors, trivectors, et cetera. These higher degree
objects are created by wedge product of vectors. Combining all vectors via
wedge product results a *pseudoscalar* of degree `D`

. Objects of any degree
can be scaled and added to form *multivectors*.

Importantly for this crate, any multivector can be represented uniquely as a
linear combination of elements of a chosen *base* with `2^D`

different
objects, meaning that we can represent multivectors as arrays of floating
point numbers of that size. The return types `[T; 1]`

, `[T; 2]`

, `[T; 4]`

,
`[T; 8]`

, etc., of the functions generated by this crate are exactly those
representations of multivectors of geometric algebras of spaces with `0`

,
`1`

, `2`

, `3`

, etc. dimensions, respectively.

The macro attaches documentation of the choice of base to the generated
function. Sticking to 3D for an example, the choice of base in this crate
is: one scalar (`S`

), three vectors (`X`

, `Y`

, `Z`

), three bivectors (`X∧Y`

,
`X∧Z`

, `Y∧X`

), and one pseudoscalar (`X∧Y∧Z`

). The generated
product interprets the 8-dimensional coefficient arrays as `[S, X, Y, X∧Y, Z, X∧Z, Y∧Z, X∧Y∧Z]`

. In the documentation attached to the respective
generated product functions, this is represented as `[[], [0], [1], [0,1], [2], [0,2], [1,2], [0,1,2]]`

.

## On the Generation

Maybe there is better way to get the products that doesn’t use the 'sledgehammer' proc macro approach. The generation is not performant, but we’re talking build time.

The generation logic is inspired by All Hail Geometric Algebra!.

Brief description of the how the macro works:

- Generates elements (base)
- Product is simple concatenation here
- Canonization with Bubblesort & a kind of duplication elimination

- Generates product sums
- Expand terms
- More products & canonization
- Adding/Subtracting based on sign returned by canonization

- A bit of formatting

More details in the source code:
`algebra_generation.rs`

.