#### phantom-algebra

Phantom-algebra is a pure OCaml library implementing strongly-typed

small tensors with dimensions 0 ≤ 4, rank ≤ 2, and limited to square matrices.

It makes it possible to manipulate vector and matrix expressions with an

uniform notation while still catching non-sensical operations at compile time

## Tutorial

For instance, this extract is valid

```
open Phantom_algebra.Core
let v = vec3 1. 2. 3.
let w = vec3 3. 2. 1.
let u = scalar 2. + cross (v + w) (v - w)
let rot = rotation u v 1.
let r = w + rot * v
```

but adding a vector to a matrix is not, and yields a type error:

```
v + rot
```

Type errors tend to be quite long to say the least, but individual type

of scalars, vectors and matrices are much simpler. However, the size of the

type of higher order function may increase exponentially due to the exotic

type construction used internally.

`Phantom-algebra`

is inspired by GLSL conventions:

addition is the usual vector addition, with scalar broadcasted

to tensors of any dimension and rank

```
let v = vec2 0. 1. + scalar 1. (* = (1. 2.) *)
```

`x * y`

is interpreted as:the external product if either

`x`

or`y`

is a scalarthe matrix product if either

`x`

or`y`

is a matrixthe component-wise (Hadamard) product otherwise

(if both`x`

and`y`

are a vector)

the cross-product of two 2d vectors yields a scalar whereas

the cross-product of two 3d vectors yields a 3d pseudo-vectors.

(other cross-product are type errors), for instance`cross (vec2 1. 1.) (vec2 (-1.) 1.) + vec4 1 0. 0. 0. = vec4 3. 2. 2. 2.`

Indices are also-strongly typed, trying to access a index beyond the

tensor dimension yields a type error.`let v = vec2 2. 3. let fine = v.%(x') let wrong = v.%(z') let m = mat2 v v let fine = m.%(xy') let also_wrong = m.%(zx') let wrong_rank_this_time = m.%(x')`

Index names follows GLSL convention with a

`'`

suffix to avoid shadowing:

either`x'`

,`y'`

,`z'`

and`w'`

,`r',`

g'`,`

b'`,`

a'`or`

s'`,`

t'`,`

p'`and`

q'`.

Similarly, slicing a rank

`k`

tensor with a rank`n`

index

yields a rank`k-n`

tensor of the same dimension, e.g`let e1 = (vec2 1. 0.) let id = mat2 e1 (vec2 0. 1.) let e1' = id.%[x'] (* this is the first row of the id matrix *) let zero = id.%[xy']`

Swizzling is supported:

`dim`

indices can be combined with the`&`

operator

to yield an objet of`r+1`

rank:`let v = vec4 0. 1. 2. 3. let w = v.%[w'&z'&'y&'x] (* slicing a vector yields a scalar, and 4 scalars grouped together become a vector *) ;; w = vec4 3. 2. 1. 0. let mat = eye d2 let s = mat.%[y'&x'] (* we are reversing the rows, and obtaining a new matrix*) ;; s = mat2 (vec2 0. 1.) (vec2 1. 0.)`

the scalar product and usual norm are supported:

```
norm2 v = (v|*|v)
```

Usual mathematics functions have been extended to operates

element-wise on tensor, they are able in the`Math`

module

```
let v = Math.cos (vec2 1. 2.)
```

Some usual matrix and vector functions are predefined

```
let id = eye d2
let rxy t = rotation (vec3 1. 0. 0.) (vec3 0. 1. 0.) t
let id = diag (vec3 1. 1. 1.)
```

The exponential function on matrices is the matrix exponentiation

```
;; exp (mat2 (0. 1.) (0. -1) ) = rxy 1.
```

Vectors can be concatened and stretched to a given dimension

```
let v = scalar 0. |+| vec2 1. 0. |+| scalar 1.
let w = vec4' (scalar 1.)
```