Phantom-algebra — a strongly-typed tensor library à la GLSL

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 scalar
  • the matrix product if either x or y is a matrix
  • the 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'ors', t', p'andq'`.

• 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.)