package phantom-algebra

Dimension for vectors and matrix, 1d vectors are considered scalars

Tensor core type:

• rank is either 2 (for matrix), 1 (for vector) or zero for scalars
• dim belongs to {1,2,3,4}.

Printer function

Type abreviations

Constructors

scalar f build a scalar from the naked scalar type. It can be abbreviated to (+s).

vec$n' v extends a vector of dimension d <= n to a vector of dimension n by repeating the last value of the vector

Vector concatenation : v |+| w is (v_0, v_1, … , v_{dim-1}, w_0, …, w_{dim-1})

x + y is the standard vector sum, except for scalar argument which are broadcasted to a constant tensor

x <+> y is the standard vector sum, without broadcasting

~-x is the standard addition inverse

x - y is the standard vector difference, except for scalar argument which are broadcasted to a constant tensor

x <-> y is the standard vector difference, without broadcasting

x * y is:

• the external product if x or y is a scalar
• the matrix product if x or y is a matrix
• the element-wise (hadamard) product otherwise

x / y is:

• the external product division if x or y is a scalar
• the right matrix division if y is a matrix and x is either a matrix or a vector
• the element-wise division if both x and y are vectors

t ** k is t * … * t k-times

exp is the algebraic exponential: exp m = 1 + m + m ** 2 / 2 + m **3 / 3! + …

(x|*|y) is the canonical scalar product

norm x is the canonical 2-norm of x

normalize x is x / scalar (norm x)

orthonomalize [v_0;...;v_n] returns a list of orthonormal vectors [w_0;...;w_k] that spans the same vector subspace as v_0;...;v_n. If the family [v_0;...;v_n] was free then k = n, otherwise k<n.

distance x y is norm (x - y)

norm_1 x is ∑ |x_i|

norm_q q x is (∑ |x_i|^q) ^ 1/q

cross v w is the cross product, it maps either two 3d vectors to a 3d pseudo-vector, or two 2d vectors to a scalar

See cross for the 2d and 3d cross-product for vectors v ^ w is the infinitesimal rotation matrix in the plane generated by v and w with an amplitude |v||w| sin θ . In other words the matrix representation of the 2-form dv ^ dw in the corresponding graded algebra.

commutator m n is m * n - n * m

anticommutator m n is m * n + n * m

trace m is ∑_i m_ii

det m is the signed volume of the convex hull of of the matrix rows

transpose m is the matrix with row and column reversed

An index of type (+'dim,'+len,+'rank,'group) index can be used to index a tensor, each type parameter informs on which kind of tensor can be used ('dim and 'rank), on the type of the resulting vector ('len and 'rank), or on with which indices it can be combined when swizzling. More precisely,

• `dim is the list of tensor dimension compatible with the index, for instance `x` works for all dimension, whereas `w` is only meaningful for a 4-vector.
• `rank is the number of coordinate specified by the index, a tensor can be indexed only if its own rank is equal or superior to the index rank. For instance, xx is a rank 2 tensor and can only index matrices, whereas x is a valid index for both vector and tensor. When slicing, the rank of the slice will be the difference between the tensor rank and the index rank. For a vector v and a matrix m, v.%x is a scalar (1 - 1 = 0) like m.%[xx] ( 2 - 2 = 0) but m.%x is a vector (2-1=0) corresponding to the first row of the matrix
• 'len is the number of indices combined in the aggregated index by swizzling, if 'len > 1 it increases the rank of the resulting tensor by one and sets its dimension to 'len. See the slice function for more information.
• 'group corresponds to the index namespace, only index of the same namespace can be combined by swizzling. Availaibles namespace are `xyzx,`rgba and `stpq.

Index concatenation

XYZW group

RGBA

STPQ group

slice t n or t.%[n] computes a slice of rank tensor_rank - index_rank, in other words for a vector v and a matrix m, v.%[x] and m.%[xx] are a scalar, whereas m.%[x] is the first row vector of the matrix m

t.%(x) returns the value of the tensor at index x

zero dim rank is the zero scalar, vector or matrix with dimension dim

id rank dim t ** 0 for any tensor of corresponding rank and dimension

eye dim is id matrix dim, the identity matrix with ones on the diagonal

diag vec is the diagonal matrix with vec on the diagonal

rotation x y θ computes the rotation matrix in the plane spanned by x y with a θ angle.

clone_2 v returns two clones x,y of the value v with the same types as the original type of  v

clone_k are not strictly required, but they are here to avoid the pattern

let a, t = clone_2 t in
let b, t = clone_2 t in
…

required by the sole use of clone_2