meliao / geometric-algebra

For the Geometric Algebra reading group.

Geometric-Algebra

Overview of data representation

The object represents a multivector by storing its coefficients in the orthonormal basis 1, e_1, ..., e_n, e_1 e_2, ..., pseudoscalar. The coefficients are stored in an one-dimensional array, and the ordering of this array is a bit counterintuitive. The ordering was chosen to simplify the geometric product (hopefully).

Each basis element can be associated with a subset I of [n]. This subset can be interpreted as a bitstring of length n, where there are 1's in the indicies identified by I and 0's everywhere else. This bitstring can be interpreted as a little-endian unsigned integer, which indicates the index of the basis element in the array.

index = bitstring[0] * (2 ** 0) + bitstring[1] * (2 ** 1) + bitstring[3] * (2 ** 3) + ... + bitstring[n] * (2 ** n)

Examples:

• e_1 corresponds to bitstring [1 0 0 ... 0] -> index 1
• e_2 corresponds to bitstring [0 1 0 0 ... 0] -> index 2
• e_3 corresponds to bitstring [0 0 1 0 ... 0] -> index 4
• e_1 e_3 e_4 corresponds to bitstrinc [1 0 1 1 0 0 ...] -> index 1 + 4 + 8 = 13

TODO

• Implement get_part(k) function
• Make a LUT for _compute_output_index() and _compute_output_parity() calls.

Testing

python -m pytest test/