Geometric algebra operators within sympy

gaexpr.py

# derived from sympy's matexpr.py. Very incomplete

from sympy.core import S
from sympy.core import Basic, Expr, Add, Mul, Symbol
from sympy.core.operations import AssocOp
from sympy.core.decorators import _sympifyit, call_highest_priority
from sympy.core.compatibility import string_types


class GaExpr(Expr):
    """
    Base class for all multivector expressions, which overloads operators.

    Note that even the `+` operator must be overloaded, so that the result of
    ``(a + b)`` itself is a :class:`GaExpr`.
    """
    _iterable = False

    is_commutative = False
    is_number = False
    is_symbol = False
    is_scalar = False

    def __neg__(self):
        return GaMul(S.NegativeOne, self).doit()

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__radd__')
    def __add__(self, other):
        return GaAdd(self, other)

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__add__')
    def __radd__(self, other):
        return GaAdd(other, self)

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__rsub__')
    def __sub__(self, other):
        return GaAdd(self, -other)

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__sub__')
    def __rsub__(self, other):
        return GaAdd(other, -self)

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__rmul__')
    def __mul__(self, other):
        return GaMul(self, other)

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__mul__')
    def __rmul__(self, other):
        return GaMul(other, self)

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__rxor__')
    def __xor__(self, other):
        return GaWedge(self, other)

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__xor__')
    def __rxor__(self, other):
        return GaWedge(other, self)

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__rxor__')
    def __or__(self, other):
        return GaDot(self, other)

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__xor__')
    def __ror__(self, other):
        return GaDot(other, self)


class GaAdd(GaExpr, Add):
    pass


class GaMul(GaExpr, Mul):
    pass


class GaDot(GaExpr):
    # not associative, so require exactly two items
    def __new__(cls, a, b, **kwargs):
        return Basic.__new__(cls, a, b, **kwargs)


class GaWedge(GaExpr, AssocOp):
    is_commutative = False
    identity = GaAdd()

    @classmethod
    def flatten(cls, args):
        c_part, nc_part, order_symbols = super(GaWedge, cls).flatten(args)
        seen = set()
        for a in nc_part:
            if a in seen:
                return [], [GaAdd()], None
            seen.add(a)
        return c_part, nc_part, order_symbols

class GaSymbol(GaExpr):
    is_symbol = True
    is_commutative = False

    def __new__(cls, name):
        if isinstance(name, string_types):
            name = Symbol(name)

        return Basic.__new__(cls, name)

    @property
    def name(self):
        return self.args[0].name


# now we monkey-patch the latex printer to know about our new types

from sympy.printing.latex import LatexPrinter as _LatexPrinter

def _print_GaWedge(self, expr):
    tex = ""
    for i, term in enumerate(expr.args):
        if i == 0:
            pass
        else:
            tex += r" \wedge "
        term_tex = self._print(term)
        if self._needs_add_brackets(term):
            term_tex = r"\left(%s\right)" % term_tex
        tex += term_tex

    return tex

def _print_GaDot(self, expr):
    tex = ""
    for i, term in enumerate(expr.args):
        if i == 0:
            pass
        else:
            tex += r" \cdot "
        term_tex = self._print(term)
        if self._needs_add_brackets(term):
            term_tex = r"\left(%s\right)" % term_tex
        tex += term_tex

    return tex

def _print_GaSymbol(self, expr):
    return self._print_Symbol(expr, style=self._settings['ga_symbol_style'])


_LatexPrinter._default_settings['ga_symbol_style'] = 'plain'
_LatexPrinter._print_GaSymbol = _print_GaSymbol
_LatexPrinter._print_GaWedge = _print_GaWedge
_LatexPrinter._print_GaDot = _print_GaDot
_LatexPrinter._print_GaAdd = _LatexPrinter._print_Add
_LatexPrinter._print_GaMul = _LatexPrinter._print_Mul

Using a very basic example: