geometric algebra (basics)

699 days ago by cjfsyntropy

Load sympy's GA Module
from sympy.galgebra.GA import * 
       
The documentation at http://docs.sympy.org/modules/galgebra/GA/GAsympy.html says you need to do this to get access to GA variables for operations.
set_main(sys.modules[__name__]);dir(sympy.galgebra.GA) 
        
['HALF', 'LaTeX_lst', 'MAIN_PROGRAM', 'MV', 'NUMPAT', 'ONE', 'TWO',
'TrigSimp', 'ZERO', '__builtins__', '__doc__', '__file__', '__name__',
'__package__', 'abs_type', 'add_type', 'build_base', 'collect',
'collect_common_factors', 'comb', 'copy', 'cp', 'diagpq', 'dualsort',
'expand', 'function_lst', 'is_quasi_unit_numpy_array', 'isint', 'israt',
'magnitude', 'make_null_array', 'make_scalars', 'make_symbols',
'mul_type', 'normalize', 'numeric', 'numpy', 'os', 'plist', 'pow_type',
'print_lst', 'reciprocal_frame', 'reduce_base', 'regrep', 'set_main',
'set_names', 'sqrfree', 'sqrt', 'string', 'sub_base', 'sym_type',
'symbol', 'sympy', 'sys', 'test_int_flgs', 'types', 'unabs',
'vector_fct']
['HALF', 'LaTeX_lst', 'MAIN_PROGRAM', 'MV', 'NUMPAT', 'ONE', 'TWO', 'TrigSimp', 'ZERO', '__builtins__', '__doc__', '__file__', '__name__', '__package__', 'abs_type', 'add_type', 'build_base', 'collect', 'collect_common_factors', 'comb', 'copy', 'cp', 'diagpq', 'dualsort', 'expand', 'function_lst', 'is_quasi_unit_numpy_array', 'isint', 'israt', 'magnitude', 'make_null_array', 'make_scalars', 'make_symbols', 'mul_type', 'normalize', 'numeric', 'numpy', 'os', 'plist', 'pow_type', 'print_lst', 'reciprocal_frame', 'reduce_base', 'regrep', 'set_main', 'set_names', 'sqrfree', 'sqrt', 'string', 'sub_base', 'sym_type', 'symbol', 'sympy', 'sys', 'test_int_flgs', 'types', 'unabs', 'vector_fct']
MV.setup('a b c d e') 
        
'Setup of a b c d e complete!'
'Setup of a b c d e complete!'
type(a) 
        
<class 'sympy.galgebra.GA.MV'>
<class 'sympy.galgebra.GA.MV'>
MV.set_str_format(1) 
       
print 'e|(a^b^c) =',e|(a^b^c) 
        
e|(a^b^c) = ((c.e))a^b
+(-(b.e))a^c
+((a.e))b^c
e|(a^b^c) = ((c.e))a^b
+(-(b.e))a^c
+((a.e))b^c
print 'a*(b^c)-b*(a^c)+c*(a^b) =',a*(b^c)-b*(a^c)+c*(a^b) 
        
a*(b^c)-b*(a^c)+c*(a^b) = (3)a^b^c
a*(b^c)-b*(a^c)+c*(a^b) = (3)a^b^c
e=a*(b^c)-b*(a^c)+c*(a^b) 
       
e 
        
<sympy.galgebra.GA.MV object at 0x5cdea90>
<sympy.galgebra.GA.MV object at 0x5cdea90>
print e 
        
(3)a^b^c
(3)a^b^c
dir(e) 
        
['I', 'Name', '__add__', '__add_ab__', '__call__', '__class__',
'__delattr__', '__dict__', '__div__', '__div_ab__', '__doc__', '__eq__',
'__format__', '__getattribute__', '__hash__', '__init__', '__module__',
'__mul__', '__neg__', '__new__', '__or__', '__pos__', '__pow__',
'__radd__', '__reduce__', '__reduce_ex__', '__repr__', '__rmul__',
'__ror__', '__rsub__', '__rxor__', '__setattr__', '__sizeof__',
'__str__', '__sub__', '__sub_ab__', '__subclasshook__', '__weakref__',
'__xor__', 'add_in_place', 'addition', 'basis', 'basis_map',
'basislabel', 'basislabel_lst', 'basisroot', 'blade_table', 'bladeflg',
'bladelabel', 'bladeprint', 'btable', 'bvec', 'cancel', 'collect',
'combine_common_factors', 'compact', 'construct_index', 'contract',
'convert', 'convert_from_blades', 'convert_to_blades', 'coord',
'coords', 'copy', 'cse', 'curvilinear_flg', 'ddt', 'debug',
'define_basis', 'define_metric', 'define_reciprocal_frame', 'diff',
'div', 'even', 'expand', 'flatten', 'g', 'gabasis', 'geometric_product',
'grad', 'grad_ext', 'grad_int', 'ibtable', 'index', 'inner_product',
'inverse_blade_table', 'is_pure', 'is_setup', 'mag2', 'max_grade',
'metric', 'metric_str', 'mtable', 'multiplication_table', 'mv', 'n',
'n1', 'n1rg', 'name', 'named', 'nbasis', 'npow', 'nrg', 'odd',
'outer_product', 'pad_zeros', 'print_bases', 'print_blades', 'printmv',
'printnm', 'project', 'puregrade', 'rebase', 'reduce_basis',
'reduce_basis_loop', 'rev', 'scalar_fct', 'scalar_mul',
'scalar_mul_inplace', 'scalar_to_symbol', 'set_coef', 'set_coords',
'set_name', 'set_str_format', 'set_value', 'setup', 'simplify',
'sqrfree', 'str_mode', 'str_rep', 'sub_in_place', 'sub_mv',
'sub_scalar', 'subs', 'substitute_base', 'subtraction', 'tables_flg',
'trigsimp', 'trim', 'vbasis', 'vdiff', 'vector_fct', 'vsyms', 'wedge',
'x']
['I', 'Name', '__add__', '__add_ab__', '__call__', '__class__', '__delattr__', '__dict__', '__div__', '__div_ab__', '__doc__', '__eq__', '__format__', '__getattribute__', '__hash__', '__init__', '__module__', '__mul__', '__neg__', '__new__', '__or__', '__pos__', '__pow__', '__radd__', '__reduce__', '__reduce_ex__', '__repr__', '__rmul__', '__ror__', '__rsub__', '__rxor__', '__setattr__', '__sizeof__', '__str__', '__sub__', '__sub_ab__', '__subclasshook__', '__weakref__', '__xor__', 'add_in_place', 'addition', 'basis', 'basis_map', 'basislabel', 'basislabel_lst', 'basisroot', 'blade_table', 'bladeflg', 'bladelabel', 'bladeprint', 'btable', 'bvec', 'cancel', 'collect', 'combine_common_factors', 'compact', 'construct_index', 'contract', 'convert', 'convert_from_blades', 'convert_to_blades', 'coord', 'coords', 'copy', 'cse', 'curvilinear_flg', 'ddt', 'debug', 'define_basis', 'define_metric', 'define_reciprocal_frame', 'diff', 'div', 'even', 'expand', 'flatten', 'g', 'gabasis', 'geometric_product', 'grad', 'grad_ext', 'grad_int', 'ibtable', 'index', 'inner_product', 'inverse_blade_table', 'is_pure', 'is_setup', 'mag2', 'max_grade', 'metric', 'metric_str', 'mtable', 'multiplication_table', 'mv', 'n', 'n1', 'n1rg', 'name', 'named', 'nbasis', 'npow', 'nrg', 'odd', 'outer_product', 'pad_zeros', 'print_bases', 'print_blades', 'printmv', 'printnm', 'project', 'puregrade', 'rebase', 'reduce_basis', 'reduce_basis_loop', 'rev', 'scalar_fct', 'scalar_mul', 'scalar_mul_inplace', 'scalar_to_symbol', 'set_coef', 'set_coords', 'set_name', 'set_str_format', 'set_value', 'setup', 'simplify', 'sqrfree', 'str_mode', 'str_rep', 'sub_in_place', 'sub_mv', 'sub_scalar', 'subs', 'substitute_base', 'subtraction', 'tables_flg', 'trigsimp', 'trim', 'vbasis', 'vdiff', 'vector_fct', 'vsyms', 'wedge', 'x']
x=a+b 
       
print(x) 
        
a
+b
a
+b
The "|" is used for inner product. In the output, it seems to be displayed as ((a.b))1 perhaps to show that it is a 1-vector.
y=a|b 
       
print y 
        
((a.b))1
((a.b))1
The wedge (or outer) product producing a bivector.
w=a^b 
       
print w 
        
a^b
a^b
The geometric product of a and b works as expected!
v=a*b 
       
print v 
        
((a.b))1
+a^b
((a.b))1
+a^b
c=(a+b)*(a+b) 
       
print c 
        
((a**2) + (b**2) + 2*(a.b))1
((a**2) + (b**2) + 2*(a.b))1
print c**2 
        
(2*(a**2) + 2*(b**2) + 4*(a.b))1
(2*(a**2) + 2*(b**2) + 4*(a.b))1