Update of galgebra for Barcelona conference. Most importantly correct…

…ions

to the < and > operators for multivectors and the input of metric signature
operations requiring a normalized pseudo-scalar.  Details are in expanded
documentation.

galgebra-master/examples/LaTeX/Dop.py

@@ -8,6 +8,7 @@
 (o3d, ex, ey, ez) = Ga.build('e', g=[1, 1, 1], coords=xyz_coords, norm=True)
 f = o3d.mv('f', 'scalar', f=True)
 lap = o3d.grad*o3d.grad
+print r'\nabla =', o3d.grad
 print r'%\nabla^{2} = \nabla . \nabla =', lap
 print r'%\lp\nabla^{2}\rp f =', lap*f
 print r'%\nabla\cdot\lp\nabla f\rp =', o3d.grad | (o3d.grad * f)

galgebra-master/examples/LaTeX/FmtChk.py

@@ -14,8 +14,6 @@
 print Fmt(l,1)
 print F.Fmt(3)
 print B.Fmt(3)
-print Fmt(l)
-print Fmt(l,1)
 
 lap = o3d.grad*o3d.grad
 print r'%\nabla^{2} = \nabla\cdot\nabla =', lap

galgebra-master/examples/LaTeX/em_waves_latex.py

@@ -1,7 +1,8 @@
 
 import sys
-from sympy import symbols,sin,cos,exp,I,Matrix,solve,simplify
-from printer import Format,xpdf,Get_Program,Print_Function
+from sympy import symbols,sin,cos,exp,I,Matrix,simplify,Eq,S
+from sympy.solvers import solve
+from printer import Format,xpdf,Get_Program,Print_Function,Fmt
 from ga import Ga
 from metric import linear_expand
 
@@ -10,7 +11,7 @@ def EM_Waves_in_Geom_Calculus():
     X = (t,x,y,z) = symbols('t x y z',real=True)
     (st4d,g0,g1,g2,g3) = Ga.build('gamma*t|x|y|z',g=[1,-1,-1,-1],coords=X)
 
-    i = st4d.i
+    i = st4d.I()
 
     B = st4d.mv('B','vector')
     E = st4d.mv('E','vector')
@@ -31,20 +32,23 @@ def EM_Waves_in_Geom_Calculus():
 
     print r'\text{Pseudo Scalar\;\;}I =',i
     print r'%I_{xyz} =',Ixyz
-    F.Fmt(3,'\\text{Electromagnetic Field Bi-Vector\\;\\;} F')
+    print F.Fmt(3,'\\text{Electromagnetic Field Bi-Vector\\;\\;} F')
     gradF = st4d.grad*F
 
     print '#Geom Derivative of Electomagnetic Field Bi-Vector'
-    gradF.Fmt(3,'grad*F = 0')
+    print gradF.Fmt(3,'grad*F = 0')
 
     gradF = gradF / (I * exp(I*KX))
-    gradF.Fmt(3,r'%\lp\bm{\nabla}F\rp /\lp i e^{iK\cdot X}\rp = 0')
+    print gradF.Fmt(3,r'%\lp\bm{\nabla}F\rp /\lp i e^{iK\cdot X}\rp = 0')
 
-    g = '1 # 0 0,# 1 0 0,0 0 1 0,0 0 0 -1'
+    g = '-1 # 0 0,# -1 0 0,0 0 -1 0,0 0 0 1'
     X = (xE,xB,xk,t) = symbols('x_E x_B x_k t',real=True)
-    (EBkst,eE,eB,ek,et) = Ga.build('e_E e_B e_k t',g=g,coords=X)
 
-    i = EBkst.i
+    # The correct signature sig=p for signature (p,q) is needed
+    # since the correct value of I**2 is needed to compute exp(I)
+    (EBkst,eE,eB,ek,et) = Ga.build('e_E e_B e_k t',g=g,coords=X,sig=1)
+
+    i = EBkst.I()
 
     E,B,k,w = symbols('E B k omega',real=True)
 
@@ -65,31 +69,30 @@ def EM_Waves_in_Geom_Calculus():
 
     gradF_reduced = (EBkst.grad*F)/(I*exp(I*KX))
 
-    gradF_reduced.Fmt(3,r'%\lp\bm{\nabla}F\rp/\lp ie^{iK\cdot X} \rp = 0')
+    print gradF_reduced.Fmt(3,r'%\lp\bm{\nabla}F\rp/\lp ie^{iK\cdot X} \rp = 0')
 
     print r'%\mbox{Previous equation requires that: }e_{E}\cdot e_{B} = 0\mbox{ if }B\ne 0\mbox{ and }k\ne 0'
 
-    gradF_reduced = gradF_reduced.subs({EBkst.g[0,1]:0})
-    gradF_reduced.Fmt(3,r'%\lp\bm{\nabla}F\rp/\lp ie^{iK\cdot X} \rp = 0')
+    gradF_reduced = gradF_reduced.subs({EBkst.g[0,1]:0}).expand()
+    print gradF_reduced.Fmt(3,r'%\lp\bm{\nabla}F\rp/\lp ie^{iK\cdot X} \rp = 0')
 
     (coefs,bases) = linear_expand(gradF_reduced.obj)
 
     eq1 = coefs[0]
     eq2 = coefs[1]
 
+    print r'0 =', eq1
+    print r'0 =', eq2
+
     B1 = solve(eq1,B)[0]
     B2 = solve(eq2,B)[0]
 
-    print r'\mbox{eq1: }B =',B1
-    print r'\mbox{eq2: }B =',B2
-
     eq3 = B1-B2
 
     print r'\mbox{eq3 = eq1-eq2: }0 =',eq3
     eq3 = simplify(eq3 / E)
     print r'\mbox{eq3 = (eq1-eq2)/E: }0 =',eq3
     print 'k =',Matrix(solve(eq3,k))
-
     print 'B =',Matrix([B1.subs(w,k),B1.subs(-w,k)])
 
     return

galgebra-master/examples/LaTeX/groups.py

@@ -71,7 +71,7 @@ def main():
     Get_Program()
     Format()
     Product_of_Rotors()
-    xpdf(paper=(8.5,11),debug=True)
+    xpdf(paper=(8.5,11))
     return
 
 if __name__ == "__main__":

galgebra-master/examples/LaTeX/latex_check.py

@@ -1,7 +1,7 @@
 from sympy import Symbol, symbols, sin, cos, Rational, expand, simplify, collect
 from printer import Format, Eprint, Get_Program, Print_Function, xpdf, Fmt
 from ga import Ga, one, zero
-from mv import Nga, com
+from mv import Nga, com, cross
 
 HALF = Rational(1,2)
 
@@ -30,24 +30,25 @@ def basic_multivector_operations_3D():
 
     A = g3d.mv('A','mv')
 
-    A.Fmt(1,'A')
-    A.Fmt(2,'A')
-    A.Fmt(3,'A')
+    print A.Fmt(1,'A')
+    print A.Fmt(2,'A')
+    print A.Fmt(3,'A')
 
-    A.even().Fmt(1,'%A_{+}')
-    A.odd().Fmt(1,'%A_{-}')
+    print A.even().Fmt(1,'%A_{+}')
+    print A.odd().Fmt(1,'%A_{-}')
 
     X = g3d.mv('X','vector')
     Y = g3d.mv('Y','vector')
 
     print 'g_{ij} = ',g3d.g
 
-    X.Fmt(1,'X')
-    Y.Fmt(1,'Y')
+    print X.Fmt(1,'X')
+    print Y.Fmt(1,'Y')
 
-    (X*Y).Fmt(2,'X*Y')
-    (X^Y).Fmt(2,'X^Y')
-    (X|Y).Fmt(2,'X|Y')
+    print (X*Y).Fmt(2,'X*Y')
+    print (X^Y).Fmt(2,'X^Y')
+    print (X|Y).Fmt(2,'X|Y')
+    print cross(X,Y).Fmt(1,r'X\times Y')
     return
 
 def basic_multivector_operations_2D():
@@ -172,6 +173,7 @@ def rounding_numerical_components():
 def noneuclidian_distance_calculation():
     Print_Function()
     from sympy import solve,sqrt
+    Fmt(1)
 
     g = '0 # #,# 0 #,# # 1'
     nel = Ga('X Y e',g=g)
@@ -278,6 +280,7 @@ def noneuclidian_distance_calculation():
 def conformal_representations_of_circles_lines_spheres_and_planes():
     Print_Function()
     global n,nbar
+    Fmt(1)
     g = '1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0'
 
     c3d = Ga('e_1 e_2 e_3 n \\bar{n}',g=g)
@@ -319,6 +322,7 @@ def conformal_representations_of_circles_lines_spheres_and_planes():
 def properties_of_geometric_objects():
     Print_Function()
     global n, nbar
+    Fmt(1)
     g = '# # # 0 0,'+ \
         '# # # 0 0,'+ \
         '# # # 0 0,'+ \
@@ -350,6 +354,7 @@ def properties_of_geometric_objects():
 
 def extracting_vectors_from_conformal_2_blade():
     Print_Function()
+    Fmt(1)
     print r'B = P1\W P2'
 
     g = '0 -1 #,'+ \
@@ -382,6 +387,7 @@ def extracting_vectors_from_conformal_2_blade():
 
 def reciprocal_frame_test():
     Print_Function()
+    Fmt(1)
     g = '1 # #,'+ \
         '# 1 #,'+ \
         '# # 1'

galgebra-master/examples/LaTeX/manifold.py

@@ -57,31 +57,6 @@
 a1 = sph2d.mv('a_1','vector')
 a2 = sph2d.mv('a_2','vector')
 
-"""
-V = Mlt('V',sph2d,(a1,),fct=True)
-T = Mlt('T',sph2d,(a1,a2),fct=True)
-
-print '#Tensors on the Unit Sphere'
-
-print 'V =', V
-#print 'T =', T
-T.Fmt(5,'T')
-print '#Tensor Contraction'
-print r'T[1,2] =', T.contract(1,2)
-print '#Tensor Evaluation'
-print r'T(a,b) =', T(a,b)
-print r'T(a,b+c) =', T(a,b+c).expand()
-print r'T(a,\alpha b) =', T(a,alpha*b)
-print '#Geometric Derivative With Respect To Slot'
-print r'\nabla_{a_{1}}T =',T.pdiff(1)
-print r'\nabla_{a_{2}}T =',T.pdiff(2)
-print '#Covariant Derivatives'
-print r'\mathcal{D}V =', V.cderiv()
-DT = T.cderiv()
-#print r'\mathcal{D}T =', DT
-DT.Fmt(3,r'\mathcal{D}T')
-print r'\mathcal{D}T[1,3](a) =', (DT.contract(1,3))(a)
-"""
 #Define curve on unit sphere manifold
 
 us = Function('u__s')(s)

galgebra-master/examples/LaTeX/products_latex.py

@@ -3,7 +3,7 @@
 from printer import Format,xpdf
 
 def main():
-    #Format()
+    Format()
 
     coords = (x,y,z) = symbols('x y z',real=True)
 
@@ -27,12 +27,12 @@ def main():
     for xi in X:
         print ''
         for yi in X:
-            print xi[1]+'*'+yi[1]+' =',xi[0]*yi[0]
-            print xi[1]+'^'+yi[1]+' =',xi[0]^yi[0]
+            print xi[1]+' * '+yi[1]+' =',xi[0]*yi[0]
+            print xi[1]+' ^ '+yi[1]+' =',xi[0]^yi[0]
             if xi[1] != 's' and yi[1] != 's':
-                print xi[1]+'|'+yi[1]+' =',xi[0]|yi[0]
-            print xi[1]+'<'+yi[1]+' =',xi[0]<yi[0]
-            print xi[1]+'>'+yi[1]+' =',xi[0]>yi[0]
+                print xi[1]+' | '+yi[1]+' =',xi[0]|yi[0]
+            print xi[1]+' < '+yi[1]+' =',xi[0]<yi[0]
+            print xi[1]+' > '+yi[1]+' =',xi[0]>yi[0]
 
     fs = o3d.mv('s','scalar',f=True)
     fv = o3d.mv('v','vector',f=True)
@@ -54,12 +54,12 @@ def main():
             if xi[1] == 'grad' and yi[1] == 'grad':
                 pass
             else:
-                print xi[1]+'*'+yi[1]+' =',xi[0]*yi[0]
-                print xi[1]+'^'+yi[1]+' =',xi[0]^yi[0]
+                print xi[1]+' * '+yi[1]+' =',xi[0]*yi[0]
+                print xi[1]+' ^ '+yi[1]+' =',xi[0]^yi[0]
                 if xi[1] != 's' and yi[1] != 's':
-                    print xi[1]+'|'+yi[1]+' =',xi[0]|yi[0]
-                print xi[1]+'<'+yi[1]+' =',xi[0]<yi[0]
-                print xi[1]+'>'+yi[1]+' =',xi[0]>yi[0]
+                    print xi[1]+' | '+yi[1]+' =',xi[0]|yi[0]
+                print xi[1]+' < '+yi[1]+' =' ,xi[0]<yi[0]
+                print xi[1]+' > '+yi[1]+' =' ,xi[0]>yi[0]
 
 
     (g2d,ex,ey) = Ga.build('e',coords=(x,y))
@@ -85,14 +85,16 @@ def main():
             if xi[1] == 'grad' and yi[1] == 'grad':
                 pass
             else:
-                print xi[1]+'*'+yi[1]+' =',xi[0]*yi[0]
-                print xi[1]+'^'+yi[1]+' =',xi[0]^yi[0]
+                print xi[1]+' * '+yi[1]+' =',xi[0]*yi[0]
+                print xi[1]+' ^ '+yi[1]+' =',xi[0]^yi[0]
                 if xi[1] != 's' and yi[1] != 's':
-                    print xi[1]+'|'+yi[1]+' =',xi[0]|yi[0]
-                print xi[1]+'<'+yi[1]+' =',xi[0]<yi[0]
-                print xi[1]+'>'+yi[1]+' =',xi[0]>yi[0]
+                    print xi[1]+' | '+yi[1]+' =',xi[0]|yi[0]
+                else:
+                    print xi[1]+' | '+yi[1]+' = Not Allowed'
+                print xi[1]+' < '+yi[1]+' =',xi[0]<yi[0]
+                print xi[1]+' > '+yi[1]+' ='  ,xi[0]>yi[0]
 
-    #xpdf(paper='letter')
+    xpdf(paper='letter')
     return
 
 if __name__ == "__main__":