latex_check.py

from __future__ import print_function
from sympy import Symbol, symbols, sin, cos, Rational, expand, simplify, collect
from galgebra.printer import Format, Eprint, Print_Function, xpdf, Fmt
from galgebra.ga import Ga, one, zero
from galgebra.mv import Nga, cross

HALF = Rational(1,2)

def F(x):
    global n,nbar
    Fx = ((x*x)*n+2*x-nbar) / 2
    return(Fx)

def make_vector(a,n = 3, ga=None):
    if isinstance(a,str):
        v = zero
        for i in range(n):
            a_i = Symbol(a+str(i+1))
            v += a_i*ga.basis[i]
        v = ga.mv(v)
        return(F(v))
    else:
        return(F(a))

def basic_multivector_operations_3D():
    Print_Function()

    g3d = Ga('e*x|y|z')
    (ex,ey,ez) = g3d.mv()

    A = g3d.mv('A','mv')

    print(A.Fmt(1,'A'))
    print(A.Fmt(2,'A'))
    print(A.Fmt(3,'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)

    print(X.Fmt(1,'X'))
    print(Y.Fmt(1,'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():
    Print_Function()
    g2d = Ga('e*x|y')
    (ex,ey) = g2d.mv()

    print('g_{ij} =',g2d.g)

    X = g2d.mv('X','vector')
    A = g2d.mv('A','spinor')

    print(X.Fmt(1,'X'))
    print(A.Fmt(1,'A'))

    print((X|A).Fmt(2,'X|A'))
    print((X<A).Fmt(2,'X<A'))
    print((A>X).Fmt(2,'A>X'))
    return

def basic_multivector_operations_2D_orthogonal():
    Print_Function()
    o2d = Ga('e*x|y',g=[1,1])
    (ex,ey) = o2d.mv()
    print('g_{ii} =',o2d.g)

    X = o2d.mv('X','vector')
    A = o2d.mv('A','spinor')

    print(X.Fmt(1,'X'))
    print(A.Fmt(1,'A'))

    print((X*A).Fmt(2,'X*A'))
    print((X|A).Fmt(2,'X|A'))
    print((X<A).Fmt(2,'X<A'))
    print((X>A).Fmt(2,'X>A'))

    print((A*X).Fmt(2,'A*X'))
    print((A|X).Fmt(2,'A|X'))
    print((A<X).Fmt(2,'A<X'))
    print((A>X).Fmt(2,'A>X'))
    return

def check_generalized_BAC_CAB_formulas():
    Print_Function()
    g4d = Ga('a b c d e')
    (a,b,c,d,e) = g4d.mv()

    print('g_{ij} =',g4d.g)

    print('\\bm{a|(b*c)} =',a|(b*c))
    print('\\bm{a|(b^c)} =',a|(b^c))
    print('\\bm{a|(b^c^d)} =',a|(b^c^d))
    print('\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',(a|(b^c))+(c|(a^b))+(b|(c^a)))
    print('\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',a*(b^c)-b*(a^c)+c*(a^b))
    print('\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c))
    print('\\bm{(a^b)|(c^d)} =',(a^b)|(c^d))
    print('\\bm{((a^b)|c)|d} =',((a^b)|c)|d)
    print('\\bm{(a^b)\\times (c^d)} =',Ga.com(a^b,c^d))
    print('\\bm{(a^b^c)(d^e)} =',((a^b^c)*(d^e)).Fmt(2))
    return

def derivatives_in_rectangular_coordinates():
    Print_Function()
    X = (x,y,z) = symbols('x y z')
    o3d = Ga('e_x e_y e_z',g=[1,1,1],coords=X)
    (ex,ey,ez) = o3d.mv()
    grad = o3d.grad

    f = o3d.mv('f','scalar',f=True)
    A = o3d.mv('A','vector',f=True)
    B = o3d.mv('B','bivector',f=True)
    C = o3d.mv('C','mv')
    print('f =',f)
    print('A =',A)
    print('B =',B)
    print('C =',C)

    print('grad*f =',grad*f)
    print('grad|A =',grad|A)
    print('grad*A =',grad*A)

    print('-I*(grad^A) =',-o3d.I()*(grad^A))
    print('grad*B =',grad*B)
    print('grad^B =',grad^B)
    print('grad|B =',grad|B)
    return

def derivatives_in_spherical_coordinates():
    Print_Function()
    X = (r,th,phi) = symbols('r theta phi')
    s3d = Ga('e_r e_theta e_phi',g=[1,r**2,r**2*sin(th)**2],coords=X,norm=True)
    (er,eth,ephi) = s3d.mv()
    grad = s3d.grad

    f = s3d.mv('f','scalar',f=True)
    A = s3d.mv('A','vector',f=True)
    B = s3d.mv('B','bivector',f=True)

    print('f =',f)
    print('A =',A)
    print('B =',B)

    print('grad*f =',grad*f)
    print('grad|A =',grad|A)
    print('-I*(grad^A) =',(-s3d.E()*(grad^A)).simplify())
    print('grad^B =',grad^B)

def rounding_numerical_components():
    Print_Function()
    o3d = Ga('e_x e_y e_z',g=[1,1,1])
    (ex,ey,ez) = o3d.mv()

    X = 1.2*ex+2.34*ey+0.555*ez
    Y = 0.333*ex+4*ey+5.3*ez

    print('X =',X)
    print('Nga(X,2) =',Nga(X,2))
    print('X*Y =',X*Y)
    print('Nga(X*Y,2) =',Nga(X*Y,2))
    return

def noneuclidian_distance_calculation():
    Print_Function()
    from sympy import solve,sqrt
    Fmt(1)

    g = '0 # #,# 0 #,# # 1'
    nel = Ga('X Y e',g=g)
    (X,Y,e) = nel.mv()

    print('g_{ij} =',nel.g)

    print('%(X\\W Y)^{2} =',(X^Y)*(X^Y))

    L = X^Y^e
    B = L*e # D&L 10.152
    Bsq = (B*B).scalar()
    print('#%L = X\\W Y\\W e \\text{ is a non-euclidian line}')
    print('B = L*e =',B)

    BeBr =B*e*B.rev()
    print('%BeB^{\\dagger} =',BeBr)
    print('%B^{2} =',B*B)
    print('%L^{2} =',L*L) # D&L 10.153
    (s,c,Binv,M,S,C,alpha) = symbols('s c (1/B) M S C alpha')

    XdotY = nel.g[0,1]
    Xdote = nel.g[0,2]
    Ydote = nel.g[1,2]

    Bhat = Binv*B # D&L 10.154
    R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2)
    print('#%s = \\f{\\sinh}{\\alpha/2} \\text{ and } c = \\f{\\cosh}{\\alpha/2}')
    print('%e^{\\alpha B/{2\\abs{B}}} =',R)

    Z = R*X*R.rev() # D&L 10.155
    Z.obj = expand(Z.obj)
    Z.obj = Z.obj.collect([Binv,s,c,XdotY])
    Z.Fmt(3,'%RXR^{\\dagger}')
    W = Z|Y # Extract scalar part of multivector
    # From this point forward all calculations are with sympy scalars
    #print '#Objective is to determine value of C = cosh(alpha) such that W = 0'
    W = W.scalar()
    print('%W = Z\\cdot Y =',W)
    W = expand(W)
    W = simplify(W)
    W = W.collect([s*Binv])

    M = 1/Bsq
    W = W.subs(Binv**2,M)
    W = simplify(W)
    Bmag = sqrt(XdotY**2-2*XdotY*Xdote*Ydote)
    W = W.collect([Binv*c*s,XdotY])

    #Double angle substitutions

    W = W.subs(2*XdotY**2-4*XdotY*Xdote*Ydote,2/(Binv**2))
    W = W.subs(2*c*s,S)
    W = W.subs(c**2,(C+1)/2)
    W = W.subs(s**2,(C-1)/2)
    W = simplify(W)
    W = W.subs(Binv,1/Bmag)
    W = expand(W)

    print('#%S = \\f{\\sinh}{\\alpha} \\text{ and } C = \\f{\\cosh}{\\alpha}')

    print('W =',W)

    Wd = collect(W,[C,S],exact=True,evaluate=False)

    Wd_1 = Wd[one]
    Wd_C = Wd[C]
    Wd_S = Wd[S]

    print('%\\text{Scalar Coefficient} =',Wd_1)
    print('%\\text{Cosh Coefficient} =',Wd_C)
    print('%\\text{Sinh Coefficient} =',Wd_S)

    print('%\\abs{B} =',Bmag)
    Wd_1 = Wd_1.subs(Bmag,1/Binv)
    Wd_C = Wd_C.subs(Bmag,1/Binv)
    Wd_S = Wd_S.subs(Bmag,1/Binv)

    lhs = Wd_1+Wd_C*C
    rhs = -Wd_S*S
    lhs = lhs**2
    rhs = rhs**2
    W = expand(lhs-rhs)
    W = expand(W.subs(1/Binv**2,Bmag**2))
    W = expand(W.subs(S**2,C**2-1))
    W = W.collect([C,C**2],evaluate=False)

    a = simplify(W[C**2])
    b = simplify(W[C])
    c = simplify(W[one])

    print('#%\\text{Require } aC^{2}+bC+c = 0')

    print('a =',a)
    print('b =',b)
    print('c =',c)

    x = Symbol('x')
    C =  solve(a*x**2+b*x+c,x)[0]
    print('%b^{2}-4ac =',simplify(b**2-4*a*c))
    print('%\\f{\\cosh}{\\alpha} = C = -b/(2a) =',expand(simplify(expand(C))))
    return

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)
    (e1,e2,e3,n,nbar) = c3d.mv()

    print('g_{ij} =',c3d.g)

    e = n+nbar
    #conformal representation of points

    A = make_vector(e1, ga=c3d)    # point a = (1,0,0)  A = F(a)
    B = make_vector(e2, ga=c3d)    # point b = (0,1,0)  B = F(b)
    C = make_vector(-e1, ga=c3d)   # point c = (-1,0,0) C = F(c)
    D = make_vector(e3, ga=c3d)    # point d = (0,0,1)  D = F(d)
    X = make_vector('x',3, ga=c3d)

    print('F(a) =',A)
    print('F(b) =',B)
    print('F(c) =',C)
    print('F(d) =',D)
    print('F(x) =',X)

    print('#a = e1, b = e2, c = -e1, and d = e3')
    print('#A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.')
    print('#Circle through a, b, and c')
    print('Circle: A^B^C^X = 0 =',(A^B^C^X))
    print('#Line through a and b')
    print('Line  : A^B^n^X = 0 =',(A^B^n^X))
    print('#Sphere through a, b, c, and d')
    print('Sphere: A^B^C^D^X = 0 =',(((A^B)^C)^D)^X)
    print('#Plane through a, b, and d')
    print('Plane : A^B^n^D^X = 0 =',(A^B^n^D^X))

    L = (A^B^e)^X

    L.Fmt(3,'Hyperbolic\\;\\; Circle: (A^B^e)^X = 0')
    return

def properties_of_geometric_objects():
    Print_Function()
    global n, nbar
    Fmt(1)
    g = '# # # 0 0,'+ \
        '# # # 0 0,'+ \
        '# # # 0 0,'+ \
        '0 0 0 0 2,'+ \
        '0 0 0 2 0'

    c3d = Ga('p1 p2 p3 n \\bar{n}',g=g)
    (p1,p2,p3,n,nbar) = c3d.mv()

    print('g_{ij} =',c3d.g)

    P1 = F(p1)
    P2 = F(p2)
    P3 = F(p3)

    print('\\text{Extracting direction of line from }L = P1\\W P2\\W n')

    L = P1^P2^n
    delta = (L|n)|nbar
    print('(L|n)|\\bar{n} =',delta)

    print('\\text{Extracting plane of circle from }C = P1\\W P2\\W P3')

    C = P1^P2^P3
    delta = ((C^n)|n)|nbar
    print('((C^n)|n)|\\bar{n}=',delta)
    print('(p2-p1)^(p3-p1)=',(p2-p1)^(p3-p1))
    return

def extracting_vectors_from_conformal_2_blade():
    Print_Function()
    Fmt(1)
    print(r'B = P1\W P2')

    g = '0 -1 #,'+ \
        '-1 0 #,'+ \
        '# # #'

    c2b = Ga('P1 P2 a',g=g)
    (P1,P2,a) = c2b.mv()

    print('g_{ij} =',c2b.g)

    B = P1^P2
    Bsq = B*B
    print('%B^{2} =',Bsq)
    ap = a-(a^B)*B
    print("a' = a-(a^B)*B =",ap)

    Ap = ap+ap*B
    Am = ap-ap*B

    print("A+ = a'+a'*B =",Ap)
    print("A- = a'-a'*B =",Am)

    print('%(A+)^{2} =',Ap*Ap)
    print('%(A-)^{2} =',Am*Am)

    aB = a|B
    print('a|B =',aB)
    return

def reciprocal_frame_test():
    Print_Function()
    Fmt(1)
    g = '1 # #,'+ \
        '# 1 #,'+ \
        '# # 1'

    ng3d = Ga('e1 e2 e3',g=g)
    (e1,e2,e3) = ng3d.mv()

    print('g_{ij} =',ng3d.g)

    E = e1^e2^e3
    Esq = (E*E).scalar()
    print('E =',E)
    print('%E^{2} =',Esq)
    Esq_inv = 1/Esq

    E1 = (e2^e3)*E
    E2 = (-1)*(e1^e3)*E
    E3 = (e1^e2)*E

    print('E1 = (e2^e3)*E =',E1)
    print('E2 =-(e1^e3)*E =',E2)
    print('E3 = (e1^e2)*E =',E3)

    w = (E1|e2)
    w = w.expand()
    print('E1|e2 =',w)

    w = (E1|e3)
    w = w.expand()
    print('E1|e3 =',w)

    w = (E2|e1)
    w = w.expand()
    print('E2|e1 =',w)

    w = (E2|e3)
    w = w.expand()
    print('E2|e3 =',w)

    w = (E3|e1)
    w = w.expand()
    print('E3|e1 =',w)

    w = (E3|e2)
    w = w.expand()
    print('E3|e2 =',w)

    w = (E1|e1)
    w = (w.expand()).scalar()
    Esq = expand(Esq)
    print('%(E1\\cdot e1)/E^{2} =',simplify(w/Esq))

    w = (E2|e2)
    w = (w.expand()).scalar()
    print('%(E2\\cdot e2)/E^{2} =',simplify(w/Esq))

    w = (E3|e3)
    w = (w.expand()).scalar()
    print('%(E3\\cdot e3)/E^{2} =',simplify(w/Esq))
    return

def signature_test():
    Print_Function()

    e3d = Ga('e1 e2 e3',g=[1,1,1])
    print('g =', e3d.g)
    print(r'%Signature = (3,0)\: I =', e3d.I(), r'\: I^{2} =', e3d.I()*e3d.I())

    e3d = Ga('e1 e2 e3',g=[2,2,2])
    print('g =', e3d.g)
    print(r'%Signature = (3,0)\: I =', e3d.I(),'|; I^{2} =', e3d.I()*e3d.I())

    sp4d = Ga('e1 e2 e3 e4',g=[1,-1,-1,-1])
    print('g =', sp4d.g)
    print(r'%Signature = (1,3)\: I =', sp4d.I(), r'\: I^{2} =', sp4d.I()*sp4d.I())

    sp4d = Ga('e1 e2 e3 e4',g=[2,-2,-2,-2])
    print('g =', sp4d.g)
    print(r'%Signature = (1,3)\: I =', sp4d.I(), r'\: I^{2} =', sp4d.I()*sp4d.I())

    e4d = Ga('e1 e2 e3 e4',g=[1,1,1,1])
    print('g =', e4d.g)
    print(r'%Signature = (4,0)\: I =', e4d.I(), r'\: I^{2} =', e4d.I()*e4d.I())

    cf3d = Ga('e1 e2 e3 e4 e5',g=[1,1,1,1,-1])
    print('g =', cf3d.g)
    print(r'%Signature = (4,1)\: I =', cf3d.I(), r'\: I^{2} =', cf3d.I()*cf3d.I())

    cf3d = Ga('e1 e2 e3 e4 e5',g=[2,2,2,2,-2])
    print('g =', cf3d.g)
    print(r'%Signature = (4,1)\: I =', cf3d.I(), r'\: I^{2} =', cf3d.I()*cf3d.I())

    return

def Fmt_test():
    Print_Function()

    e3d = Ga('e1 e2 e3',g=[1,1,1])

    v = e3d.mv('v','vector')
    B = e3d.mv('B','bivector')
    M = e3d.mv('M','mv')

    Fmt(2)

    print('#Global $Fmt = 2$')

    print('v =',v)
    print('B =',B)
    print('M =',M)

    print('#Using $.Fmt()$ Function')

    print('v.Fmt(3) =',v.Fmt(3))
    print('B.Fmt(3) =',B.Fmt(3))
    print('M.Fmt(2) =',M.Fmt(2))
    print('M.Fmt(1) =',M.Fmt(1))

    print('#Global $Fmt = 1$')

    Fmt(1)

    print('v =',v)
    print('B =',B)
    print('M =',M)

    return

def main():
    Format()

    basic_multivector_operations_3D()
    basic_multivector_operations_2D()
    basic_multivector_operations_2D_orthogonal()
    check_generalized_BAC_CAB_formulas()
    rounding_numerical_components()
    derivatives_in_rectangular_coordinates()
    derivatives_in_spherical_coordinates()
    noneuclidian_distance_calculation()
    conformal_representations_of_circles_lines_spheres_and_planes()
    properties_of_geometric_objects()
    extracting_vectors_from_conformal_2_blade()
    reciprocal_frame_test()
    signature_test()
    Fmt_test()

    # xpdf()
    xpdf(pdfprog=None)
    return

if __name__ == "__main__":
    main()