Python Ga.mv方法代码示例

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
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', f=True)
    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

    print 'grad<A =', grad < A
    print 'grad>A =', grad > A
    print 'grad<B =', grad < B
    print 'grad>B =', grad > B
    print 'grad<C =', grad < C
    print 'grad>C =', grad > C

    return

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
def basic_multivector_operations_3D():
    Print_Function()

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

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

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

    A.even().Fmt(1,'%A_{+}')
    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')

    (X*Y).Fmt(2,'X*Y')
    (X^Y).Fmt(2,'X^Y')
    (X|Y).Fmt(2,'X|Y')
    return

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
def properties_of_geometric_objects():
    Print_Function()
    global n, nbar

    g = '# # # 0 0,'+ \
        '# # # 0 0,'+ \
        '# # # 0 0,'+ \
        '0 0 0 0 2,'+ \
        '0 0 0 2 0'

    c3d = Ga('p1 p2 p3 n nbar',g=g)

    (p1,p2,p3,n,nbar) = c3d.mv()

    print 'g_{ij} =\n',c3d.g

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

    print 'Extracting direction of line from L = P1^P2^n'

    L = P1^P2^n
    delta = (L|n)|nbar
    print '(L|n)|nbar =',delta

    print 'Extracting plane of circle from C = P1^P2^P3'

    C = P1^P2^P3
    delta = ((C^n)|n)|nbar
    print '((C^n)|n)|nbar =',delta
    print '(p2-p1)^(p3-p1) =',(p2-p1)^(p3-p1)

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
def extracting_vectors_from_conformal_2_blade():
    Print_Function()

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

    e2b = Ga('P1 P2 a',g=g)

    (P1,P2,a) = e2b.mv()

    print 'g_{ij} =\n',e2b.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

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
def test_reciprocal_frame():
    """
    Test of formula for general reciprocal frame of three vectors.
    Let three independent vectors be e1, e2, and e3. The reciprocal
    vectors E1, E2, and E3 obey the relations:

    e_i.E_j = delta_ij*(e1^e2^e3)**2
    """
    g = '1 # #,'+ \
        '# 1 #,'+ \
        '# # 1'

    g3dn = Ga('e1 e2 e3',g=g)

    (e1,e2,e3) = g3dn.mv()

    E = e1^e2^e3
    Esq = (E*E).scalar()
    Esq_inv = 1 / Esq

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

    w = (E1|e2)
    w = w.expand()
    assert w.scalar() == 0

    w = (E1|e3)
    w = w.expand()
    assert w.scalar() == 0

    w = (E2|e1)
    w = w.expand()
    assert w.scalar() == 0

    w = (E2|e3)
    w = w.expand()
    assert w.scalar() == 0

    w = (E3|e1)
    w = w.expand()
    assert w.scalar() == 0

    w = (E3|e2)
    w = w.expand()
    assert w.scalar() == 0

    w = (E1|e1)
    w = (w.expand()).scalar()
    Esq = expand(Esq)
    assert simplify(w/Esq) == 1

    w = (E2|e2)
    w = (w.expand()).scalar()
    assert simplify(w/Esq) == 1

    w = (E3|e3)
    w = (w.expand()).scalar()
    assert simplify(w/Esq) == 1

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
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')

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

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

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
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.i*(grad^A)).simplify()
    print 'grad^B =',grad^B

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
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

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
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')

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

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

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

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
class LieAlgebra(object):

    def __init__(self,n):
        self.n = n
        e = ''
        ebar = ''
        for i in range(1,n+1):
            e += ' e_' + str(i)
            if GaLatexPrinter.latex_flg:
                ebar += r' \bar{e}_' + str(i)
            else:
                ebar += ' ebar_' + str(i)

        g = n * [1] + n * [-1]
        basis = e[1:] + ebar

        self.Ga = Ga(basis, g=g)
        self.basis = self.Ga.mv()

        self.e = self.basis[:n]
        self.ebar = self.basis[n:]

        self.w = []
        self.wstar = []

        for i in range(n):
            self.w.append(self.e[i] + self.ebar[i])
            self.wstar.append(self.e[i] - self.ebar[i])

        self.Nu_bais = self.w + self.wstar
        self.Eij = []
        self.Fij = []
        self.Ki = []
        for i in range(n):
            self.Ki.append(self.e[i] * self.ebar[i])
            print r'%F_{'+str(i)+'} =',self.Ki[-1] * self.Ki[-1].rev()
            for j in range(i):
                self.Eij.append(self.e[i] * self.e[j] - self.ebar[i] * self.ebar[j])
                self.Fij.append(self.e[i] * self.ebar[j] - self.ebar[i] * self.e[j])
                print r'%E_{'+str(i)+str(j)+'} =',self.Eij[-1] * self.Eij[-1].rev()
                print r'%F_{'+str(i)+str(j)+'} =',self.Fij[-1] * self.Fij[-1].rev()

        print 'K_{i} =',self.Ki
        print 'E_{ij} =',self.Eij
        print 'F_{ij} =',self.Fij

        E = self.Eij[0]/2

        for i in range(2*n):
            print E
            E *= self.Eij[0]/2

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
def check_generalized_BAC_CAB_formulas():
    Print_Function()
    g4d = Ga('a b c d')
    (a,b,c,d) = 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)} =',Com(a^b,c^d)
    return

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
def LieBasis(n):
    g = n * [1] + n * [-1]
    basis = ''
    for i in range(1,n+1):
        basis += 'e_' +str(i) + ' '
    for i in range(1,n+1):
        basis += r'\bar{e}_' +str(i) + ' '
    basis = basis[:-1]

    LieGA = Ga(basis,g=g)
    bases = LieGA.mv()
    e = bases[:n]
    ebar = bases[n:]
    print e
    print ebar

    print LieGA.g

    E = []
    F = []
    K = []
    indexes = []

    for i in range(n):
        K.append(e[i]*ebar[i])
        for j in range(n):
            if i < j:
                indexes.append((i+1,j+1))
                E.append(e[i]*e[j]-ebar[i]*ebar[j])
                F.append(e[i]*ebar[j]-ebar[i]*e[j])

    print indexes
    print 'E =',E
    print 'F =',F
    print 'K =',K,'\n'

    for k in range(len(indexes)):
        k_i = indexes[k][0]
        k_j = indexes[k][1]
        for l in range(len(indexes)):
            l_i = indexes[l][0]
            l_j = indexes[l][1]
            print 'E_'+str(k_i)+str(k_j)+' x F_'+str(l_i)+str(l_j)+' = '+str(Com(E[k],F[l]))

    return

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
def conformal_representations_of_circles_lines_spheres_and_planes():
    global n,nbar
    Print_Function()

    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'

    cnfml3d = Ga('e_1 e_2 e_3 n nbar',g=g)

    (e1,e2,e3,n,nbar) = cnfml3d.mv()

    print 'g_{ij} =\n',cnfml3d.g

    e = n+nbar
    #conformal representation of points

    A = make_vector(e1,ga=cnfml3d)    # point a = (1,0,0)  A = F(a)
    B = make_vector(e2,ga=cnfml3d)    # point b = (0,1,0)  B = F(b)
    C = make_vector(-e1,ga=cnfml3d)   # point c = (-1,0,0) C = F(c)
    D = make_vector(e3,ga=cnfml3d)    # point d = (0,0,1)  D = F(d)
    X = make_vector('x',3,ga=cnfml3d)

    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

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
def basic_multivector_operations():
    Print_Function()
    g3d = Ga('e*x|y|z')
    (ex, ey, ez) = g3d.mv()

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

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

    X = g3d.mv('X', 'vector')
    Y = g3d.mv('Y', 'vector')

    print 'g_{ij} =\n', g3d.g

    X.Fmt(1, 'X')
    Y.Fmt(1, 'Y')

    (X * Y).Fmt(2, 'X*Y')
    (X ^ Y).Fmt(2, 'X^Y')
    (X | Y).Fmt(2, 'X|Y')

    g2d = Ga('e*x|y')

    (ex, ey) = g2d.mv()

    print 'g_{ij} =\n', g2d.g

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

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

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

    o2d = Ga('e*x|y', g=[1, 1])

    (ex, ey) = o2d.mv()

    print 'g_{ii} =\n', o2d.g

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

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

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

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

# 需要导入模块: from sympy.galgebra.ga import Ga [as 别名]
# 或者: from sympy.galgebra.ga.Ga import mv [as 别名]
def reciprocal_frame_test():
    Print_Function()

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

    g3dn = Ga('e1 e2 e3',g=g)

    (e1,e2,e3) = g3dn.mv()

    print 'g_{ij} =\n',g3dn.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|e1)/E**2 =',simplify(w/Esq)

    w = (E2|e2)
    w = (w.expand()).scalar()
    print '(E2|e2)/E**2 =',simplify(w/Esq)

    w = (E3|e3)
    w = (w.expand()).scalar()
    print '(E3|e3)/E**2 =',simplify(w/Esq)
    return