Python Ga.mv方法代码示例

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

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

# 需要导入模块: from ga import Ga [as 别名]
# 或者: from ga.Ga import mv [as 别名]
    def test_3_2_2(self):
        """
        Computing the contraction explicitly.
        """

        Ga.dual_mode("Iinv+")

        R = Ga('e*1|2|3')
        A_blades = [R.mv('A', i, 'grade') for i in range(R.n + 1)]
        B_blades = [R.mv('B', i, 'grade') for i in range(R.n + 1)]
        C_blades = [R.mv('C', i, 'grade') for i in range(R.n + 1)]

        # scalar and blades of various grades
        A = A_blades[0]
        for B in B_blades:
            self.assertEquals(A < B, A * B)

        A = A_blades[0]
        for B in B_blades:
            self.assertEquals(B < A, 0 if B.pure_grade() > 0 else A * B)

        # vectors
        A = A_blades[1]
        B = B_blades[1]
        self.assertEquals(A < B, A | B)

        # vector and the outer product of 2 blades of various grades (scalars, vectors, 2-vectors...)
        A = A_blades[1]
        for B, C in product(B_blades, C_blades):
            self.assertEquals(A < (B ^ C), ((A < B) ^ C) + (-1)**B.pure_grade() * (B ^ (A < C)))

        # vector and the outer product of 2 blades of various grades (scalars, vectors, 2-vectors...)
        for A, B, C in product(A_blades, B_blades, C_blades):
            self.assertEquals((A ^ B) < C, A < (B < C))

        # distributive properties
        for A, B, C in product(A_blades, B_blades, C_blades):
            self.assertEquals((A + B) < C, (A < C) + (B < C))

        for A, B, C in product(A_blades, B_blades, C_blades):
            self.assertEquals(A < (B + C), (A < B) + (A < C))

        alpha = Symbol("alpha")
        for A, B in product(A_blades, B_blades):
            self.assertEquals((alpha * A) < B, alpha * (A < B))
            self.assertEquals((alpha * A) < B, A < (alpha * B))

        a = R.mv('a', 1, 'grade')
        for A_minus1, B in product(A_blades[:-1], B_blades):
            A = A_minus1 ^ a
            self.assertEquals(A < B, (A_minus1 ^ a) < B)
            self.assertEquals(A < B, A_minus1 < (a < B))

# 需要导入模块: from ga import Ga [as 别名]
# 或者: from 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 \\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

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

# 需要导入模块: from ga import Ga [as 别名]
# 或者: from 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 ga import Ga [as 别名]
# 或者: from 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 ga import Ga [as 别名]
# 或者: from ga.Ga import mv [as 别名]
    def test_3_4(self):
        """
        The other contraction.
        """

        Ga.dual_mode("Iinv+")

        R = Ga('e*1|2|3')
        A_blades = [R.mv('A', i, 'grade') for i in range(R.n + 1)]
        B_blades = [R.mv('B', i, 'grade') for i in range(R.n + 1)]

        for A, B in product(A_blades, B_blades):
            self.assertEquals(B > A, ((-1) ** (A.pure_grade() * (B.pure_grade() - 1))) * (A < B))

# 需要导入模块: from ga import Ga [as 别名]
# 或者: from 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 ga import Ga [as 别名]
# 或者: from 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 ga import Ga [as 别名]
# 或者: from ga.Ga import mv [as 别名]
    def test_3_5_4(self):
        """
        The duality relationships.
        """

        Ga.dual_mode("Iinv+")

        R = Ga('e*1|2|3')
        A_blades = [R.mv('A', i, 'grade') for i in range(R.n + 1)]
        B_blades = [R.mv('B', i, 'grade') for i in range(R.n + 1)]

        for A, B in product(A_blades, B_blades):
            self.assertEquals((A ^ B).dual(), A < B.dual())

        for A, B in product(A_blades, B_blades):
            self.assertEquals((A < B).dual(), A ^ B.dual())

# 需要导入模块: from ga import Ga [as 别名]
# 或者: from 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 ga import Ga [as 别名]
# 或者: from ga.Ga import mv [as 别名]
    def test_3_5_2(self):
        """
        The inverse of a blade.
        """

        Ga.dual_mode("Iinv+")

        R = Ga('e*1|2|3')
        A_blades = [R.mv('A', i, 'grade') for i in range(R.n + 1)]

        for A in A_blades:
            self.assertEquals(A.inv(), ((-1) ** (A.pure_grade() * (A.pure_grade() - 1) / 2)) * (A / A.norm2()))

        for A in A_blades:
            self.assertEquals(A < A.inv(), 1)

        A = A_blades[1]
        self.assertEquals(A.inv(), A / A.norm2())

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

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