Python GA.MV类代码示例

def test_extract_plane_and_line():
    """
    Show that conformal trivector encodes planes and lines. See D&L section
    10.4.2
    """
    metric = '# # # 0 0,'+ \
             '# # # 0 0,'+ \
             '# # # 0 0,'+ \
             '0 0 0 0 2,'+ \
             '0 0 0 2 0'

    MV.setup('p1 p2 p3 n nbar',metric,debug=0)
    MV.set_str_format(1)

    ZERO_MV = MV()

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

    #Line through p1 and p2
    L = P1^P2^n
    delta = (L|n)|nbar
    delta_test = 2*p1-2*p2
    diff = delta-delta_test
    diff.compact()
    assert diff == ZERO_MV

    #Plane through p1, p2, and p3
    C = P1^P2^P3
    delta = ((C^n)|n)|nbar
    delta_test = 2*(p1^p2)-2*(p1^p3)+2*(p2^p3)
    diff = delta-delta_test
    diff.compact()
    assert diff == ZERO_MV

def test_vector_extraction():
    """
    Show that conformal bivector encodes two points. See D&L Section 10.4.1
    """
    metric = ' 0 -1 #,'+ \
             '-1  0 #,'+ \
             ' #  # #,'

    MV.setup('P1 P2 a',metric)
    """
    P1 and P2 are null vectors and hence encode points in conformal space.
    Show that P1 and P2 can be extracted from the bivector B = P1^P2. a is a
    third vector in the conformal space with a.B not 0.
    """
    ZERO_MV = MV()
    B = P1^P2
    Bsq = B*B
    ap = a-(a^B)*B
    Ap = ap+ap*B
    Am = ap-ap*B
    Ap_test = (-2*P2dota)*P1
    Am_test = (-2*P1dota)*P2
    Ap.compact()
    Am.compact()
    Ap_test.compact()
    Am_test.compact()
    assert Ap == Ap_test
    assert Am == Am_test
    Ap2 = Ap*Ap
    Am2 = Am*Am
    Ap2.compact()
    Am2.compact()
    assert Ap2 == ZERO_MV
    assert Am2 == ZERO_MV

def test_metrics():
    """
    Test specific metrics (diagpq, arbitrary_metric, arbitrary_metric_conformal)
    """
    from sympy.galgebra.GA import diagpq, arbitrary_metric, arbitrary_metric_conformal
    metric = diagpq(3)
    p1, p2, p3 = MV.setup('p1 p2 p3', metric, debug=0)
    MV.set_str_format(1)
    x1, y1, z1 = sympy.symbols('x1 y1 z1')
    x2, y2, z2 = sympy.symbols('x2 y2 z2')
    v1 = x1*p1 + y1*p2 + z1*p3
    v2 = x2*p1 + y2*p2 + z2*p3
    prod1 = v1*v2
    prod2 = (v1|v2) + (v1^v2)
    diff = prod1 - prod2
    diff.compact()
    assert diff == ZERO
    metric = arbitrary_metric(3)
    p1, p2, p3 = MV.setup('p1 p2 p3', metric, debug=0)
    v1 = x1*p1 + y1*p2 + z1*p3
    v2 = x2*p1 + y2*p2 + z2*p3
    prod1 = v1*v2
    prod2 = (v1|v2) + (v1^v2)
    diff = prod1 - prod2
    diff.compact()
    assert diff == ZERO
    metric = arbitrary_metric_conformal(3)
    p1, p2, p3 = MV.setup('p1 p2 p3', metric, debug=0)
    v1 = x1*p1 + y1*p2 + z1*p3
    v2 = x2*p1 + y2*p2 + z2*p3
    prod1 = v1*v2
    prod2 = (v1|v2) + (v1^v2)
    diff = prod1 - prod2
    diff.compact()
    assert diff == ZERO

def test_substitution():

    MV.setup("e_x e_y e_z", "1 0 0, 0 1 0, 0 0 1", offset=1)
    make_symbols("x y z")

    X = x * e_x + y * e_y + z * e_z
    Y = X.subs([(x, 2), (y, 3), (z, 4)])
    assert Y == 2 * e_x + 3 * e_y + 4 * e_z

def test_substitution():

    MV.setup('e_x e_y e_z','1 0 0, 0 1 0, 0 0 1',offset=1)
    make_symbols('x y z')

    X = x*e_x+y*e_y+z*e_z
    Y = X.subs([(x,2),(y,3),(z,4)])
    assert Y == 2*e_x+3*e_y+4*e_z

def test_derivative():
    coords = make_symbols('x y z')
    MV.setup('e','1 0 0, 0 1 0, 0 0 1',coords=coords)
    X = x*e_x+y*e_y+z*e_z
    a = MV('a','vector')

    assert ((X|a).grad()) == a
    assert ((X*X).grad()) == 2*X
    assert (X*X*X).grad() == 5*X*X
    assert X.grad_int() == 3

def test_str():
    MV.setup('e_1 e_2 e_3','1 0 0, 0 1 0, 0 0 1')

    X = MV('x')
    assert str(X) == 'x+x__0*e_1+x__1*e_2+x__2*e_3+x__01*e_1e_2+x__02*e_1e_3+x__12*e_2e_3+x__012*e_1e_2e_3'
    Y = MV('y','spinor')
    assert str(Y) == 'y+y__01*e_1e_2+y__02*e_1e_3+y__12*e_2e_3'
    Z = X+Y
    assert str(Z) == 'x+y+x__0*e_1+x__1*e_2+x__2*e_3+(x__01+y__01)*e_1e_2+(x__02+y__02)*e_1e_3+(x__12+y__12)*e_2e_3+x__012*e_1e_2e_3'
    assert str(e_1|e_1) == '1'

def test_derivative():
    coords = make_symbols("x y z")
    MV.setup("e", "1 0 0, 0 1 0, 0 0 1", coords=coords)
    X = x * e_x + y * e_y + z * e_z
    a = MV("a", "vector")

    assert ((X | a).grad()) == a
    assert ((X * X).grad()) == 2 * X
    assert (X * X * X).grad() == 5 * X * X
    assert X.grad_int() == 3

def test_rmul():
    """
    Test for commutative scalar multiplication.  Leftover from when sympy and
    numpy were not working together and __mul__ and __rmul__ would not give the
    same answer.
    """
    MV.setup('x y z')
    make_symbols('a b c')
    assert 5*x == x*5
    assert HALF*x == x*HALF
    assert a*x == x*a

def test_geometry():
    """
    Test conformal geometric description of circles, lines, spheres, and planes.
    """
    metric = "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"

    MV.setup("e0 e1 e2 n nbar", metric, debug=0)
    e = n + nbar
    # conformal representation of points
    ZERO_MV = MV()

    A = make_vector(e0)  # point a = (1,0,0)  A = F(a)
    B = make_vector(e1)  # point b = (0,1,0)  B = F(b)
    C = make_vector(-1 * e0)  # point c = (-1,0,0) C = F(c)
    D = make_vector(e2)  # point d = (0,0,1)  D = F(d)
    X = make_vector("x", 3)

    Circle = A ^ B ^ C ^ X
    Line = A ^ B ^ n ^ X
    Sphere = A ^ B ^ C ^ D ^ X
    Plane = A ^ B ^ n ^ D ^ X

    # Circle through a, b, and c
    Circle_test = (
        -x2 * (e0 ^ e1 ^ e2 ^ n)
        + x2 * (e0 ^ e1 ^ e2 ^ nbar)
        + HALF * (-1 + x0 ** 2 + x1 ** 2 + x2 ** 2) * (e0 ^ e1 ^ n ^ nbar)
    )
    diff = Circle - Circle_test
    diff.compact()
    assert diff == ZERO_MV

    # Line through a and b
    Line_test = (
        -x2 * (e0 ^ e1 ^ e2 ^ n)
        + HALF * (-1 + x0 + x1) * (e0 ^ e1 ^ n ^ nbar)
        + (HALF * x2) * (e0 ^ e2 ^ n ^ nbar)
        + (-HALF * x2) * (e1 ^ e2 ^ n ^ nbar)
    )
    diff = Line - Line_test
    diff.compact()
    assert diff == ZERO_MV

    # Sphere through a, b, c, and d
    Sphere_test = HALF * (1 - x0 ** 2 - x1 ** 2 - x2 ** 2) * (e0 ^ e1 ^ e2 ^ n ^ nbar)
    diff = Sphere - Sphere_test
    diff.compact()
    assert diff == ZERO_MV

    # Plane through a, b, and d
    Plane_test = HALF * (1 - x0 - x1 - x2) * (e0 ^ e1 ^ e2 ^ n ^ nbar)
    diff = Plane - Plane_test
    diff.compact()
    assert diff == ZERO_MV

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
    """
    metric = '1 # #,'+ \
             '# 1 #,'+ \
             '# # 1,'

    MV.setup('e1 e2 e3',metric)
    E = e1^e2^e3
    Esq = (E*E)()
    Esq_inv = 1/Esq
    E1 = (e2^e3)*E
    E2 = (-1)*(e1^e3)*E
    E3 = (e1^e2)*E
    w = (E1|e2)
    w.collect(MV.g)
    w = w().expand()
    w = (E1|e3)
    w.collect(MV.g)
    w = w().expand()
    assert w == 0
    w = (E2|e1)
    w.collect(MV.g)
    w = w().expand()
    assert w == 0
    w = (E2|e3)
    w.collect(MV.g)
    w = w().expand()
    assert w == 0
    w = (E3|e1)
    w.collect(MV.g)
    w = w().expand()
    assert w == 0
    w = (E3|e2)
    w.collect(MV.g)
    w = w().expand()
    assert w == 0
    w = (E1|e1)
    w = w().expand()
    Esq = Esq.expand()
    assert w/Esq == 1
    w = (E2|e2)
    w = w().expand()
    assert w/Esq == 1
    w = (E3|e3)
    w = w().expand()
    assert w/Esq == 1

def test_str():
    MV.setup("e_1 e_2 e_3", "1 0 0, 0 1 0, 0 0 1")

    X = MV("x")
    assert str(X) == "x+x__0*e_1+x__1*e_2+x__2*e_3+x__01*e_1e_2+x__02*e_1e_3+x__12*e_2e_3+x__012*e_1e_2e_3"
    Y = MV("y", "spinor")
    assert str(Y) == "y+y__01*e_1e_2+y__02*e_1e_3+y__12*e_2e_3"
    Z = X + Y
    assert (
        str(Z)
        == "x+y+x__0*e_1+x__1*e_2+x__2*e_3+(x__01+y__01)*e_1e_2+(x__02+y__02)*e_1e_3+(x__12+y__12)*e_2e_3+x__012*e_1e_2e_3"
    )
    assert str(e_1 | e_1) == "1"

def test_contraction():
    """
    Test for inner product and left and right contraction
    """

    MV.setup('e_1 e_2 e_3','1 0 0, 0 1 0, 0 0 1',offset=1)

    assert ((e_1^e_3)|e_1) == -e_3
    assert ((e_1^e_3)>e_1) == -e_3
    assert (e_1|(e_1^e_3)) == e_3
    assert (e_1<(e_1^e_3)) == e_3
    assert ((e_1^e_3)<e_1) == 0
    assert (e_1>(e_1^e_3)) == 0

def test_constructor():
    """
    Test various multivector constructors
    """
    MV.setup("e_1 e_2 e_3", "[1,1,1]")
    make_symbols("x")
    assert str(S(1)) == "1"
    assert str(S(x)) == "x"
    assert str(MV("a", "scalar")) == "a"
    assert str(MV("a", "vector")) == "a__0*e_1+a__1*e_2+a__2*e_3"
    assert str(MV("a", "pseudo")) == "a*e_1e_2e_3"
    assert str(MV("a", "spinor")) == "a+a__01*e_1e_2+a__02*e_1e_3+a__12*e_2e_3"
    assert str(MV("a")) == "a+a__0*e_1+a__1*e_2+a__2*e_3+a__01*e_1e_2+a__02*e_1e_3+a__12*e_2e_3+a__012*e_1e_2e_3"
    assert str(MV([2, "a"], "grade")) == "a__01*e_1e_2+a__02*e_1e_3+a__12*e_2e_3"
    assert str(MV("a", "grade2")) == "a__01*e_1e_2+a__02*e_1e_3+a__12*e_2e_3"

def test_constructor():
    """
    Test various multivector constructors
    """
    MV.setup('e_1 e_2 e_3','[1,1,1]')
    make_symbols('x')
    assert str(S(1)) == '1'
    assert str(S(x)) == 'x'
    assert str(MV('a','scalar')) == 'a'
    assert str(MV('a','vector')) == 'a__0*e_1+a__1*e_2+a__2*e_3'
    assert str(MV('a','pseudo')) == 'a*e_1e_2e_3'
    assert str(MV('a','spinor')) == 'a+a__01*e_1e_2+a__02*e_1e_3+a__12*e_2e_3'
    assert str(MV('a')) == 'a+a__0*e_1+a__1*e_2+a__2*e_3+a__01*e_1e_2+a__02*e_1e_3+a__12*e_2e_3+a__012*e_1e_2e_3'
    assert str(MV([2,'a'],'grade')) == 'a__01*e_1e_2+a__02*e_1e_3+a__12*e_2e_3'
    assert str(MV('a','grade2')) == 'a__01*e_1e_2+a__02*e_1e_3+a__12*e_2e_3'