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The reference D&L is "Geometric Algebra for Physicists" by Doran and Lasenby
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from sympy.external import import_module
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numpy = import_module('numpy')
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sys.path.append('../')
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from sympy.galgebra.GA import MV, ZERO, HALF, S
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Conformal Mapping Function from 3D Euclidean space to 5D conformal space
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where the images of all maps are null vectors.
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Fx = HALF*((x*x)*n + 2*x - nbar)
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Test for commutative scalar multiplication. Leftover from when sympy and
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numpy were not working together and __mul__ and __rmul__ would not give the
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x, y, z = MV.setup('x y z')
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a, b, c = sympy.symbols('a b c')
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assert HALF*x == x*HALF
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def test_contraction():
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Test for inner product and left and right contraction
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e_1, e_2, e_3 = MV.setup('e_1 e_2 e_3', '1 0 0, 0 1 0, 0 0 1', offset=1)
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assert ((e_1 ^ e_3) | e_1) == -e_3
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assert ((e_1 ^ e_3) > e_1) == -e_3
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assert (e_1 | (e_1 ^ e_3)) == e_3
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assert (e_1 < (e_1 ^ e_3)) == e_3
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assert ((e_1 ^ e_3) < e_1) == 0
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assert (e_1 > (e_1 ^ e_3)) == 0
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def test_substitution():
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e_x, e_y, e_z = MV.setup('e_x e_y e_z', '1 0 0, 0 1 0, 0 0 1', offset=1)
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x, y, z = sympy.symbols('x y z')
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X = x*e_x + y*e_y + z*e_z
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Y = X.subs([(x, 2), (y, 3), (z, 4)])
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assert Y == 2*e_x + 3*e_y + 4*e_z
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def test_vector_extraction():
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Show that conformal bivector encodes two points. See D&L Section 10.4.1
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metric = ' 0 -1 #,' + \
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P1, P2, a = MV.setup('P1 P2 a', metric)
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P1 and P2 are null vectors and hence encode points in conformal space.
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Show that P1 and P2 can be extracted from the bivector B = P1^P2. a is a
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third vector in the conformal space with a.B not 0.
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P1dota = sympy.Symbol('(P1.a)')
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P2dota = sympy.Symbol('(P2.a)')
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Ap_test = (-2*P2dota)*P1
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Am_test = (-2*P1dota)*P2
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Test specific metrics (diagpq, arbitrary_metric, arbitrary_metric_conformal)
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from sympy.galgebra.GA import diagpq, arbitrary_metric, arbitrary_metric_conformal
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p1, p2, p3 = MV.setup('p1 p2 p3', metric, debug=0)
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x1, y1, z1 = sympy.symbols('x1 y1 z1')
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x2, y2, z2 = sympy.symbols('x2 y2 z2')
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v1 = x1*p1 + y1*p2 + z1*p3
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v2 = x2*p1 + y2*p2 + z2*p3
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prod2 = (v1|v2) + (v1^v2)
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metric = arbitrary_metric(3)
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p1, p2, p3 = MV.setup('p1 p2 p3', metric, debug=0)
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v1 = x1*p1 + y1*p2 + z1*p3
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v2 = x2*p1 + y2*p2 + z2*p3
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prod2 = (v1|v2) + (v1^v2)
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metric = arbitrary_metric_conformal(3)
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p1, p2, p3 = MV.setup('p1 p2 p3', metric, debug=0)
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v1 = x1*p1 + y1*p2 + z1*p3
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v2 = x2*p1 + y2*p2 + z2*p3
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prod2 = (v1|v2) + (v1^v2)
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Test conformal geometric description of circles, lines, spheres, and planes.
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metric = '1 0 0 0 0,' + \
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e0, e1, e2, n, nbar = MV.setup('e0 e1 e2 n nbar', metric, debug=0)
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#conformal representation of points
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A = F(e0, n, nbar) # point a = (1,0,0) A = F(a)
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B = F(e1, n, nbar) # point b = (0,1,0) B = F(b)
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C = F(-1*e0, n, nbar) # point c = (-1,0,0) C = F(c)
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D = F(e2, n, nbar) # point d = (0,0,1) D = F(d)
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x0, x1, x2 = sympy.symbols('x0 x1 x2')
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X = F(MV([x0, x1, x2], 'vector'), n, nbar)
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Circle = A ^ B ^ C ^ X
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Sphere = A ^ B ^ C ^ D ^ X
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Plane = A ^ B ^ n ^ D ^ X
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#Circle through a, b, and c
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Circle_test = -x2*(e0 ^ e1 ^ e2 ^ n) + x2*(
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e0 ^ e1 ^ e2 ^ nbar) + HALF*(-1 + x0**2 + x1**2 + x2**2)*(e0 ^ e1 ^ n ^ nbar)
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diff = Circle - Circle_test
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#Line through a and b
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Line_test = -x2*(e0 ^ e1 ^ e2 ^ n) + \
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HALF*(-1 + x0 + x1)*(e0 ^ e1 ^ n ^ nbar) + \
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(HALF*x2)*(e0 ^ e2 ^ n ^ nbar) + \
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(-HALF*x2)*(e1 ^ e2 ^ n ^ nbar)
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diff = Line - Line_test
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#Sphere through a, b, c, and d
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Sphere_test = HALF*(1 - x0**2 - x1**2 - x2**2)*(e0 ^ e1 ^ e2 ^ n ^ nbar)
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diff = Sphere - Sphere_test
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#Plane through a, b, and d
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Plane_test = HALF*(1 - x0 - x1 - x2)*(e0 ^ e1 ^ e2 ^ n ^ nbar)
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diff = Plane - Plane_test
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def test_extract_plane_and_line():
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Show that conformal trivector encodes planes and lines. See D&L section
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metric = '# # # 0 0,' + \
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p1, p2, p3, n, nbar = MV.setup('p1 p2 p3 n nbar', metric, debug=0)
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#Line through p1 and p2
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delta = (L | n) | nbar
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delta_test = 2*p1 - 2*p2
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diff = delta - delta_test
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#Plane through p1, p2, and p3
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delta = ((C ^ n) | n) | nbar
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delta_test = 2*(p1 ^ p2) - 2*(p1 ^ p3) + 2*(p2 ^ p3)
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diff = delta - delta_test
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def test_reciprocal_frame():
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Test of formula for general reciprocal frame of three vectors.
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Let three independent vectors be e1, e2, and e3. The reciprocal
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vectors E1, E2, and E3 obey the relations:
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e_i.E_j = delta_ij*(e1^e2^e3)**2
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metric = '1 # #,' + \
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e1, e2, e3 = MV.setup('e1 e2 e3', metric)
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E2 = (-1)*(e1 ^ e3)*E
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def test_derivative():
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coords = x, y, z = sympy.symbols('x y z')
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e_x, e_y, e_z = MV.setup('e', '1 0 0, 0 1 0, 0 0 1', coords=coords)
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X = x*e_x + y*e_y + z*e_z
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a = MV('a', 'vector')
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assert ((X | a).grad()) == a
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assert ((X*X).grad()) == 2*X
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assert (X*X*X).grad() == 5*X*X
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assert X.grad_int() == 3
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e_1, e_2, e_3 = MV.setup('e_1 e_2 e_3', '1 0 0, 0 1 0, 0 0 1')
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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'
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Y = MV('y', 'spinor')
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assert str(Y) == 'y+y__01*e_1e_2+y__02*e_1e_3+y__12*e_2e_3'
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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'
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assert str(e_1 | e_1) == '1'
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MV.setup('e_1 e_2 e_3', '[1,1,1]')
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assert str(MV.metric) == '[[1 0 0]\n [0 1 0]\n [0 0 1]]'
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def test_constructor():
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Test various multivector constructors
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e_1, e_2, e_3 = MV.setup('e_1 e_2 e_3', '[1,1,1]')
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x = sympy.symbols('x')
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assert str(S(1)) == '1'
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assert str(S(x)) == 'x'
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assert str(MV('a', 'scalar')) == 'a'
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assert str(MV('a', 'vector')) == 'a__0*e_1+a__1*e_2+a__2*e_3'
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assert str(MV('a', 'pseudo')) == 'a*e_1e_2e_3'
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assert str(MV('a', 'spinor')) == 'a+a__01*e_1e_2+a__02*e_1e_3+a__12*e_2e_3'
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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'
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MV([2, 'a'], 'grade')) == 'a__01*e_1e_2+a__02*e_1e_3+a__12*e_2e_3'
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assert str(MV('a', 'grade2')) == 'a__01*e_1e_2+a__02*e_1e_3+a__12*e_2e_3'
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def test__print_Mul_Add():
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from sympy.galgebra.latex_ex import LatexPrinter
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from sympy import symbols
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n, m = symbols('n,m', negative=True)
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assert l._print_Mul(n*m) == 'm n'
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assert l._print_Mul(-2*m) == '- 2 m'
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assert l._print_Mul(2*m) == '2 m'
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assert l._print_Add(-5 + 4*z) == '-5 + 4 z'
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assert l._print_Add(-5 - 4*z) == '-5 - 4 z'
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assert l._print_Add(n - 2) == '-2 + n'
added
removed
lib/py/sympy/galgebra/tests/test_GA.py
