def test_str():
e_1, e_2, e_3 = 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__1*e_1 + x__2*e_2 + x__3*e_3 + x__12*e_1^e_2 + x__13*e_1^e_3 + x__23*e_2^e_3 + x__123**e_1^e_2^e_3'
Y = MV('y', 'spinor')
assert str(Y) == 'y + y__12*e_1^e_2 + y__13*e_1^e_3 + y__23*e_2^e_3'
Z = X + Y
assert str(Z) == 'x + y + x__1*e_1 + x__2*e_2 + x__3*e_3 + (x__12 + y__12)*e_1^e_2 + (x__13 + y__13)*e_1^e_3 + (x__23 + y__23)*e_2^e_3 + x__123*e_1^e_2^e_3'
assert str(e_1 | e_1) == '1'
def test_derivative():
coords = x, y, z = symbols('x y z')
e_x, e_y, e_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_metrics_xfail():
from sympy.galgebra.ga import arbitrary_metric_conformal
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
assert diff == MV(S.Zero)
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'
e0, e1, e2, n, nbar = MV.setup('e0 e1 e2 n nbar', metric, debug=0)
e = n + nbar
#conformal representation of points
A = F(e0, n, nbar) # point a = (1,0,0) A = F(a)
B = F(e1, n, nbar) # point b = (0,1,0) B = F(b)
C = F(-1 * e0, n, nbar) # point c = (-1,0,0) C = F(c)
D = F(e2, n, nbar) # point d = (0,0,1) D = F(d)
x0, x1, x2 = symbols('x0 x1 x2')
X = F(MV([x0, x1, x2], 'vector'), n, nbar)
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) + Rational(
1, 2) * (-1 + x0**2 + x1**2 + x2**2) * (e0 ^ e1 ^ n ^ nbar)
diff = Circle - Circle_test
assert diff == S.Zero
#Line through a and b
Line_test = -x2*(e0 ^ e1 ^ e2 ^ n) + \
Rational(1, 2)*(-1 + x0 + x1)*(e0 ^ e1 ^ n ^ nbar) + \
(Rational(1, 2)*x2)*(e0 ^ e2 ^ n ^ nbar) + \
(-Rational(1, 2)*x2)*(e1 ^ e2 ^ n ^ nbar)
diff = Line - Line_test
assert diff == S.Zero
#Sphere through a, b, c, and d
Sphere_test = Rational(
1, 2) * (1 - x0**2 - x1**2 - x2**2) * (e0 ^ e1 ^ e2 ^ n ^ nbar)
diff = Sphere - Sphere_test
assert diff == S.Zero
#Plane through a, b, and d
Plane_test = Rational(1,
2) * (1 - x0 - x1 - x2) * (e0 ^ e1 ^ e2 ^ n ^ nbar)
diff = Plane - Plane_test
assert diff == S.Zero
def test_basic_multivector_operations():
with GA_Printer():
(ex, ey, ez) = MV.setup('e*x|y|z')
A = MV('A', 'mv')
assert str(A) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z'
assert str(A) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z'
assert str(A) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z'
X = MV('X', 'vector')
Y = MV('Y', 'vector')
assert str(X) == 'X__x*e_x + X__y*e_y + X__z*e_z'
assert str(Y) == 'Y__x*e_x + Y__y*e_y + Y__z*e_z'
assert str((X*Y)) == '(e_x.e_x)*X__x*Y__x + (e_x.e_y)*X__x*Y__y + (e_x.e_y)*X__y*Y__x + (e_x.e_z)*X__x*Y__z + (e_x.e_z)*X__z*Y__x + (e_y.e_y)*X__y*Y__y + (e_y.e_z)*X__y*Y__z + (e_y.e_z)*X__z*Y__y + (e_z.e_z)*X__z*Y__z + (X__x*Y__y - X__y*Y__x)*e_x^e_y + (X__x*Y__z - X__z*Y__x)*e_x^e_z + (X__y*Y__z - X__z*Y__y)*e_y^e_z'
assert str((X ^ Y)) == '(X__x*Y__y - X__y*Y__x)*e_x^e_y + (X__x*Y__z - X__z*Y__x)*e_x^e_z + (X__y*Y__z - X__z*Y__y)*e_y^e_z'
assert str((X | Y)) == '(e_x.e_x)*X__x*Y__x + (e_x.e_y)*X__x*Y__y + (e_x.e_y)*X__y*Y__x + (e_x.e_z)*X__x*Y__z + (e_x.e_z)*X__z*Y__x + (e_y.e_y)*X__y*Y__y + (e_y.e_z)*X__y*Y__z + (e_y.e_z)*X__z*Y__y + (e_z.e_z)*X__z*Y__z'
(ex, ey) = MV.setup('e*x|y')
X = MV('X', 'vector')
A = MV('A', 'spinor')
assert str(X) == 'X__x*e_x + X__y*e_y'
assert str(A) == 'A + A__xy*e_x^e_y'
assert str((X | A)) == '(-A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y))*e_x + (A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y))*e_y'
assert str((X < A)) == '(-A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y))*e_x + (A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y))*e_y'
assert str((A > X)) == '(A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y))*e_x + (-A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y))*e_y'
(ex, ey) = MV.setup('e*x|y', metric='[1,1]')
X = MV('X', 'vector')
A = MV('A', 'spinor')
assert str(X) == 'X__x*e_x + X__y*e_y'
assert str(A) == 'A + A__xy*e_x^e_y'
assert str((X*A)) == '(A*X__x - A__xy*X__y)*e_x + (A*X__y + A__xy*X__x)*e_y'
assert str((X | A)) == '-A__xy*X__y*e_x + A__xy*X__x*e_y'
assert str((X < A)) == '-A__xy*X__y*e_x + A__xy*X__x*e_y'
assert str((X > A)) == 'A*X__x*e_x + A*X__y*e_y'
assert str((A*X)) == '(A*X__x + A__xy*X__y)*e_x + (A*X__y - A__xy*X__x)*e_y'
assert str((A | X)) == 'A__xy*X__y*e_x - A__xy*X__x*e_y'
assert str((A < X)) == 'A*X__x*e_x + A*X__y*e_y'
assert str((A > X)) == 'A__xy*X__y*e_x - A__xy*X__x*e_y'
return
def test_derivatives_in_rectangular_coordinates():
GA_Printer.on()
X = (x, y, z) = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('e_x e_y e_z', metric='[1,1,1]', coords=X)
f = MV('f', 'scalar', fct=True)
A = MV('A', 'vector', fct=True)
B = MV('B', 'grade2', fct=True)
C = MV('C', 'mv', fct=True)
assert str(f) == 'f'
assert str(A) == 'A__x*e_x + A__y*e_y + A__z*e_z'
assert str(B) == 'B__xy*e_x^e_y + B__xz*e_x^e_z + B__yz*e_y^e_z'
assert str(
C
) == 'C + C__x*e_x + C__y*e_y + C__z*e_z + C__xy*e_x^e_y + C__xz*e_x^e_z + C__yz*e_y^e_z + C__xyz*e_x^e_y^e_z'
assert str(grad * f) == 'D{x}f*e_x + D{y}f*e_y + D{z}f*e_z'
assert str(grad | A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(
grad * A
) == 'D{x}A__x + D{y}A__y + D{z}A__z + (-D{y}A__x + D{x}A__y)*e_x^e_y + (-D{z}A__x + D{x}A__z)*e_x^e_z + (-D{z}A__y + D{y}A__z)*e_y^e_z'
assert str(
-MV.I * (grad ^ A)
) == '(-D{z}A__y + D{y}A__z)*e_x + (D{z}A__x - D{x}A__z)*e_y + (-D{y}A__x + D{x}A__y)*e_z'
assert str(
grad * B
) == '(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z + (D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z'
assert str(grad ^ B) == '(D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z'
assert str(
grad | B
) == '(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z'
assert str(grad < A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(grad > A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(
grad < B
) == '(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z'
assert str(grad > B) == '0'
assert str(
grad < C
) == 'D{x}C__x + D{y}C__y + D{z}C__z + (-(D{y}C__xy + D{z}C__xz))*e_x + (D{x}C__xy - D{z}C__yz)*e_y + (D{x}C__xz + D{y}C__yz)*e_z + D{z}C__xyz*e_x^e_y - D{y}C__xyz*e_x^e_z + D{x}C__xyz*e_y^e_z'
assert str(
grad > C
) == 'D{x}C__x + D{y}C__y + D{z}C__z + D{x}C*e_x + D{y}C*e_y + D{z}C*e_z'
GA_Printer.off()
return
def Distorted_manifold_with_scalar_function():
Print_Function()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('ex ey ez', metric='[1,1,1]', coords=coords)
mfvar = (u, v) = symbols('u v')
X = 2 * u * ex + 2 * v * ey + (u**3 + v**3 / 2) * ez
MF = Manifold(X, mfvar, I=MV.I)
(eu, ev) = MF.Basis()
g = (v + 1) * log(u)
dg = MF.Grad(g)
print('g =', g)
print('dg =', dg)
print('dg(1,0) =', dg.subs({u: 1, v: 0}))
G = u * eu + v * ev
dG = MF.Grad(G)
print('G =', G)
print('P(G) =', MF.Proj(G))
print('zcoef =', simplify(2 * (u**2 + v**2) * (-4 * u**2 - 4 * v**2 - 1)))
print('dG =', dG)
print('P(dG) =', MF.Proj(dG))
PS = u * v * eu ^ ev
print('PS =', PS)
print('dPS =', MF.Grad(PS))
print('P(dPS) =', MF.Proj(MF.Grad(PS)))
return
def Test_Reciprocal_Frame():
Print_Function()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('ex ey ez', metric='[1,1,1]', coords=coords)
mfvar = (u, v) = symbols('u v')
eu = ex + ey
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('Frame', (eu, ev), 'Reciprocal Frame', (eu_r, ev_r))
print('eu.eu_r =', eu | eu_r)
print('eu.ev_r =', eu | ev_r)
print('ev.eu_r =', ev | eu_r)
print('ev.ev_r =', ev | ev_r)
eu = ex + ey + ez
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('Frame', (eu, ev), 'Reciprocal Frame', (eu_r, ev_r))
print('eu.eu_r =', eu | eu_r)
print('eu.ev_r =', eu | ev_r)
print('ev.eu_r =', ev | eu_r)
print('ev.ev_r =', ev | ev_r)
return
def Distorted_manifold_with_scalar_function():
Print_Function()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('ex ey ez', metric='[1,1,1]', coords=coords)
mfvar = (u, v) = symbols('u v')
X = 2*u*ex + 2*v*ey + (u**3 + v**3/2)*ez
MF = Manifold(X, mfvar, I=MV.I)
(eu, ev) = MF.Basis()
g = (v + 1)*log(u)
dg = MF.Grad(g)
print('g =', g)
print('dg =', dg)
print('dg(1,0) =', dg.subs({u: 1, v: 0}))
G = u*eu + v*ev
dG = MF.Grad(G)
print('G =', G)
print('P(G) =', MF.Proj(G))
print('zcoef =', simplify(2*(u**2 + v**2)*(-4*u**2 - 4*v**2 - 1)))
print('dG =', dG)
print('P(dG) =', MF.Proj(dG))
PS = u*v*eu ^ ev
print('PS =', PS)
print('dPS =', MF.Grad(PS))
print('P(dPS) =', MF.Proj(MF.Grad(PS)))
return
def test_vector_extraction():
"""
Show that conformal bivector encodes two points. See D&L Section 10.4.1
"""
metric = ' 0 -1 #,' + \
'-1 0 #,' + \
' # # #,'
P1, P2, a = 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.
"""
B = P1 ^ P2
Bsq = B * B
ap = a - (a ^ B) * B
Ap = ap + ap * B
Am = ap - ap * B
P1dota = Symbol('(P1.a)')
P2dota = Symbol('(P2.a)')
Ap_test = (-2 * P2dota) * P1
Am_test = (-2 * P1dota) * P2
assert Ap == Ap_test
assert Am == Am_test
Ap2 = Ap * Ap
Am2 = Am * Am
assert Ap2 == S.Zero
assert Am2 == S.Zero
def Test_Reciprocal_Frame():
Print_Function()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('ex ey ez', metric='[1,1,1]', coords=coords)
mfvar = (u, v) = symbols('u v')
eu = ex + ey
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('Frame', (eu, ev), 'Reciprocal Frame', (eu_r, ev_r))
print('eu.eu_r =', eu | eu_r)
print('eu.ev_r =', eu | ev_r)
print('ev.eu_r =', ev | eu_r)
print('ev.ev_r =', ev | ev_r)
eu = ex + ey + ez
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('Frame', (eu, ev), 'Reciprocal Frame', (eu_r, ev_r))
print('eu.eu_r =', eu | eu_r)
print('eu.ev_r =', eu | ev_r)
print('ev.eu_r =', ev | eu_r)
print('ev.ev_r =', ev | ev_r)
return
def test_derivatives_in_spherical_coordinates():
with GA_Printer():
X = (r, th, phi) = symbols("r theta phi")
curv = [[r * cos(phi) * sin(th), r * sin(phi) * sin(th), r * cos(th)], [1, r, r * sin(th)]]
(er, eth, ephi, grad) = MV.setup("e_r e_theta e_phi", metric="[1,1,1]", coords=X, curv=curv)
f = MV("f", "scalar", fct=True)
A = MV("A", "vector", fct=True)
B = MV("B", "grade2", fct=True)
assert str(f) == "f"
assert str(A) == "A__r*e_r + A__theta*e_theta + A__phi*e_phi"
assert str(B) == "B__rtheta*e_r^e_theta + B__rphi*e_r^e_phi + B__thetaphi*e_theta^e_phi"
assert str(grad * f) == "D{r}f*e_r + D{theta}f/r*e_theta + D{phi}f/(r*sin(theta))*e_phi"
assert (
str(grad | A)
== "D{r}A__r + 2*A__r/r + A__theta*cos(theta)/(r*sin(theta)) + D{theta}A__theta/r + D{phi}A__phi/(r*sin(theta))"
)
assert (
str(-MV.I * (grad ^ A))
== "((A__phi*cos(theta)/sin(theta) + D{theta}A__phi - D{phi}A__theta/sin(theta))/r)*e_r + (-D{r}A__phi - A__phi/r + D{phi}A__r/(r*sin(theta)))*e_theta + (D{r}A__theta + A__theta/r - D{theta}A__r/r)*e_phi"
)
assert (
str(grad ^ B)
== "(D{r}B__thetaphi - B__rphi*cos(theta)/(r*sin(theta)) + 2*B__thetaphi/r - D{theta}B__rphi/r + D{phi}B__rtheta/(r*sin(theta)))*e_r^e_theta^e_phi"
)
return
def test_vector_extraction():
"""
Show that conformal bivector encodes two points. See D&L Section 10.4.1
"""
metric = ' 0 -1 #,' + \
'-1 0 #,' + \
' # # #,'
P1, P2, a = 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.
"""
B = P1 ^ P2
Bsq = B*B
ap = a - (a ^ B)*B
Ap = ap + ap*B
Am = ap - ap*B
P1dota = Symbol('(P1.a)')
P2dota = Symbol('(P2.a)')
Ap_test = (-2*P2dota)*P1
Am_test = (-2*P1dota)*P2
assert Ap == Ap_test
assert Am == Am_test
Ap2 = Ap*Ap
Am2 = Am*Am
assert Ap2 == S.Zero
assert Am2 == S.Zero
def test_conformal_representations_of_circles_lines_spheres_and_planes():
global n, nbar
with GA_Printer():
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'
(e1, e2, e3, n, nbar) = MV.setup('e_1 e_2 e_3 n nbar', metric)
e = n + nbar
#conformal representation of points
A = make_vector(e1)
B = make_vector(e2)
C = make_vector(-e1)
D = make_vector(e3)
X = make_vector('x', 3)
assert str(A) == 'e_1 + 1/2*n - 1/2*nbar'
assert str(B) == 'e_2 + 1/2*n - 1/2*nbar'
assert str(C) == '-e_1 + 1/2*n - 1/2*nbar'
assert str(D) == 'e_3 + 1/2*n - 1/2*nbar'
assert str(X) == 'x1*e_1 + x2*e_2 + x3*e_3 + ((x1**2 + x2**2 + x3**2)/2)*n - 1/2*nbar'
assert str((A ^ B ^ C ^ X)) == '-x3*e_1^e_2^e_3^n + x3*e_1^e_2^e_3^nbar + ((x1**2 + x2**2 + x3**2 - 1)/2)*e_1^e_2^n^nbar'
assert str((A ^ B ^ n ^ X)) == '-x3*e_1^e_2^e_3^n + ((x1 + x2 - 1)/2)*e_1^e_2^n^nbar + x3/2*e_1^e_3^n^nbar - x3/2*e_2^e_3^n^nbar'
assert str((((A ^ B) ^ C) ^ D) ^ X) == '((-x1**2 - x2**2 - x3**2 + 1)/2)*e_1^e_2^e_3^n^nbar'
assert str((A ^ B ^ n ^ D ^ X)) == '((-x1 - x2 - x3 + 1)/2)*e_1^e_2^e_3^n^nbar'
L = (A ^ B ^ e) ^ X
assert str(L) == '-x3*e_1^e_2^e_3^n - x3*e_1^e_2^e_3^nbar + (-x1**2/2 + x1 - x2**2/2 + x2 - x3**2/2 - 1/2)*e_1^e_2^n^nbar + x3*e_1^e_3^n^nbar - x3*e_2^e_3^n^nbar'
return
def test_reciprocal_frame_test():
with GA_Printer():
metric = '1 # #,' + \
'# 1 #,' + \
'# # 1,'
(e1, e2, e3) = MV.setup('e1 e2 e3', metric)
E = e1 ^ e2 ^ e3
Esq = (E*E).scalar()
assert str(E) == 'e1^e2^e3'
assert str(Esq) == '(e1.e2)**2 - 2*(e1.e2)*(e1.e3)*(e2.e3) + (e1.e3)**2 + (e2.e3)**2 - 1'
Esq_inv = 1/Esq
E1 = (e2 ^ e3)*E
E2 = (-1)*(e1 ^ e3)*E
E3 = (e1 ^ e2)*E
assert str(E1) == '((e2.e3)**2 - 1)*e1 + ((e1.e2) - (e1.e3)*(e2.e3))*e2 + (-(e1.e2)*(e2.e3) + (e1.e3))*e3'
assert str(E2) == '((e1.e2) - (e1.e3)*(e2.e3))*e1 + ((e1.e3)**2 - 1)*e2 + (-(e1.e2)*(e1.e3) + (e2.e3))*e3'
assert str(E3) == '(-(e1.e2)*(e2.e3) + (e1.e3))*e1 + (-(e1.e2)*(e1.e3) + (e2.e3))*e2 + ((e1.e2)**2 - 1)*e3'
w = (E1 | e2)
w = w.expand()
assert str(w) == '0'
w = (E1 | e3)
w = w.expand()
assert str(w) == '0'
w = (E2 | e1)
w = w.expand()
assert str(w) == '0'
w = (E2 | e3)
w = w.expand()
assert str(w) == '0'
w = (E3 | e1)
w = w.expand()
assert str(w) == '0'
w = (E3 | e2)
w = w.expand()
assert str(w) == '0'
w = (E1 | e1)
w = (w.expand()).scalar()
Esq = expand(Esq)
assert str(simplify(w/Esq)) == '1'
w = (E2 | e2)
w = (w.expand()).scalar()
assert str(simplify(w/Esq)) == '1'
w = (E3 | e3)
w = (w.expand()).scalar()
assert str(simplify(w/Esq)) == '1'
return
def test_extracting_vectors_from_conformal_2_blade():
with GA_Printer():
metric = ' 0 -1 #,' + \
'-1 0 #,' + \
' # # #,'
(P1, P2, a) = MV.setup('P1 P2 a', metric)
B = P1 ^ P2
Bsq = B*B
assert str(Bsq) == '1'
ap = a - (a ^ B)*B
assert str(ap) == '-(P2.a)*P1 - (P1.a)*P2'
Ap = ap + ap*B
Am = ap - ap*B
assert str(Ap) == '-2*(P2.a)*P1'
assert str(Am) == '-2*(P1.a)*P2'
assert str(Ap*Ap) == '0'
assert str(Am*Am) == '0'
aB = a | B
assert str(aB) == '-(P2.a)*P1 + (P1.a)*P2'
return
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'
p1, p2, p3, n, nbar = MV.setup('p1 p2 p3 n nbar', metric, debug=0)
P1 = F(p1, n, nbar)
P2 = F(p2, n, nbar)
P3 = F(p3, n, nbar)
#Line through p1 and p2
L = P1 ^ P2 ^ n
delta = (L | n) | nbar
delta_test = 2 * p1 - 2 * p2
diff = delta - delta_test
assert diff == S.Zero
#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
assert diff == S.Zero
def test_extracting_vectors_from_conformal_2_blade():
GA_Printer.on()
metric = ' 0 -1 #,' + \
'-1 0 #,' + \
' # # #,'
(P1, P2, a) = MV.setup('P1 P2 a', metric)
B = P1 ^ P2
Bsq = B * B
assert str(Bsq) == '1'
ap = a - (a ^ B) * B
assert str(ap) == '-(P2.a)*P1 - (P1.a)*P2'
Ap = ap + ap * B
Am = ap - ap * B
assert str(Ap) == '-2*(P2.a)*P1'
assert str(Am) == '-2*(P1.a)*P2'
assert str(Ap * Ap) == '0'
assert str(Am * Am) == '0'
aB = a | B
assert str(aB) == '-(P2.a)*P1 + (P1.a)*P2'
GA_Printer.off()
return
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'
p1, p2, p3, n, nbar = MV.setup('p1 p2 p3 n nbar', metric, debug=0)
P1 = F(p1, n, nbar)
P2 = F(p2, n, nbar)
P3 = F(p3, n, nbar)
#Line through p1 and p2
L = P1 ^ P2 ^ n
delta = (L | n) | nbar
delta_test = 2*p1 - 2*p2
diff = delta - delta_test
assert diff == S.Zero
#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
assert diff == S.Zero
def test_derivatives_in_rectangular_coordinates():
with GA_Printer():
X = (x, y, z) = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('e_x e_y e_z', metric='[1,1,1]', coords=X)
f = MV('f', 'scalar', fct=True)
A = MV('A', 'vector', fct=True)
B = MV('B', 'grade2', fct=True)
C = MV('C', 'mv', fct=True)
assert str(f) == 'f'
assert str(A) == 'A__x*e_x + A__y*e_y + A__z*e_z'
assert str(B) == 'B__xy*e_x^e_y + B__xz*e_x^e_z + B__yz*e_y^e_z'
assert str(C) == 'C + C__x*e_x + C__y*e_y + C__z*e_z + C__xy*e_x^e_y + C__xz*e_x^e_z + C__yz*e_y^e_z + C__xyz*e_x^e_y^e_z'
assert str(grad*f) == 'D{x}f*e_x + D{y}f*e_y + D{z}f*e_z'
assert str(grad | A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(grad*A) == 'D{x}A__x + D{y}A__y + D{z}A__z + (-D{y}A__x + D{x}A__y)*e_x^e_y + (-D{z}A__x + D{x}A__z)*e_x^e_z + (-D{z}A__y + D{y}A__z)*e_y^e_z'
assert str(-MV.I*(grad ^ A)) == '(-D{z}A__y + D{y}A__z)*e_x + (D{z}A__x - D{x}A__z)*e_y + (-D{y}A__x + D{x}A__y)*e_z'
assert str(grad*B) == '(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z + (D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z'
assert str(grad ^ B) == '(D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z'
assert str(grad | B) == '(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z'
assert str(grad < A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(grad > A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(grad < B) == '(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z'
assert str(grad > B) == '0'
assert str(grad < C) == 'D{x}C__x + D{y}C__y + D{z}C__z + (-(D{y}C__xy + D{z}C__xz))*e_x + (D{x}C__xy - D{z}C__yz)*e_y + (D{x}C__xz + D{y}C__yz)*e_z + D{z}C__xyz*e_x^e_y - D{y}C__xyz*e_x^e_z + D{x}C__xyz*e_y^e_z'
assert str(grad > C) == 'D{x}C__x + D{y}C__y + D{z}C__z + D{x}C*e_x + D{y}C*e_y + D{z}C*e_z'
return
def test_extracting_vectors_from_conformal_2_blade():
with GA_Printer():
metric = " 0 -1 #," + "-1 0 #," + " # # #,"
(P1, P2, a) = MV.setup("P1 P2 a", metric)
B = P1 ^ P2
Bsq = B * B
assert str(Bsq) == "1"
ap = a - (a ^ B) * B
assert str(ap) == "-(P2.a)*P1 - (P1.a)*P2"
Ap = ap + ap * B
Am = ap - ap * B
assert str(Ap) == "-2*(P2.a)*P1"
assert str(Am) == "-2*(P1.a)*P2"
assert str(Ap * Ap) == "0"
assert str(Am * Am) == "0"
aB = a | B
assert str(aB) == "-(P2.a)*P1 + (P1.a)*P2"
return
def test_substitution():
e_x, e_y, e_z = MV.setup('e_x e_y e_z', '1 0 0, 0 1 0, 0 0 1')
x, y, z = 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():
e_x, e_y, e_z = MV.setup('e_x e_y e_z', '1 0 0, 0 1 0, 0 0 1')
x, y, z = 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():
e_x, e_y, e_z = MV.setup("e_x e_y e_z", "1 0 0, 0 1 0, 0 0 1")
x, y, z = 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_reciprocal_frame_test():
with GA_Printer():
metric = "1 # #," + "# 1 #," + "# # 1,"
(e1, e2, e3) = MV.setup("e1 e2 e3", metric)
E = e1 ^ e2 ^ e3
Esq = (E * E).scalar()
assert str(E) == "e1^e2^e3"
assert str(Esq) == "(e1.e2)**2 - 2*(e1.e2)*(e1.e3)*(e2.e3) + (e1.e3)**2 + (e2.e3)**2 - 1"
Esq_inv = 1 / Esq
E1 = (e2 ^ e3) * E
E2 = (-1) * (e1 ^ e3) * E
E3 = (e1 ^ e2) * E
assert str(E1) == "((e2.e3)**2 - 1)*e1 + ((e1.e2) - (e1.e3)*(e2.e3))*e2 + (-(e1.e2)*(e2.e3) + (e1.e3))*e3"
assert str(E2) == "((e1.e2) - (e1.e3)*(e2.e3))*e1 + ((e1.e3)**2 - 1)*e2 + (-(e1.e2)*(e1.e3) + (e2.e3))*e3"
assert str(E3) == "(-(e1.e2)*(e2.e3) + (e1.e3))*e1 + (-(e1.e2)*(e1.e3) + (e2.e3))*e2 + ((e1.e2)**2 - 1)*e3"
w = E1 | e2
w = w.expand()
assert str(w) == "0"
w = E1 | e3
w = w.expand()
assert str(w) == "0"
w = E2 | e1
w = w.expand()
assert str(w) == "0"
w = E2 | e3
w = w.expand()
assert str(w) == "0"
w = E3 | e1
w = w.expand()
assert str(w) == "0"
w = E3 | e2
w = w.expand()
assert str(w) == "0"
w = E1 | e1
w = (w.expand()).scalar()
Esq = expand(Esq)
assert str(simplify(w / Esq)) == "1"
w = E2 | e2
w = (w.expand()).scalar()
assert str(simplify(w / Esq)) == "1"
w = E3 | e3
w = (w.expand()).scalar()
assert str(simplify(w / Esq)) == "1"
return
def Plot_Mobius_Strip_Manifold():
Print_Function()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('ex ey ez', metric='[1,1,1]', coords=coords)
mfvar = (u, v) = symbols('u v')
X = (cos(u) + v*cos(u/2)*cos(u))*ex + (sin(u) + v*cos(u/2)*sin(u))*ey + v*sin(u/2)*ez
MF = Manifold(X, mfvar, True, I=MV.I)
MF.Plot2DSurface([0.0, 6.28, 48], [-0.3, 0.3, 12], surf=False, skip=[4, 4], tan=0.15)
return
def test_str():
e_1, e_2, e_3 = 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__1*e_1 + x__2*e_2 + x__3*e_3 + x__12*e_1^e_2 + x__13*e_1^e_3 + x__23*e_2^e_3 + x__123**e_1^e_2^e_3'
Y = MV('y', 'spinor')
assert str(Y) == 'y + y__12*e_1^e_2 + y__13*e_1^e_3 + y__23*e_2^e_3'
Z = X + Y
assert str(Z) == 'x + y + x__1*e_1 + x__2*e_2 + x__3*e_3 + (x__12 + y__12)*e_1^e_2 + (x__13 + y__13)*e_1^e_3 + (x__23 + y__23)*e_2^e_3 + x__123*e_1^e_2^e_3'
assert str(e_1 | e_1) == '1'
def test_derivative():
coords = x, y, z = symbols('x y z')
e_x, e_y, e_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_metrics_xfail():
from sympy.galgebra.ga import arbitrary_metric_conformal
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
assert diff == MV(S.Zero)
def test_derivative():
coords = x, y, z = symbols("x y z")
e_x, e_y, e_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_constructor():
"""
Test various multivector constructors
"""
e_1, e_2, e_3 = MV.setup('e_1 e_2 e_3', '[1,1,1]')
assert str(MV('a', 'scalar')) == 'a'
assert str(MV('a', 'vector')) == 'a__1*e_1 + a__2*e_2 + a__3*e_3'
assert str(MV('a', 'pseudo')) == 'a__123*e_1^e_2^e_3'
assert str(MV('a', 'spinor')) == 'a + a__12*e_1^e_2 + a__13*e_1^e_3 + a__23*e_2^e_3'
assert str(MV('a')) == 'a + a__1*e_1 + a__2*e_2 + a__3*e_3 + a__12*e_1^e_2 + a__13*e_1^e_3 + a__23*e_2^e_3 + a__123*e_1^e_2^e_3'
assert str(MV([2, 'a'], 'grade')) == 'a__12*e_1^e_2 + a__13*e_1^e_3 + a__23*e_2^e_3'
assert str(MV('a', 'grade2')) == 'a__12*e_1^e_2 + a__13*e_1^e_3 + a__23*e_2^e_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.
"""
x, y, z = MV.setup('x y z')
a, b, c = symbols('a b c')
assert 5 * x == x * 5
assert Rational(1, 2) * x == x * Rational(1, 2)
assert a * x == x * a
def make_vector(a, m=3):
global n, nbar
if isinstance(a, str):
sym_str = ''
for i in range(m):
sym_str += a + str(i + 1) + ' '
sym_lst = list(symbols(sym_str))
sym_lst.append(S.Zero)
sym_lst.append(S.Zero)
a = MV(sym_lst, 'vector')
return F(a, n, nbar)
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.
"""
x, y, z = MV.setup('x y z')
a, b, c = symbols('a b c')
assert 5*x == x*5
assert Rational(1, 2)*x == x*Rational(1, 2)
assert a*x == x*a
def test_derivatives_in_spherical_coordinates():
with GA_Printer():
X = (r, th, phi) = symbols('r theta phi')
curv = [[r*cos(phi)*sin(th), r*sin(phi)*sin(th), r*cos(th)], [1, r, r*sin(th)]]
(er, eth, ephi, grad) = MV.setup('e_r e_theta e_phi', metric='[1,1,1]', coords=X, curv=curv)
f = MV('f', 'scalar', fct=True)
A = MV('A', 'vector', fct=True)
B = MV('B', 'grade2', fct=True)
assert str(f) == 'f'
assert str(A) == 'A__r*e_r + A__theta*e_theta + A__phi*e_phi'
assert str(B) == 'B__rtheta*e_r^e_theta + B__rphi*e_r^e_phi + B__thetaphi*e_theta^e_phi'
assert str(grad*f) == 'D{r}f*e_r + D{theta}f/r*e_theta + D{phi}f/(r*sin(theta))*e_phi'
assert str(grad | A) == 'D{r}A__r + 2*A__r/r + A__theta*cos(theta)/(r*sin(theta)) + D{theta}A__theta/r + D{phi}A__phi/(r*sin(theta))'
assert str(-MV.I*(grad ^ A)) == '((A__phi*cos(theta)/sin(theta) + D{theta}A__phi - D{phi}A__theta/sin(theta))/r)*e_r + (-D{r}A__phi - A__phi/r + D{phi}A__r/(r*sin(theta)))*e_theta + (D{r}A__theta + A__theta/r - D{theta}A__r/r)*e_phi'
assert str(grad ^ B) == '(D{r}B__thetaphi - B__rphi*cos(theta)/(r*sin(theta)) + 2*B__thetaphi/r - D{theta}B__rphi/r + D{phi}B__rtheta/(r*sin(theta)))*e_r^e_theta^e_phi'
return
def main():
enhance_print()
(ex, ey, ez) = MV.setup('e*x|y|z', metric='[1,1,1]')
u = MV('u', 'vector')
v = MV('v', 'vector')
w = MV('w', 'vector')
print(u)
print(v)
print(w)
uv = u ^ v
print(uv)
print(uv.is_blade())
uvw = u ^ v ^ w
print(uvw)
print(uvw.is_blade())
print(simplify((uv * uv).scalar()))
return
def test_derivatives_in_rectangular_coordinates():
with GA_Printer():
X = (x, y, z) = symbols("x y z")
(ex, ey, ez, grad) = MV.setup("e_x e_y e_z", metric="[1,1,1]", coords=X)
f = MV("f", "scalar", fct=True)
A = MV("A", "vector", fct=True)
B = MV("B", "grade2", fct=True)
C = MV("C", "mv", fct=True)
assert str(f) == "f"
assert str(A) == "A__x*e_x + A__y*e_y + A__z*e_z"
assert str(B) == "B__xy*e_x^e_y + B__xz*e_x^e_z + B__yz*e_y^e_z"
assert (
str(C)
== "C + C__x*e_x + C__y*e_y + C__z*e_z + C__xy*e_x^e_y + C__xz*e_x^e_z + C__yz*e_y^e_z + C__xyz*e_x^e_y^e_z"
)
assert str(grad * f) == "D{x}f*e_x + D{y}f*e_y + D{z}f*e_z"
assert str(grad | A) == "D{x}A__x + D{y}A__y + D{z}A__z"
assert (
str(grad * A)
== "D{x}A__x + D{y}A__y + D{z}A__z + (-D{y}A__x + D{x}A__y)*e_x^e_y + (-D{z}A__x + D{x}A__z)*e_x^e_z + (-D{z}A__y + D{y}A__z)*e_y^e_z"
)
assert (
str(-MV.I * (grad ^ A))
== "(-D{z}A__y + D{y}A__z)*e_x + (D{z}A__x - D{x}A__z)*e_y + (-D{y}A__x + D{x}A__y)*e_z"
)
assert (
str(grad * B)
== "(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z + (D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z"
)
assert str(grad ^ B) == "(D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z"
assert (
str(grad | B)
== "(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z"
)
assert str(grad < A) == "D{x}A__x + D{y}A__y + D{z}A__z"
assert str(grad > A) == "D{x}A__x + D{y}A__y + D{z}A__z"
assert (
str(grad < B)
== "(-(D{y}B__xy + D{z}B__xz))*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z"
)
assert str(grad > B) == "0"
assert (
str(grad < C)
== "D{x}C__x + D{y}C__y + D{z}C__z + (-(D{y}C__xy + D{z}C__xz))*e_x + (D{x}C__xy - D{z}C__yz)*e_y + (D{x}C__xz + D{y}C__yz)*e_z + D{z}C__xyz*e_x^e_y - D{y}C__xyz*e_x^e_z + D{x}C__xyz*e_y^e_z"
)
assert str(grad > C) == "D{x}C__x + D{y}C__y + D{z}C__z + D{x}C*e_x + D{y}C*e_y + D{z}C*e_z"
return
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,'
e1, e2, e3 = 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_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,'
e1, e2, e3 = 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 DD(self, v, f, opstr=False):
mf_comp = []
for e in self.rbasis:
mf_comp.append((v | e).scalar() / self.E_sq)
result = MV()
op = ''
for (coord, comp) in zip(self.coords, mf_comp):
result += comp * (f.diff(coord))
if opstr:
op += '(' + str(comp) + ')D{' + str(coord) + '}+'
if opstr:
return str(result), op[:-1]
return result
def test_contraction():
"""
Test for inner product and left and right contraction
"""
e_1, e_2, e_3 = MV.setup('e_1 e_2 e_3', '1 0 0, 0 1 0, 0 0 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_rounding_numerical_components():
with GA_Printer():
(ex, ey, ez) = MV.setup('e_x e_y e_z', metric='[1,1,1]')
X = 1.2*ex + 2.34*ey + 0.555*ez
Y = 0.333*ex + 4*ey + 5.3*ez
assert str(X) == '1.20000000000000*e_x + 2.34000000000000*e_y + 0.555000000000000*e_z'
assert str(Nga(X, 2)) == '1.2*e_x + 2.3*e_y + 0.55*e_z'
assert str(X*Y) == '12.7011000000000 + 4.02078000000000*e_x^e_y + 6.17518500000000*e_x^e_z + 10.1820000000000*e_y^e_z'
assert str(Nga(X*Y, 2)) == '13. + 4.0*e_x^e_y + 6.2*e_x^e_z + 10.*e_y^e_z'
return
def test_contraction():
"""
Test for inner product and left and right contraction
"""
e_1, e_2, e_3 = MV.setup('e_1 e_2 e_3', '1 0 0, 0 1 0, 0 0 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_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"
e0, e1, e2, n, nbar = MV.setup("e0 e1 e2 n nbar", metric, debug=0)
e = n + nbar
# conformal representation of points
A = F(e0, n, nbar) # point a = (1,0,0) A = F(a)
B = F(e1, n, nbar) # point b = (0,1,0) B = F(b)
C = F(-1 * e0, n, nbar) # point c = (-1,0,0) C = F(c)
D = F(e2, n, nbar) # point d = (0,0,1) D = F(d)
x0, x1, x2 = symbols("x0 x1 x2")
X = F(MV([x0, x1, x2], "vector"), n, nbar)
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)
+ Rational(1, 2) * (-1 + x0 ** 2 + x1 ** 2 + x2 ** 2) * (e0 ^ e1 ^ n ^ nbar)
)
diff = Circle - Circle_test
assert diff == S.Zero
# Line through a and b
Line_test = (
-x2 * (e0 ^ e1 ^ e2 ^ n)
+ Rational(1, 2) * (-1 + x0 + x1) * (e0 ^ e1 ^ n ^ nbar)
+ (Rational(1, 2) * x2) * (e0 ^ e2 ^ n ^ nbar)
+ (-Rational(1, 2) * x2) * (e1 ^ e2 ^ n ^ nbar)
)
diff = Line - Line_test
assert diff == S.Zero
# Sphere through a, b, c, and d
Sphere_test = Rational(1, 2) * (1 - x0 ** 2 - x1 ** 2 - x2 ** 2) * (e0 ^ e1 ^ e2 ^ n ^ nbar)
diff = Sphere - Sphere_test
assert diff == S.Zero
# Plane through a, b, and d
Plane_test = Rational(1, 2) * (1 - x0 - x1 - x2) * (e0 ^ e1 ^ e2 ^ n ^ nbar)
diff = Plane - Plane_test
assert diff == S.Zero
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)
x1, y1, z1 = symbols('x1 y1 z1')
x2, y2, z2 = 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
assert diff == MV(S.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
assert diff == MV(S.Zero)
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)
x1, y1, z1 = symbols('x1 y1 z1')
x2, y2, z2 = 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
assert diff == MV(S.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
assert diff == MV(S.Zero)
def test_constructor():
"""
Test various multivector constructors
"""
e_1, e_2, e_3 = MV.setup("e_1 e_2 e_3", "[1,1,1]")
assert str(MV("a", "scalar")) == "a"
assert str(MV("a", "vector")) == "a__1*e_1 + a__2*e_2 + a__3*e_3"
assert str(MV("a", "pseudo")) == "a__123*e_1^e_2^e_3"
assert str(MV("a", "spinor")) == "a + a__12*e_1^e_2 + a__13*e_1^e_3 + a__23*e_2^e_3"
assert (
str(MV("a"))
== "a + a__1*e_1 + a__2*e_2 + a__3*e_3 + a__12*e_1^e_2 + a__13*e_1^e_3 + a__23*e_2^e_3 + a__123*e_1^e_2^e_3"
)
assert str(MV([2, "a"], "grade")) == "a__12*e_1^e_2 + a__13*e_1^e_3 + a__23*e_2^e_3"
assert str(MV("a", "grade2")) == "a__12*e_1^e_2 + a__13*e_1^e_3 + a__23*e_2^e_3"
def test_check_generalized_BAC_CAB_formulas():
with GA_Printer():
(a, b, c, d, e) = MV.setup("a b c d e")
assert str(a | (b * c)) == "-(a.c)*b + (a.b)*c"
assert str(a | (b ^ c)) == "-(a.c)*b + (a.b)*c"
assert str(a | (b ^ c ^ d)) == "(a.d)*b^c - (a.c)*b^d + (a.b)*c^d"
assert str((a | (b ^ c)) + (c | (a ^ b)) + (b | (c ^ a))) == "0"
assert str(a * (b ^ c) - b * (a ^ c) + c * (a ^ b)) == "3*a^b^c"
assert str(a * (b ^ c ^ d) - b * (a ^ c ^ d) + c * (a ^ b ^ d) - d * (a ^ b ^ c)) == "4*a^b^c^d"
assert str((a ^ b) | (c ^ d)) == "-(a.c)*(b.d) + (a.d)*(b.c)"
assert str(((a ^ b) | c) | d) == "-(a.c)*(b.d) + (a.d)*(b.c)"
assert str(Com(a ^ b, c ^ d)) == "-(b.d)*a^c + (b.c)*a^d + (a.d)*b^c - (a.c)*b^d"
assert str((a | (b ^ c)) | (d ^ e)) == "(-(a.b)*(c.e) + (a.c)*(b.e))*d + ((a.b)*(c.d) - (a.c)*(b.d))*e"
return
def test_check_generalized_BAC_CAB_formulas():
with GA_Printer():
(a, b, c, d, e) = MV.setup('a b c d e')
assert str(a | (b*c)) == '-(a.c)*b + (a.b)*c'
assert str(a | (b ^ c)) == '-(a.c)*b + (a.b)*c'
assert str(a | (b ^ c ^ d)) == '(a.d)*b^c - (a.c)*b^d + (a.b)*c^d'
assert str((a | (b ^ c)) + (c | (a ^ b)) + (b | (c ^ a))) == '0'
assert str(a*(b ^ c) - b*(a ^ c) + c*(a ^ b)) == '3*a^b^c'
assert str(a*(b ^ c ^ d) - b*(a ^ c ^ d) + c*(a ^ b ^ d) - d*(a ^ b ^ c)) == '4*a^b^c^d'
assert str((a ^ b) | (c ^ d)) == '-(a.c)*(b.d) + (a.d)*(b.c)'
assert str(((a ^ b) | c) | d) == '-(a.c)*(b.d) + (a.d)*(b.c)'
assert str(Com(a ^ b, c ^ d)) == '-(b.d)*a^c + (b.c)*a^d + (a.d)*b^c - (a.c)*b^d'
assert str((a | (b ^ c)) | (d ^ e)) == '(-(a.b)*(c.e) + (a.c)*(b.e))*d + ((a.b)*(c.d) - (a.c)*(b.d))*e'
return
def Simple_manifold_with_scalar_function_derivative():
Print_Function()
coords = (x, y, z) = symbols('x y z')
basis = (e1, e2, e3, grad) = MV.setup('e_1 e_2 e_3', metric='[1,1,1]', coords=coords)
# Define surface
mfvar = (u, v) = symbols('u v')
X = u*e1 + v*e2 + (u**2 + v**2)*e3
print(X)
MF = Manifold(X, mfvar)
# Define field on the surface.
g = (v + 1)*log(u)
# Method 1: Using old Manifold routines.
VectorDerivative = (MF.rbasis[0]/MF.E_sq)*diff(g, u) + (MF.rbasis[1]/MF.E_sq)*diff(g, v)
print('Vector derivative =', VectorDerivative.subs({u: 1, v: 0}))
# Method 2: Using new Manifold routines.
dg = MF.Grad(g)
print('Vector derivative =', dg.subs({u: 1, v: 0}))
return
def test_properties_of_geometric_objects():
GA_Printer.on()
metric = '# # # 0 0,' + \
'# # # 0 0,' + \
'# # # 0 0,' + \
'0 0 0 0 2,' + \
'0 0 0 2 0'
(p1, p2, p3, n, nbar) = MV.setup('p1 p2 p3 n nbar', metric)
P1 = F(p1, n, nbar)
P2 = F(p2, n, nbar)
P3 = F(p3, n, nbar)
L = P1 ^ P2 ^ n
delta = (L | n) | nbar
assert str(delta) == '2*p1 - 2*p2'
C = P1 ^ P2 ^ P3
delta = ((C ^ n) | n) | nbar
assert str(delta) == '2*p1^p2 - 2*p1^p3 + 2*p2^p3'
assert str((p2 - p1) ^ (p3 - p1)) == 'p1^p2 - p1^p3 + p2^p3'
GA_Printer.off()
return
def test_noneuclidian_distance_calculation():
from sympy import solve, sqrt
GA_Printer.on()
metric = '0 # #,# 0 #,# # 1'
(X, Y, e) = MV.setup('X Y e', metric)
assert str((X ^ Y) * (X ^ Y)) == '(X.Y)**2'
L = X ^ Y ^ e
B = L * e
assert str(B) == 'X^Y - (Y.e)*X^e + (X.e)*Y^e'
Bsq = B * B
assert str(Bsq) == '(X.Y)*((X.Y) - 2*(X.e)*(Y.e))'
Bsq = Bsq.scalar()
assert str(B) == 'X^Y - (Y.e)*X^e + (X.e)*Y^e'
BeBr = B * e * B.rev()
assert str(BeBr) == '((X.Y)*(-(X.Y) + 2*(X.e)*(Y.e)))*e'
assert str(B * B) == '(X.Y)*((X.Y) - 2*(X.e)*(Y.e))'
assert str(L * L) == '(X.Y)*((X.Y) - 2*(X.e)*(Y.e))'
(s, c, Binv, M, BigS, BigC, alpha, XdotY, Xdote,
Ydote) = symbols('s c (1/B) M S C alpha (X.Y) (X.e) (Y.e)')
Bhat = Binv * B
R = c + s * Bhat
assert str(R) == 'c + (1/B)*s*X^Y - (1/B)*(Y.e)*s*X^e + (1/B)*(X.e)*s*Y^e'
Z = R * X * R.rev()
Z.obj = expand(Z.obj)
Z.obj = Z.obj.collect([Binv, s, c, XdotY])
assert str(
Z
) == '((1/B)**2*(X.Y)**2*s**2 - 2*(1/B)**2*(X.Y)*(X.e)*(Y.e)*s**2 + 2*(1/B)*(X.Y)*c*s - 2*(1/B)*(X.e)*(Y.e)*c*s + c**2)*X + 2*(1/B)*(X.e)**2*c*s*Y + (2*(1/B)*(X.Y)*(X.e)*s*(-(1/B)*(X.Y)*s + 2*(1/B)*(X.e)*(Y.e)*s - c))*e'
W = Z | Y
# From this point forward all calculations are with sympy scalars
W = W.scalar()
assert str(
W
) == '(1/B)**2*(X.Y)**3*s**2 - 4*(1/B)**2*(X.Y)**2*(X.e)*(Y.e)*s**2 + 4*(1/B)**2*(X.Y)*(X.e)**2*(Y.e)**2*s**2 + 2*(1/B)*(X.Y)**2*c*s - 4*(1/B)*(X.Y)*(X.e)*(Y.e)*c*s + (X.Y)*c**2'
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, BigS)
W = W.subs(c**2, (BigC + 1) / 2)
W = W.subs(s**2, (BigC - 1) / 2)
W = simplify(W)
W = expand(W)
W = W.subs(1 / Binv, Bmag)
assert str(
W
) == '(X.Y)*C - (X.e)*(Y.e)*C + (X.e)*(Y.e) + S*sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
Wd = collect(W, [BigC, BigS], exact=True, evaluate=False)
Wd_1 = Wd[S.One]
Wd_C = Wd[BigC]
Wd_S = Wd[BigS]
assert str(Wd_1) == '(X.e)*(Y.e)'
assert str(Wd_C) == '(X.Y) - (X.e)*(Y.e)'
assert str(Wd_S) == 'sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
assert str(Bmag) == 'sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
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 * BigC
rhs = -Wd_S * BigS
lhs = lhs**2
rhs = rhs**2
W = expand(lhs - rhs)
W = expand(W.subs(1 / Binv**2, Bmag**2))
W = expand(W.subs(BigS**2, BigC**2 - 1))
W = W.collect([BigC, BigC**2], evaluate=False)
a = simplify(W[BigC**2])
b = simplify(W[BigC])
c = simplify(W[S.One])
assert str(a) == '(X.e)**2*(Y.e)**2'
assert str(b) == '2*(X.e)*(Y.e)*((X.Y) - (X.e)*(Y.e))'
assert str(c) == '(X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e) + (X.e)**2*(Y.e)**2'
x = Symbol('x')
C = solve(a * x**2 + b * x + c, x)[0]
assert str(expand(simplify(expand(C)))) == '-(X.Y)/((X.e)*(Y.e)) + 1'
GA_Printer.off()
return
def test_metric():
MV.setup('e_1 e_2 e_3', '[1,1,1]')
assert MV.metric == Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])