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_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_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_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 main():
Get_Program(True)
with GA_Printer():
enhance_print()
Test_Reciprocal_Frame()
Distorted_manifold_with_scalar_function()
Simple_manifold_with_scalar_function_derivative()
#Plot_Mobius_Strip_Manifold()
return
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_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 main():
Get_Program(True)
with GA_Printer():
enhance_print()
basic_multivector_operations()
check_generalized_BAC_CAB_formulas()
derivatives_in_rectangular_coordinates()
derivatives_in_spherical_coordinates()
rounding_numerical_components()
noneuclidian_distance_calculation()
conformal_representations_of_circles_lines_spheres_and_planes()
properties_of_geometric_objects()
extracting_vectors_from_conformal_2_blade()
reciprocal_frame_test()
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 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_properties_of_geometric_objects():
with GA_Printer():
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'
return
def test_noneuclidian_distance_calculation():
from sympy import solve, sqrt
with GA_Printer():
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'
return
def __str__(self):
return GA_Printer().doprint(self)
