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_derivatives_in_spherical_coordinates():
GA_Printer.on()
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'
GA_Printer.off()
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 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 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_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 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_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_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_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 __init__(self, x, coords, debug=False, I=None):
"""
coords: list of coordinate variables
x: vector fuction of coordinate variables (parametric surface)
"""
self.I = I
self.x = x
self.coords = coords
self.basis = []
self.basis_str = []
self.embedded_basis = []
for u in coords:
tv = x.diff(u)
self.basis.append(tv)
(coefs, bases) = linear_expand(tv.obj)
tc = {}
for (coef, base) in zip(coefs, bases):
str_base = str(base)
tc[str_base] = coef
if str_base not in self.embedded_basis:
self.embedded_basis.append(str_base)
self.basis_str.append(tc)
self.gij = []
for base1 in self.basis:
tmp = []
for base2 in self.basis:
tmp.append(simplify(trigsimp((base1 | base2).scalar())))
self.gij.append(tmp)
for tv in self.basis_str:
for base in self.embedded_basis:
if base not in tv:
tv[base] = 0
self.dim = len(self.basis)
indexes = tuple(range(self.dim))
self.index = [()]
for i in indexes:
self.index.append(tuple(combinations(indexes, i + 1)))
self.index = tuple(self.index)
self.MFbasis = [[MV.ONE], self.basis]
for igrade in self.index[2:]:
grade = []
for iblade in igrade:
blade = MV(1, 'scalar')
for ibasis in iblade:
blade ^= self.basis[ibasis]
blade = blade.trigsimp(deep=True, recursive=True)
grade.append(blade)
self.MFbasis.append(grade)
self.E = self.MFbasis[-1][0]
self.E_sq = trigsimp((self.E * self.E).scalar(),
deep=True,
recursive=True)
duals = copy.copy(self.MFbasis[-2])
duals.reverse()
sgn = 1
self.rbasis = []
for dual in duals:
recpv = (sgn * dual * self.E).trigsimp(deep=True, recursive=True)
self.rbasis.append(recpv)
sgn = -sgn
self.dbasis = []
for base in self.basis:
dbase = []
for coord in self.coords:
d = base.diff(coord).trigsimp(deep=True, recursive=True)
dbase.append(d)
self.dbasis.append(dbase)
self.surface = {}
(coefs, bases) = linear_expand(self.x.obj)
for (coef, base) in zip(coefs, bases):
self.surface[str(base)] = coef
self.grad = MV()
self.grad.is_grad = True
self.grad.blade_rep = True
self.grad.igrade = 1
self.grad.rcpr_bases_MV = []
for rbase in self.rbasis:
self.grad.rcpr_bases_MV.append(rbase / self.E_sq)
self.grad.rcpr_bases_MV = tuple(self.grad.rcpr_bases_MV)
self.grad.coords = self.coords
self.grad.norm = self.E_sq
self.grad.connection = {}
if debug:
oprint('x', self.x, 'coords', self.coords, 'basis vectors',
self.basis, 'index', self.index, 'basis blades',
self.MFbasis, 'E', self.E, 'E**2', self.E_sq, '*basis',
duals, 'rbasis', self.rbasis, 'basis derivatives',
self.dbasis, 'surface', self.surface, 'basis strings',
self.basis_str, 'embedding basis', self.embedded_basis,
'metric tensor', self.gij)
