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_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_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_derivative():
coords = x, y, z = sympy.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 make_vector(a,n = 3):
if type(a) == types.StringType:
sym_str = ''
for i in range(n):
sym_str += a+str(i)+' '
sym_lst = make_symbols(sym_str)
sym_lst.append(ZERO)
sym_lst.append(ZERO)
a = MV(sym_lst,'vector')
return(F(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'
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 = sympy.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) + HALF * (-1 + x0**2 + x1**2 +
x2**2) * (e0 ^ e1 ^ n ^ nbar)
diff = Circle - Circle_test
diff.compact()
assert diff == ZERO
#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
#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
#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
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_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'
