def test_rmul():
#set_main(sys.modules[__name__])
MV.setup('x y z')
make_symbols('a b c')
assert 5*x == x*5
assert HALF*x == x*HALF
assert a*x == x*a
def test_rmul():
#set_main(sys.modules[__name__])
MV.setup('x y z')
make_symbols('a b c')
assert 5 * x == x * 5
assert HALF * x == x * HALF
assert a * x == x * a
def test_extract_plane_and_line():
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)
L = P1^P2^n
delta = (L|n)|nbar
delta_test = 2*p1-2*p2
diff = delta-delta_test
diff.compact()
assert diff == ZERO_MV
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_vector_extraction():
metric = ' 0 -1 #,'+ \
'-1 0 #,'+ \
' # # #,'
MV.setup('P1 P2 a',metric)
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():
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)
L = P1 ^ P2 ^ n
delta = (L | n) | nbar
delta_test = 2 * p1 - 2 * p2
diff = delta - delta_test
diff.compact()
assert diff == ZERO_MV
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_vector_extraction():
metric = ' 0 -1 #,'+ \
'-1 0 #,'+ \
' # # #,'
MV.setup('P1 P2 a', metric)
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_geometry():
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_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_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_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_test = HALF * (1 - x0 - x1 - x2) * (e0 ^ e1 ^ e2 ^ n ^ nbar)
diff = Plane - Plane_test
diff.compact()
assert diff == ZERO_MV
def test_reciprocal_frame():
metric = '1 # #,'+ \
'# 1 #,'+ \
'# # 1,'
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():
metric = '1 # #,'+ \
'# 1 #,'+ \
'# # 1,'
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_geometry():
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_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_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_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_test = HALF*(1-x0-x1-x2)*(e0^e1^e2^n^nbar)
diff = Plane-Plane_test
diff.compact()
assert diff == ZERO_MV
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_noneuclidian():
global s,c,Binv,M,S,C,alpha
#set_main(sys.modules[__name__])
metric = '0 # #,'+ \
'# 0 #,'+ \
'# # 1,'
MV.setup('X Y e',metric,debug=0)
MV.set_str_format(1)
L = X^Y^e
B = L*e
Bsq = (B*B)()
BeBr =B*e*B.rev()
make_symbols('s c Binv M S C alpha')
Bhat = Binv*B # Normalize translation generator
R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2)
Z = R*X*R.rev()
Z.expand()
Z.collect([Binv,s,c,XdotY])
W = Z|Y
W.expand()
W.collect([s*Binv])
M = 1/Bsq
W.subs(Binv**2,M)
W.simplify()
Bmag = sympy.sqrt(XdotY**2-2*XdotY*Xdote*Ydote)
W.collect([Binv*c*s,XdotY])
W.subs(2*XdotY**2-4*XdotY*Xdote*Ydote,2/(Binv**2))
W.subs(2*c*s,S)
W.subs(c**2,(C+1)/2)
W.subs(s**2,(C-1)/2)
W.simplify()
W.subs(1/Binv,Bmag)
W = W().expand()
#print '(R*X*R.rev()).Y =',W
Wd = collect(W,[C,S],exact=True,evaluate=False)
#print 'Wd =',Wd
Wd_1 = Wd[ONE]
Wd_C = Wd[C]
Wd_S = Wd[S]
#print '|B| =',Bmag
Wd_1 = Wd_1.subs(Bmag,1/Binv)
Wd_C = Wd_C.subs(Bmag,1/Binv)
Wd_S = Wd_S.subs(Bmag,1/Binv)
#print 'Wd[ONE] =',Wd_1
#print 'Wd[C] =',Wd_C
#print 'Wd[S] =',Wd_S
lhs = Wd_1+Wd_C*C
rhs = -Wd_S*S
lhs = lhs**2
rhs = rhs**2
W = (lhs-rhs).expand()
W = (W.subs(1/Binv**2,Bmag**2)).expand()
#print 'W =',W
W = (W.subs(S**2,C**2-1)).expand()
W = collect(W,[C**2,C],evaluate=False)
#print 'W =',W
a = W[C**2]
b = W[(C**2)**(sympify(1)/2)]
c = W[ONE]
#print 'a =',a
#print 'b =',b
#print 'c =',c
D = (b**2-4*a*c).expand()
#print 'Setting to 0 and solving for C gives:'
#print 'Descriminant D = b^2-4*a*c =',D
C = (-b/(2*a)).expand()
#print 'C = cosh(alpha) = -b/(2*a) =',C
"""
Wd = collect(W,[C,S],evaluate=False)
lhs = Wd[ONE]+Wd[C]*C
rhs = -Wd[S]*S
lhs = lhs**2
rhs = rhs**2
W = (lhs-rhs).expand()
W = (W.subs(S**2,C**2-1)).expand()
W = collect(W,[C**2,C],evaluate=False)
a = W[C**2]
b = W[abs(C)]
c = W[ONE]
D = (b**2-4*a*c).expand()
C = (-b/(2*a)).expand()
"""
assert C == 1-XdotY/(Xdote*Ydote)
def test_noneuclidian():
global s, c, Binv, M, S, C, alpha
#set_main(sys.modules[__name__])
metric = '0 # #,'+ \
'# 0 #,'+ \
'# # 1,'
MV.setup('X Y e', metric, debug=0)
MV.set_str_format(1)
L = X ^ Y ^ e
B = L * e
Bsq = (B * B)()
BeBr = B * e * B.rev()
make_symbols('s c Binv M S C alpha')
Bhat = Binv * B # Normalize translation generator
R = c + s * Bhat # Rotor R = exp(alpha*Bhat/2)
Z = R * X * R.rev()
Z.expand()
Z.collect([Binv, s, c, XdotY])
W = Z | Y
W.expand()
W.collect([s * Binv])
M = 1 / Bsq
W.subs(Binv**2, M)
W.simplify()
Bmag = sympy.sqrt(XdotY**2 - 2 * XdotY * Xdote * Ydote)
W.collect([Binv * c * s, XdotY])
W.subs(2 * XdotY**2 - 4 * XdotY * Xdote * Ydote, 2 / (Binv**2))
W.subs(2 * c * s, S)
W.subs(c**2, (C + 1) / 2)
W.subs(s**2, (C - 1) / 2)
W.simplify()
W.subs(1 / Binv, Bmag)
W = W().expand()
#print '(R*X*R.rev()).Y =',W
Wd = collect(W, [C, S], exact=True, evaluate=False)
#print 'Wd =',Wd
Wd_1 = Wd[ONE]
Wd_C = Wd[C]
Wd_S = Wd[S]
#print '|B| =',Bmag
Wd_1 = Wd_1.subs(Bmag, 1 / Binv)
Wd_C = Wd_C.subs(Bmag, 1 / Binv)
Wd_S = Wd_S.subs(Bmag, 1 / Binv)
#print 'Wd[ONE] =',Wd_1
#print 'Wd[C] =',Wd_C
#print 'Wd[S] =',Wd_S
lhs = Wd_1 + Wd_C * C
rhs = -Wd_S * S
lhs = lhs**2
rhs = rhs**2
W = (lhs - rhs).expand()
W = (W.subs(1 / Binv**2, Bmag**2)).expand()
#print 'W =',W
W = (W.subs(S**2, C**2 - 1)).expand()
W = collect(W, [C**2, C], evaluate=False)
#print 'W =',W
a = W[C**2]
b = W[(C**2)**(sympify(1) / 2)]
c = W[ONE]
#print 'a =',a
#print 'b =',b
#print 'c =',c
D = (b**2 - 4 * a * c).expand()
#print 'Setting to 0 and solving for C gives:'
#print 'Descriminant D = b^2-4*a*c =',D
C = (-b / (2 * a)).expand()
#print 'C = cosh(alpha) = -b/(2*a) =',C
"""
Wd = collect(W,[C,S],evaluate=False)
lhs = Wd[ONE]+Wd[C]*C
rhs = -Wd[S]*S
lhs = lhs**2
rhs = rhs**2
W = (lhs-rhs).expand()
W = (W.subs(S**2,C**2-1)).expand()
W = collect(W,[C**2,C],evaluate=False)
a = W[C**2]
b = W[abs(C)]
c = W[ONE]
D = (b**2-4*a*c).expand()
C = (-b/(2*a)).expand()
"""
assert C == 1 - XdotY / (Xdote * Ydote)
return(Fx)
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))
if __name__ == '__main__':
MV.setup('a b c d e',debug=0)
MV.set_str_format(1)
print 'e|(a^b) =',e|(a^b)
print 'e|(a^b^c) =',e|(a^b^c)
print 'a*(b^c)-b*(a^c)+c*(a^b) =',a*(b^c)-b*(a^c)+c*(a^b)
print 'e|(a^b^c^d) =',e|(a^b^c^d)
print -d*(a^b^c)+c*(a^b^d)-b*(a^c^d)+a*(b^c^d)
print (a^b)|(c^d)
#sys.exit(0)
"""
Note that is A and B are multivectors:
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))
if __name__ == '__main__':
MV.setup('a b c d e', debug=0)
MV.set_str_format(1)
print 'e|(a^b) =', e | (a ^ b)
print 'e|(a^b^c) =', e | (a ^ b ^ c)
print 'a*(b^c)-b*(a^c)+c*(a^b) =', a * (b ^ c) - b * (a ^ c) + c * (a ^ b)
print 'e|(a^b^c^d) =', e | (a ^ b ^ c ^ d)
print -d * (a ^ b ^ c) + c * (a ^ b ^ d) - b * (a ^ c ^ d) + a * (b ^ c
^ d)
print(a ^ b) | (c ^ d)
#sys.exit(0)
"""
Note that is A and B are multivectors:
