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 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 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)
"""
X and Y are conformal mappings of two vectors on the Poincare disk
"""
L = X^Y^e
"""
B is the bivector generator of translations on the circle defined by L
"""
B = L*e
Bsq = (B*B)()
print 'L = X^Y^e is a non-euclidian line'
print 'B = L*e =',B
BeBr =B*e*B.rev()
print 'B*e*B.rev() =',BeBr
print 'B^2 =',Bsq
print 'L^2 =',(L*L)()
make_symbols('s c Binv M S C alpha')
"""
s = sinh(alpha/2)
c = cosh(alpha/2)
Binv = 1/sqrt(B*B) It can be shown that B*B > 0
"""
Bhat = Binv*B # Normalize translation generator
R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2)
print 's = sinh(alpha/2) and c = cosh(alpha/2)'
print 'R = exp(alpha*B/(2*|B|)) =',R
Z = R*X*R.rev()
Z.expand()
Z.collect([Binv,s,c,XdotY])
print 'R*X*R.rev() =',Z
#W = Z^Y
W = Z|Y
"""
X and Y are conformal mappings of two vectors on the Poincare disk
"""
L = X ^ Y ^ e
"""
B is the bivector generator of translations on the circle defined by L
"""
B = L * e
Bsq = (B * B)()
print 'L = X^Y^e is a non-euclidian line'
print 'B = L*e =', B
BeBr = B * e * B.rev()
print 'B*e*B.rev() =', BeBr
print 'B^2 =', Bsq
print 'L^2 =', (L * L)()
make_symbols('s c Binv M S C alpha')
"""
s = sinh(alpha/2)
c = cosh(alpha/2)
Binv = 1/sqrt(B*B) It can be shown that B*B > 0
"""
Bhat = Binv * B # Normalize translation generator
R = c + s * Bhat # Rotor R = exp(alpha*Bhat/2)
print 's = sinh(alpha/2) and c = cosh(alpha/2)'
print 'R = exp(alpha*B/(2*|B|)) =', R
Z = R * X * R.rev()
Z.expand()
Z.collect([Binv, s, c, XdotY])
print 'R*X*R.rev() =', Z
#W = Z^Y
W = Z | Y