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_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'
p1, p2, p3, n, nbar = MV.setup('p1 p2 p3 n nbar', metric, debug=0)
MV.set_str_format(1)
P1 = F(p1, n, nbar)
P2 = F(p2, n, nbar)
P3 = F(p3, n, nbar)
#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
#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
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)
MV.set_str_format(1)
x1, y1, z1 = sympy.symbols('x1 y1 z1')
x2, y2, z2 = sympy.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
diff.compact()
assert diff == 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
diff.compact()
assert diff == ZERO
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
diff.compact()
assert diff == ZERO
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__':
ti = time.time()
MV.setup('a b c d e')
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)
print 'Example: non-euclidian distance calculation'
metric = '0 # #,# 0 #,# # 1'
MV.setup('X Y e',metric)
MV.set_str_format(1)
L = X^Y^e
def test_noneuclidian():
"""
Test of complex geometric algebra manipulation to derive distance function
for 2-D hyperbolic non-euclidian space. See D&L Section 10.6.2
"""
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()
(s,c,Binv,M,S,C,alpha) = sympy.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]
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
#cosh(alpha) = 1-X.Y/((X.e)(Y.e))
#alpha is noneuclidian distance
assert C == 1-XdotY/(Xdote*Ydote)
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__':
ti = time.time()
MV.setup('a b c d e')
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)
# FIXME: currently broken
"""
print 'Example: non-euclidian distance calculation'
metric = '0 # #,# 0 #,# # 1'