def Fmt_test():
Print_Function()
e3d = Ga('e1 e2 e3',g=[1,1,1])
v = e3d.mv('v','vector')
B = e3d.mv('B','bivector')
M = e3d.mv('M','mv')
Fmt(2)
print '#Global $Fmt = 2$'
print 'v =',v
print 'B =',B
print 'M =',M
print '#Using $.Fmt()$ Function'
print 'v.Fmt(3) =',v.Fmt(3)
print 'B.Fmt(3) =',B.Fmt(3)
print 'M.Fmt(2) =',M.Fmt(2)
print 'M.Fmt(1) =',M.Fmt(1)
print '#Global $Fmt = 1$'
Fmt(1)
print 'v =',v
print 'B =',B
print 'M =',M
return
def extracting_vectors_from_conformal_2_blade():
Print_Function()
Fmt(1)
print r'B = P1\W P2'
g = '0 -1 #,'+ \
'-1 0 #,'+ \
'# # #'
c2b = Ga('P1 P2 a',g=g)
(P1,P2,a) = c2b.mv()
print 'g_{ij} =',c2b.g
B = P1^P2
Bsq = B*B
print '%B^{2} =',Bsq
ap = a-(a^B)*B
print "a' = a-(a^B)*B =",ap
Ap = ap+ap*B
Am = ap-ap*B
print "A+ = a'+a'*B =",Ap
print "A- = a'-a'*B =",Am
print '%(A+)^{2} =',Ap*Ap
print '%(A-)^{2} =',Am*Am
aB = a|B
print 'a|B =',aB
return
def properties_of_geometric_objects():
Print_Function()
global n, nbar
Fmt(1)
g = '# # # 0 0,'+ \
'# # # 0 0,'+ \
'# # # 0 0,'+ \
'0 0 0 0 2,'+ \
'0 0 0 2 0'
c3d = Ga('p1 p2 p3 n \\bar{n}',g=g)
(p1,p2,p3,n,nbar) = c3d.mv()
print 'g_{ij} =',c3d.g
P1 = F(p1)
P2 = F(p2)
P3 = F(p3)
print '\\text{Extracting direction of line from }L = P1\\W P2\\W n'
L = P1^P2^n
delta = (L|n)|nbar
print '(L|n)|\\bar{n} =',delta
print '\\text{Extracting plane of circle from }C = P1\\W P2\\W P3'
C = P1^P2^P3
delta = ((C^n)|n)|nbar
print '((C^n)|n)|\\bar{n}=',delta
print '(p2-p1)^(p3-p1)=',(p2-p1)^(p3-p1)
return
def conformal_representations_of_circles_lines_spheres_and_planes():
Print_Function()
global n,nbar
Fmt(1)
g = '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'
c3d = Ga('e_1 e_2 e_3 n \\bar{n}',g=g)
(e1,e2,e3,n,nbar) = c3d.mv()
print 'g_{ij} =',c3d.g
e = n+nbar
#conformal representation of points
A = make_vector(e1, ga=c3d) # point a = (1,0,0) A = F(a)
B = make_vector(e2, ga=c3d) # point b = (0,1,0) B = F(b)
C = make_vector(-e1, ga=c3d) # point c = (-1,0,0) C = F(c)
D = make_vector(e3, ga=c3d) # point d = (0,0,1) D = F(d)
X = make_vector('x',3, ga=c3d)
print 'F(a) =',A
print 'F(b) =',B
print 'F(c) =',C
print 'F(d) =',D
print 'F(x) =',X
print '#a = e1, b = e2, c = -e1, and d = e3'
print '#A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.'
print '#Circle through a, b, and c'
print 'Circle: A^B^C^X = 0 =',(A^B^C^X)
print '#Line through a and b'
print 'Line : A^B^n^X = 0 =',(A^B^n^X)
print '#Sphere through a, b, c, and d'
print 'Sphere: A^B^C^D^X = 0 =',(((A^B)^C)^D)^X
print '#Plane through a, b, and d'
print 'Plane : A^B^n^D^X = 0 =',(A^B^n^D^X)
L = (A^B^e)^X
L.Fmt(3,'Hyperbolic\\;\\; Circle: (A^B^e)^X = 0')
return
Format()
xyz_coords = (x, y, z) = symbols('x y z', real=True)
(o3d, ex, ey, ez) = Ga.build('e', g=[1, 1, 1], coords=xyz_coords, norm=True)
f = o3d.mv('f', 'scalar', f=True)
lap = o3d.grad * o3d.grad
print r'%\nabla^{2} = \nabla\cdot\nabla =', lap
print r'%\lp\nabla^{2}\rp f =', lap * f
print r'%\nabla\cdot\lp\nabla f\rp =', o3d.grad | (o3d.grad * f)
sph_coords = (r, th, phi) = symbols('r theta phi', real=True)
(sp3d, er, eth, ephi) = Ga.build('e',
g=[1, r**2, r**2 * sin(th)**2],
coords=sph_coords,
norm=True)
f = sp3d.mv('f', 'scalar', f=True)
lap = sp3d.grad * sp3d.grad
print r'%\nabla^{2} = \nabla\cdot\nabla =', lap
print r'%\lp\nabla^{2}\rp f =', lap * f
print r'%\nabla\cdot\lp\nabla f\rp =', sp3d.grad | (sp3d.grad * f)
print Fmt([o3d.grad, o3d.grad])
F = sp3d.mv('F', 'vector', f=True)
print F.title
print F
F.fmt = 3
print F.title
print F
print F.title
print Fmt((F, F))
xpdf(paper=(6, 7))
from sympy import symbols, sin
from printer import Format, xpdf, Fmt
from ga import Ga
import sys
Format()
xyz_coords = (x, y, z) = symbols('x y z', real=True)
(o3d, ex, ey, ez) = Ga.build('e', g=[1, 1, 1], coords=xyz_coords, norm=True)
f = o3d.mv('f', 'scalar', f=True)
F = o3d.mv('F', 'vector', f=True)
B = o3d.mv('B', 'bivector', f=True)
l = [f, F, B]
print(Fmt(l))
print(Fmt(l, 1))
print(F.Fmt(3))
print(B.Fmt(3))
lap = o3d.grad * o3d.grad
print(r'%\nabla^{2} = \nabla\cdot\nabla =', lap)
dop = lap + o3d.grad
print(dop.Fmt(fmt=3, dop_fmt=3))
xpdf(paper=(6, 7))
from sympy import symbols, sin
from printer import Format, xpdf, Fmt
from ga import Ga
import sys
Format()
xyz_coords = (x, y, z) = symbols('x y z', real=True)
(o3d, ex, ey, ez) = Ga.build('e', g=[1, 1, 1], coords=xyz_coords, norm=True)
f = o3d.mv('f', 'scalar', f=True)
F = o3d.mv('F', 'vector', f=True)
B = o3d.mv('B', 'bivector', f=True)
l = [f, F, B]
print Fmt(l)
print Fmt(l, 1)
print F.Fmt(3)
print B.Fmt(3)
lap = o3d.grad * o3d.grad
print r'%\nabla^{2} = \nabla\cdot\nabla =', lap
dop = lap + o3d.grad
print dop.Fmt(fmt=3, dop_fmt=3)
xpdf(paper=(6, 7))
def reciprocal_frame_test():
Print_Function()
Fmt(1)
g = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
ng3d = Ga('e1 e2 e3',g=g)
(e1,e2,e3) = ng3d.mv()
print 'g_{ij} =',ng3d.g
E = e1^e2^e3
Esq = (E*E).scalar()
print 'E =',E
print '%E^{2} =',Esq
Esq_inv = 1/Esq
E1 = (e2^e3)*E
E2 = (-1)*(e1^e3)*E
E3 = (e1^e2)*E
print 'E1 = (e2^e3)*E =',E1
print 'E2 =-(e1^e3)*E =',E2
print 'E3 = (e1^e2)*E =',E3
w = (E1|e2)
w = w.expand()
print 'E1|e2 =',w
w = (E1|e3)
w = w.expand()
print 'E1|e3 =',w
w = (E2|e1)
w = w.expand()
print 'E2|e1 =',w
w = (E2|e3)
w = w.expand()
print 'E2|e3 =',w
w = (E3|e1)
w = w.expand()
print 'E3|e1 =',w
w = (E3|e2)
w = w.expand()
print 'E3|e2 =',w
w = (E1|e1)
w = (w.expand()).scalar()
Esq = expand(Esq)
print '%(E1\\cdot e1)/E^{2} =',simplify(w/Esq)
w = (E2|e2)
w = (w.expand()).scalar()
print '%(E2\\cdot e2)/E^{2} =',simplify(w/Esq)
w = (E3|e3)
w = (w.expand()).scalar()
print '%(E3\\cdot e3)/E^{2} =',simplify(w/Esq)
return
def noneuclidian_distance_calculation():
Print_Function()
from sympy import solve,sqrt
Fmt(1)
g = '0 # #,# 0 #,# # 1'
nel = Ga('X Y e',g=g)
(X,Y,e) = nel.mv()
print 'g_{ij} =',nel.g
print '%(X\\W Y)^{2} =',(X^Y)*(X^Y)
L = X^Y^e
B = L*e # D&L 10.152
Bsq = (B*B).scalar()
print '#%L = X\\W Y\\W e \\text{ is a non-euclidian line}'
print 'B = L*e =',B
BeBr =B*e*B.rev()
print '%BeB^{\\dagger} =',BeBr
print '%B^{2} =',B*B
print '%L^{2} =',L*L # D&L 10.153
(s,c,Binv,M,S,C,alpha) = symbols('s c (1/B) M S C alpha')
XdotY = nel.g[0,1]
Xdote = nel.g[0,2]
Ydote = nel.g[1,2]
Bhat = Binv*B # D&L 10.154
R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2)
print '#%s = \\f{\\sinh}{\\alpha/2} \\text{ and } c = \\f{\\cosh}{\\alpha/2}'
print '%e^{\\alpha B/{2\\abs{B}}} =',R
Z = R*X*R.rev() # D&L 10.155
Z.obj = expand(Z.obj)
Z.obj = Z.obj.collect([Binv,s,c,XdotY])
Z.Fmt(3,'%RXR^{\\dagger}')
W = Z|Y # Extract scalar part of multivector
# From this point forward all calculations are with sympy scalars
#print '#Objective is to determine value of C = cosh(alpha) such that W = 0'
W = W.scalar()
print '%W = Z\\cdot Y =',W
W = expand(W)
W = simplify(W)
W = W.collect([s*Binv])
M = 1/Bsq
W = W.subs(Binv**2,M)
W = simplify(W)
Bmag = sqrt(XdotY**2-2*XdotY*Xdote*Ydote)
W = W.collect([Binv*c*s,XdotY])
#Double angle substitutions
W = W.subs(2*XdotY**2-4*XdotY*Xdote*Ydote,2/(Binv**2))
W = W.subs(2*c*s,S)
W = W.subs(c**2,(C+1)/2)
W = W.subs(s**2,(C-1)/2)
W = simplify(W)
W = W.subs(1/Binv,Bmag)
W = expand(W)
print '#%S = \\f{\\sinh}{\\alpha} \\text{ and } C = \\f{\\cosh}{\\alpha}'
print 'W =',W
Wd = collect(W,[C,S],exact=True,evaluate=False)
Wd_1 = Wd[one]
Wd_C = Wd[C]
Wd_S = Wd[S]
print '%\\text{Scalar Coefficient} =',Wd_1
print '%\\text{Cosh Coefficient} =',Wd_C
print '%\\text{Sinh Coefficient} =',Wd_S
print '%\\abs{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)
lhs = Wd_1+Wd_C*C
rhs = -Wd_S*S
lhs = lhs**2
rhs = rhs**2
W = expand(lhs-rhs)
W = expand(W.subs(1/Binv**2,Bmag**2))
W = expand(W.subs(S**2,C**2-1))
W = W.collect([C,C**2],evaluate=False)
a = simplify(W[C**2])
b = simplify(W[C])
c = simplify(W[one])
print '#%\\text{Require } aC^{2}+bC+c = 0'
print 'a =',a
print 'b =',b
print 'c =',c
x = Symbol('x')
C = solve(a*x**2+b*x+c,x)[0]
print '%b^{2}-4ac =',simplify(b**2-4*a*c)
print '%\\f{\\cosh}{\\alpha} = C = -b/(2a) =',expand(simplify(expand(C)))
return
from ga import Ga
import sys
Format()
xyz_coords = (x, y, z) = symbols('x y z', real=True)
(o3d, ex, ey, ez) = Ga.build('e', g=[1, 1, 1], coords=xyz_coords, norm=True)
f = o3d.mv('f', 'scalar', f=True)
lap = o3d.grad*o3d.grad
print(r'\nabla =', o3d.grad)
print(r'%\nabla^{2} = \nabla . \nabla =', lap)
print(r'%\lp\nabla^{2}\rp f =', lap*f)
print(r'%\nabla\cdot\lp\nabla f\rp =', o3d.grad | (o3d.grad * f))
sph_coords = (r, th, phi) = symbols('r theta phi', real=True)
(sp3d, er, eth, ephi) = Ga.build('e', g=[1, r**2, r**2 * sin(th)**2], coords=sph_coords, norm=True)
f = sp3d.mv('f', 'scalar', f=True)
lap = sp3d.grad*sp3d.grad
print(r'%\nabla^{2} = \nabla\cdot\nabla =', lap)
print(r'%\lp\nabla^{2}\rp f =', lap*f)
print(r'%\nabla\cdot\lp\nabla f\rp =', sp3d.grad | (sp3d.grad * f))
print(Fmt([o3d.grad, o3d.grad]))
F = sp3d.mv('F', 'vector', f=True)
print(F.title)
print(F)
F.fmt = 3
print(F.title)
print(F)
print(F.title)
print(Fmt((F,F)))
xpdf(paper=(6, 7))