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
from __future__ import print_function
from sympy import symbols, sin
from galgebra.printer import Format, xpdf, Fmt
from galgebra.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))
xpdf(pdfprog=None, 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(Binv, 1 / 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
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))
xpdf(pdfprog=None, paper=(6, 7))
