def test_reciprocal_frame_test():
GA_Printer.on()
metric = '1 # #,'+ \
'# 1 #,'+ \
'# # 1,'
(e1,e2,e3) = MV.setup('e1 e2 e3',metric)
E = e1^e2^e3
Esq = (E*E).scalar()
assert str(E) == 'e1^e2^e3'
assert str(Esq) == '(e1.e2)**2 - 2*(e1.e2)*(e1.e3)*(e2.e3) + (e1.e3)**2 + (e2.e3)**2 - 1'
Esq_inv = 1/Esq
E1 = (e2^e3)*E
E2 = (-1)*(e1^e3)*E
E3 = (e1^e2)*E
assert str(E1) == '((e2.e3)**2 - 1)*e1 + ((e1.e2) - (e1.e3)*(e2.e3))*e2 + (-(e1.e2)*(e2.e3) + (e1.e3))*e3'
assert str(E2) == '((e1.e2) - (e1.e3)*(e2.e3))*e1 + ((e1.e3)**2 - 1)*e2 + (-(e1.e2)*(e1.e3) + (e2.e3))*e3'
assert str(E3) == '(-(e1.e2)*(e2.e3) + (e1.e3))*e1 + (-(e1.e2)*(e1.e3) + (e2.e3))*e2 + ((e1.e2)**2 - 1)*e3'
w = (E1|e2)
w = w.expand()
assert str(w) == '0'
w = (E1|e3)
w = w.expand()
assert str(w) == '0'
w = (E2|e1)
w = w.expand()
assert str(w) == '0'
w = (E2|e3)
w = w.expand()
assert str(w) == '0'
w = (E3|e1)
w = w.expand()
assert str(w) == '0'
w = (E3|e2)
w = w.expand()
assert str(w) == '0'
w = (E1|e1)
w = (w.expand()).scalar()
Esq = expand(Esq)
assert str(simplify(w/Esq)) == '1'
w = (E2|e2)
w = (w.expand()).scalar()
assert str(simplify(w/Esq)) == '1'
w = (E3|e3)
w = (w.expand()).scalar()
assert str(simplify(w/Esq)) == '1'
GA_Printer.off()
return
def test_reciprocal_frame_test():
metric = '1 # #,'+ \
'# 1 #,'+ \
'# # 1,'
(e1, e2, e3) = MV.setup('e1 e2 e3', metric)
E = e1 ^ e2 ^ e3
Esq = (E * E).scalar()
assert str(E) == 'e1^e2^e3'
assert str(
Esq
) == '(e1.e2)**2 - 2*(e1.e2)*(e1.e3)*(e2.e3) + (e1.e3)**2 + (e2.e3)**2 - 1'
Esq_inv = 1 / Esq
E1 = (e2 ^ e3) * E
E2 = (-1) * (e1 ^ e3) * E
E3 = (e1 ^ e2) * E
assert str(
E1
) == '((e2.e3)**2 - 1)*e1 + ((e1.e2) - (e1.e3)*(e2.e3))*e2 + (-(e1.e2)*(e2.e3) + (e1.e3))*e3'
assert str(
E2
) == '((e1.e2) - (e1.e3)*(e2.e3))*e1 + ((e1.e3)**2 - 1)*e2 + (-(e1.e2)*(e1.e3) + (e2.e3))*e3'
assert str(
E3
) == '(-(e1.e2)*(e2.e3) + (e1.e3))*e1 + (-(e1.e2)*(e1.e3) + (e2.e3))*e2 + ((e1.e2)**2 - 1)*e3'
w = (E1 | e2)
w = w.expand()
assert str(w) == '0'
w = (E1 | e3)
w = w.expand()
assert str(w) == '0'
w = (E2 | e1)
w = w.expand()
assert str(w) == '0'
w = (E2 | e3)
w = w.expand()
assert str(w) == '0'
w = (E3 | e1)
w = w.expand()
assert str(w) == '0'
w = (E3 | e2)
w = w.expand()
assert str(w) == '0'
w = (E1 | e1)
w = (w.expand()).scalar()
Esq = expand(Esq)
assert str(simplify(w / Esq)) == '1'
w = (E2 | e2)
w = (w.expand()).scalar()
assert str(simplify(w / Esq)) == '1'
w = (E3 | e3)
w = (w.expand()).scalar()
assert str(simplify(w / Esq)) == '1'
return
def test_noneuclidian_distance_calculation():
from sympy import solve,sqrt
GA_Printer.on()
metric = '0 # #,# 0 #,# # 1'
(X,Y,e) = MV.setup('X Y e',metric)
assert str((X^Y)*(X^Y)) == '(X.Y)**2'
L = X^Y^e
B = L*e
assert str(B) == 'X^Y - (Y.e)*X^e + (X.e)*Y^e'
Bsq = B*B
assert str(Bsq) == '(X.Y)*((X.Y) - 2*(X.e)*(Y.e))'
Bsq = Bsq.scalar()
assert str(B) == 'X^Y - (Y.e)*X^e + (X.e)*Y^e'
BeBr =B*e*B.rev()
assert str(BeBr) == '((X.Y)*(-(X.Y) + 2*(X.e)*(Y.e)))*e'
assert str(B*B) == '(X.Y)*((X.Y) - 2*(X.e)*(Y.e))'
assert str(L*L) == '(X.Y)*((X.Y) - 2*(X.e)*(Y.e))'
(s,c,Binv,M,S,C,alpha,XdotY,Xdote,Ydote) = symbols('s c (1/B) M S C alpha (X.Y) (X.e) (Y.e)')
Bhat = Binv*B
R = c+s*Bhat
assert str(R) == 'c + (1/B)*s*X^Y - (1/B)*(Y.e)*s*X^e + (1/B)*(X.e)*s*Y^e'
Z = R*X*R.rev()
Z.obj = expand(Z.obj)
Z.obj = Z.obj.collect([Binv,s,c,XdotY])
assert str(Z) == '((1/B)**2*(X.Y)**2*s**2 - 2*(1/B)**2*(X.Y)*(X.e)*(Y.e)*s**2 + 2*(1/B)*(X.Y)*c*s - 2*(1/B)*(X.e)*(Y.e)*c*s + c**2)*X + 2*(1/B)*(X.e)**2*c*s*Y + (2*(1/B)*(X.Y)*(X.e)*s*(-(1/B)*(X.Y)*s + 2*(1/B)*(X.e)*(Y.e)*s - c))*e'
W = Z|Y
# From this point forward all calculations are with sympy scalars
W = W.scalar()
assert str(W) == '(1/B)**2*(X.Y)**3*s**2 - 4*(1/B)**2*(X.Y)**2*(X.e)*(Y.e)*s**2 + 4*(1/B)**2*(X.Y)*(X.e)**2*(Y.e)**2*s**2 + 2*(1/B)*(X.Y)**2*c*s - 4*(1/B)*(X.Y)*(X.e)*(Y.e)*c*s + (X.Y)*c**2'
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)
assert str(W) == '(X.Y)*C - (X.e)*(Y.e)*C + (X.e)*(Y.e) + S*sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
Wd = collect(W,[C,S],exact=True,evaluate=False)
Wd_1 = Wd[ONE]
Wd_C = Wd[C]
Wd_S = Wd[S]
assert str(Wd_1) == '(X.e)*(Y.e)'
assert str(Wd_C) == '(X.Y) - (X.e)*(Y.e)'
assert str(Wd_S) == 'sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
assert str(Bmag) == 'sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
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])
assert str(a) == '(X.e)**2*(Y.e)**2'
assert str(b) == '2*(X.e)*(Y.e)*((X.Y) - (X.e)*(Y.e))'
assert str(c) == '(X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e) + (X.e)**2*(Y.e)**2'
x = Symbol('x')
C = solve(a*x**2+b*x+c,x)[0]
assert str(expand(simplify(expand(C)))) == '-(X.Y)/((X.e)*(Y.e)) + 1'
GA_Printer.off()
return
def test_noneuclidian_distance_calculation():
from sympy import solve, sqrt
metric = '0 # #,# 0 #,# # 1'
(X, Y, e) = MV.setup('X Y e', metric)
assert str((X ^ Y) * (X ^ Y)) == '(X.Y)**2'
L = X ^ Y ^ e
B = L * e
assert str(B) == 'X^Y - (Y.e)*X^e + (X.e)*Y^e'
Bsq = B * B
assert str(Bsq) == '(X.Y)*((X.Y) - 2*(X.e)*(Y.e))'
Bsq = Bsq.scalar()
assert str(B) == 'X^Y - (Y.e)*X^e + (X.e)*Y^e'
BeBr = B * e * B.rev()
assert str(BeBr) == '((X.Y)*(-(X.Y) + 2*(X.e)*(Y.e)))*e'
assert str(B * B) == '(X.Y)*((X.Y) - 2*(X.e)*(Y.e))'
assert str(L * L) == '(X.Y)*((X.Y) - 2*(X.e)*(Y.e))'
(s, c, Binv, M, S, C, alpha, XdotY, Xdote,
Ydote) = symbols('s c (1/B) M S C alpha (X.Y) (X.e) (Y.e)')
Bhat = Binv * B
R = c + s * Bhat
assert str(R) == 'c + (1/B)*s*X^Y - (1/B)*(Y.e)*s*X^e + (1/B)*(X.e)*s*Y^e'
Z = R * X * R.rev()
Z.obj = expand(Z.obj)
Z.obj = Z.obj.collect([Binv, s, c, XdotY])
assert str(
Z
) == '((1/B)**2*(X.Y)**2*s**2 - 2*(1/B)**2*(X.Y)*(X.e)*(Y.e)*s**2 + 2*(1/B)*(X.Y)*c*s - 2*(1/B)*(X.e)*(Y.e)*c*s + c**2)*X + 2*(1/B)*(X.e)**2*c*s*Y + (2*(1/B)*(X.Y)*(X.e)*s*(-(1/B)*(X.Y)*s + 2*(1/B)*(X.e)*(Y.e)*s - c))*e'
W = Z | Y
# From this point forward all calculations are with sympy scalars
W = W.scalar()
assert str(
W
) == '(1/B)**2*(X.Y)**3*s**2 - 4*(1/B)**2*(X.Y)**2*(X.e)*(Y.e)*s**2 + 4*(1/B)**2*(X.Y)*(X.e)**2*(Y.e)**2*s**2 + 2*(1/B)*(X.Y)**2*c*s - 4*(1/B)*(X.Y)*(X.e)*(Y.e)*c*s + (X.Y)*c**2'
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)
assert str(
W
) == '(X.Y)*C - (X.e)*(Y.e)*C + (X.e)*(Y.e) + S*sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
Wd = collect(W, [C, S], exact=True, evaluate=False)
Wd_1 = Wd[ONE]
Wd_C = Wd[C]
Wd_S = Wd[S]
assert str(Wd_1) == '(X.e)*(Y.e)'
assert str(Wd_C) == '(X.Y) - (X.e)*(Y.e)'
assert str(Wd_S) == 'sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
assert str(Bmag) == 'sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
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])
assert str(a) == '(X.e)**2*(Y.e)**2'
assert str(b) == '2*(X.e)*(Y.e)*((X.Y) - (X.e)*(Y.e))'
assert str(c) == '(X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e) + (X.e)**2*(Y.e)**2'
x = Symbol('x')
C = solve(a * x**2 + b * x + c, x)[0]
assert str(expand(simplify(expand(C)))) == '-(X.Y)/((X.e)*(Y.e)) + 1'
return