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 test_extract_plane_and_line(): 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) L = P1^P2^n delta = (L|n)|nbar delta_test = 2*p1-2*p2 diff = delta-delta_test diff.compact() assert diff == ZERO_MV 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_vector_extraction(): metric = ' 0 -1 #,'+ \ '-1 0 #,'+ \ ' # # #,' MV.setup('P1 P2 a',metric) ZERO_MV = MV() B = P1^P2 Bsq = B*B ap = a-(a^B)*B Ap = ap+ap*B Am = ap-ap*B Ap_test = (-2*P2dota)*P1 Am_test = (-2*P1dota)*P2 Ap.compact() Am.compact() Ap_test.compact() Am_test.compact() assert Ap == Ap_test assert Am == Am_test Ap2 = Ap*Ap Am2 = Am*Am Ap2.compact() Am2.compact() assert Ap2 == ZERO_MV assert Am2 == ZERO_MV
def test_extract_plane_and_line(): 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) L = P1 ^ P2 ^ n delta = (L | n) | nbar delta_test = 2 * p1 - 2 * p2 diff = delta - delta_test diff.compact() assert diff == ZERO_MV 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_vector_extraction(): metric = ' 0 -1 #,'+ \ '-1 0 #,'+ \ ' # # #,' MV.setup('P1 P2 a', metric) ZERO_MV = MV() B = P1 ^ P2 Bsq = B * B ap = a - (a ^ B) * B Ap = ap + ap * B Am = ap - ap * B Ap_test = (-2 * P2dota) * P1 Am_test = (-2 * P1dota) * P2 Ap.compact() Am.compact() Ap_test.compact() Am_test.compact() assert Ap == Ap_test assert Am == Am_test Ap2 = Ap * Ap Am2 = Am * Am Ap2.compact() Am2.compact() assert Ap2 == ZERO_MV assert Am2 == ZERO_MV
def test_geometry(): metric = '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' MV.setup('e0 e1 e2 n nbar', metric, debug=0) e = n + nbar #conformal representation of points ZERO_MV = MV() A = make_vector(e0) # point a = (1,0,0) A = F(a) B = make_vector(e1) # point b = (0,1,0) B = F(b) C = make_vector(-1 * e0) # point c = (-1,0,0) C = F(c) D = make_vector(e2) # point d = (0,0,1) D = F(d) X = make_vector('x', 3) Circle = A ^ B ^ C ^ X Line = A ^ B ^ n ^ X Sphere = A ^ B ^ C ^ D ^ X Plane = A ^ B ^ n ^ D ^ X Circle_test = -x2 * (e0 ^ e1 ^ e2 ^ n) + x2 * ( e0 ^ e1 ^ e2 ^ nbar) + HALF * (-1 + x0**2 + x1**2 + x2**2) * (e0 ^ e1 ^ n ^ nbar) diff = Circle - Circle_test diff.compact() assert diff == ZERO_MV Line_test = -x2*(e0^e1^e2^n)+HALF*(-1+x0+x1)*(e0^e1^n^nbar)+(HALF*x2)*(e0^e2^n^nbar)+\ (-HALF*x2)*(e1^e2^n^nbar) diff = Line - Line_test diff.compact() assert diff == ZERO_MV Sphere_test = HALF * (1 - x0**2 - x1**2 - x2**2) * (e0 ^ e1 ^ e2 ^ n ^ nbar) diff = Sphere - Sphere_test diff.compact() assert diff == ZERO_MV Plane_test = HALF * (1 - x0 - x1 - x2) * (e0 ^ e1 ^ e2 ^ n ^ nbar) diff = Plane - Plane_test diff.compact() assert diff == ZERO_MV
def test_reciprocal_frame(): metric = '1 # #,'+ \ '# 1 #,'+ \ '# # 1,' MV.setup('e1 e2 e3',metric) E = e1^e2^e3 Esq = (E*E)() Esq_inv = 1/Esq E1 = (e2^e3)*E E2 = (-1)*(e1^e3)*E E3 = (e1^e2)*E w = (E1|e2) w.collect(MV.g) w = w().expand() w = (E1|e3) w.collect(MV.g) w = w().expand() assert w == 0 w = (E2|e1) w.collect(MV.g) w = w().expand() assert w == 0 w = (E2|e3) w.collect(MV.g) w = w().expand() assert w == 0 w = (E3|e1) w.collect(MV.g) w = w().expand() assert w == 0 w = (E3|e2) w.collect(MV.g) w = w().expand() assert w == 0 w = (E1|e1) w = w().expand() Esq = Esq.expand() assert w/Esq == 1 w = (E2|e2) w = w().expand() assert w/Esq == 1 w = (E3|e3) w = w().expand() assert w/Esq == 1
def test_reciprocal_frame(): metric = '1 # #,'+ \ '# 1 #,'+ \ '# # 1,' MV.setup('e1 e2 e3', metric) E = e1 ^ e2 ^ e3 Esq = (E * E)() Esq_inv = 1 / Esq E1 = (e2 ^ e3) * E E2 = (-1) * (e1 ^ e3) * E E3 = (e1 ^ e2) * E w = (E1 | e2) w.collect(MV.g) w = w().expand() w = (E1 | e3) w.collect(MV.g) w = w().expand() assert w == 0 w = (E2 | e1) w.collect(MV.g) w = w().expand() assert w == 0 w = (E2 | e3) w.collect(MV.g) w = w().expand() assert w == 0 w = (E3 | e1) w.collect(MV.g) w = w().expand() assert w == 0 w = (E3 | e2) w.collect(MV.g) w = w().expand() assert w == 0 w = (E1 | e1) w = w().expand() Esq = Esq.expand() assert w / Esq == 1 w = (E2 | e2) w = w().expand() assert w / Esq == 1 w = (E3 | e3) w = w().expand() assert w / Esq == 1
def test_geometry(): metric = '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' MV.setup('e0 e1 e2 n nbar',metric,debug=0) e = n+nbar #conformal representation of points ZERO_MV = MV() A = make_vector(e0) # point a = (1,0,0) A = F(a) B = make_vector(e1) # point b = (0,1,0) B = F(b) C = make_vector(-1*e0) # point c = (-1,0,0) C = F(c) D = make_vector(e2) # point d = (0,0,1) D = F(d) X = make_vector('x',3) Circle = A^B^C^X Line = A^B^n^X Sphere = A^B^C^D^X Plane = A^B^n^D^X Circle_test = -x2*(e0^e1^e2^n)+x2*(e0^e1^e2^nbar)+HALF*(-1+x0**2+x1**2+x2**2)*(e0^e1^n^nbar) diff = Circle-Circle_test diff.compact() assert diff == ZERO_MV Line_test = -x2*(e0^e1^e2^n)+HALF*(-1+x0+x1)*(e0^e1^n^nbar)+(HALF*x2)*(e0^e2^n^nbar)+\ (-HALF*x2)*(e1^e2^n^nbar) diff = Line-Line_test diff.compact() assert diff == ZERO_MV Sphere_test = HALF*(1-x0**2-x1**2-x2**2)*(e0^e1^e2^n^nbar) diff = Sphere-Sphere_test diff.compact() assert diff == ZERO_MV Plane_test = HALF*(1-x0-x1-x2)*(e0^e1^e2^n^nbar) diff = Plane-Plane_test diff.compact() assert diff == ZERO_MV
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)
return(Fx) 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)) if __name__ == '__main__': MV.setup('a b c d e',debug=0) 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) #sys.exit(0) """ Note that is A and B are multivectors:
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)) if __name__ == '__main__': MV.setup('a b c d e', debug=0) 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) #sys.exit(0) """ Note that is A and B are multivectors: