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index.htm
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<html>
<head>
<link rel="stylesheet" type="text/css" href="https://sagecell.sagemath.org/static/sagecell_embed.css">
</head>
<body>
<script src="https://sagecell.sagemath.org/static/embedded_sagecell.js"></script>
<script>sagecell.makeSagecell({"inputLocation": ".sage"});</script>
<p>Davis, C., Geyer, S., Johnson, W., & Xie, Z. (2018). Inverse problem of central configurations in the collinear 5-body problem.
<em>Journal of Mathematical Physics, 59.</em> <a href="https://doi.org/10.1063/1.5011680">doi:10.1063/1.5011680</a></p>
<h2><a href="E14">E<sub>14</sub> Region Interactive Graph</a></h2>
<div class="sage">
<script type="text/x-sage">
# https://sagecell.sagemath.org/?z=eJyFVl1v2jAUfR4S_8FqHxoncRInzqZWYpq0h75Qqb-giIVA0Qggx9q6_frd668YCKwVikOOzzn3wzes5aEj_XLTZv2f7sdht20y2e6WanvYk213PEhFVNurhfty0S0_tt1yOrmXZEaKTEwn08mvpYweeqLInHTVA51OVvAs6pOSvpVxpMy1h496K0lCRAzP7LfRGYxZGJCs-Rxo1AgI9msiuOV0IKOEjXIDEvCU5isAcCRuLoj7gf3UIZJet5uI-JS6RM8cFeGjJTQidMWTniZaIWRCGRWaQVg8RCDi82hBFAV9LCehA0USnURwmkCRON-J9l2j70tdm1C_BgSCGwfuTzJ2BhPI6RXP7Gt9NpIa3Oj5h9wEJQgIsPvw_9t2r1q5bBT0Xrsmi0jO-t121cqoSIvs8fExJb1qj4t--7edFVlRcEqfphMCf7gdrxteYelyMCFNTJDVfAV6Qa2cC0AxqXObkLMCewJ22bUeArlgBhSk0QEdA7XGysqUGM2IMTaeKCZ9o1_SyQQTe1rOQMw6uQzE7cMO4njbQwmxtNqWqIYuGHYYSNjNV41qgSDOGgm9QW_MZnjIaxhk6MrQ3JdZ5dbN3lmEYRK4SwgcVC8PjDa70LJxYEAyh67nt6qIDPKTlcTZN_A1V_ia__FZ182Z6yZ0zX2coNns82blk9BxPHsYNm7jTdzQGJfQ5XFXUXNCDbI0yHJuBDyyvEAKgxQaKQak0EgIWMRqKI3bVZtd9dwG7nbVbpcaOyMukrU-lkEclnatT0Xg2sJ1VwYWHVw3V2AiaBfhZ8D9Ztl1yzocW4Mfw1PNg5YyjUSHWTEcL2PB8MWga0eVIXnxRwenjs71KAmeMU8CCaNmWmuSZzOugMvzvmCI1Vynt6vy5-EJ-f7eNj_97VHCvCR3L_yJ3KWjL_gIGp41jDN4V3Vljq8TpPQjIelEHpV2WeNSmREzg18F9EymvC3DWUNZx3PzUkcVP4JBReDFKFzjr27yS0sfDqfODx8diD-YKORn85iUuCnlIrH1YyAjmA5IM2Km1Bkxccz1bWaYnJbbp9qE4V-RJnHKTXOIK9BypX-FFpHtBsmPu4OCJq5y7EGGJyfHfvtKihTaPy3SzxQWPSwqSofdr9io7YeCX3wPacSKrE559qUehyiAlIAAmHCI_v3wO3p18V8Y-AdknnbG&lang=sage
#from sage.symbolic.relation import test_relation_maxima
#r = 0.4
var('s t L m3')
d = (s+2)^2*(t+2)^2*s^2*t^2 + 4*(s+t+2)^2*((s+2)^2*(t+2)^2-s^2*t^2)
f1L = t^2*((s+2)^2*(t+2)^2*(s^2 + (t+1)*(s+t+2)^2) - 4*(s+t+2)^2*((s+2)^2 + s^2))/d - 1
f1c = t^2*((s+2)^2*(s^2*(t+2)^2 + 4*(s+t+2)^2) - (s+t+2)^2*((s+2)^2*(t+2)^2+4*s^2))/d - 1
f2L = 1 + s + t^2*(4*s^2*(s+t+2)^2*(1+s)+4*(s+2)^2*(t+2)^2 - (t+2)^2*(s^2*(1+s)*(s+2)^2 + 4*(t+1)*(s+t+2)^2))/d
f2c = t^2*(4*(s+t+2)^2*(s^2+(t+2)^2) - (s+2)^2*(t+2)^2*(4+s^2))/d + 1
f5L = s^2*(1+s)*(s+2)^2 - 4*(s+2)^2 - 4*s^2
f5c = s^2*(s+2)^2 + 4*(s+2)^2 - 4*s^2
f4L = (t+2)^2*((t+1)*(s+t+2)^2+s^2)-s^2*(s+t+2)^2*(1+s)
f4c = s^2*(t+2)^2 - (s+t+2)^2*((t+2)^2+s^2)
@interact
def _(r=slider(0,0.999, step_size=0.001)):
g13 = 1/(1+r)^2 + t^2/d*((4*(s+2)^2*(s+t+2)^2/(1-r)^2) + 4*s^2*(s+t+2)^2/(1+r)^2 -(s+2)^2*(t+2)^2*(s+t+2)^2/(t+1-r)^2 -s^2*(s+2)^2*(t+2)^2/(1+r)^2)
g23 = t^2/d*(4*(t+2)^2*(s+t+2)^2/(1+t-r)^2 + s^2*(s+2)^2*(t+2)^2/(r+1+s)^2 - 4*(s+2)^2*(t+2)^2/(1-r)^2 - 4*s^2*(s+t+2)^2/(r+1+s)^2) - 1/(r+s+1)^2
g43 = s^2*(s+t+2)^2/(r+s+1)^2 - (t+2)^2*((s+t+2)^2/(1+t-r)^2 + s^2/(r+1)^2)
g53 = 4*(s+2)^2/(1-r)^2 + 4*s^2/(1+r)^2 - s^2*(s+2)^2/(r+s+1)^2
#2.3
cn = s^2*(f1L/(r+s+1)^2 + f2L/(r+1)^2) - t^2/d*(f4L*4*(s+2)^2/(r-1)^2 + f5L*(t+2)^2*(s+t+2)^2/(t+1-r)^2) - r
cd = t^2/d*(f4c*4*(s+2)^2/(r-1)^2 + f5c*(t+2)^2*(s+t+2)^2/(t+1-r)^2) - s^2*(f1c/(r+s+1)^2 + f2c/(r+1)^2) - 1
c = cn/cd
#m1 = ((f1L + f1c*c)*L + g13*m3)*s^2
#m2 = ((f2L + f2c*c)*L + g23*m3)*s^2
#m4 = ((f4L + f4c*c)*L + g43*m3)/d*4*t^2*(s+2)^2
#m5 = ((f5L + f5c*c)*L + g53*m3)/d*t^2*(t+2)^2*(s+t+2)^2
f13 = f1L + f1c*c
#f23 = f2L + f2c*c
f43 = f4L + f4c*c
#f53 = f5L + f5c*c
#2.4
#gamma5 = (t+2)^2*(s+t+2)^2
#f3L = s^2*(f1L + f2L) + t^2/d*(4*(s+2)^2*f4L + gamma5*f5L)
#fM3 = s^2*(g13 + g23) + t^2/d*(4*(s+2)^2*g43 + gamma5*g53) + 1
#G3 = 1/fM3
#M = f3L*L + m3/G3
# Check
#print "M1: ", test_relation_maxima(L*(-c-1-s)+m2/s^2+ m3/(r+1+s)^2+m4/(2+s)^2+m5/(2+t+s)^2 == 0)
#print "M2: ", test_relation_maxima(L*(-1-c)-m1/s^2 + m3/(1+r)^2 + m4/4 + m5/(2+t)^2 == 0)
#print "M3: ", test_relation_maxima(L*(r-c)-m1/(r+s+1)^2 - m2/(r+1)^2+m4/(r-1)^2 + m5/(t+1-r)^2 == 0)
#print "M4: ", test_relation_maxima(L*(1-c)-m1/(s+2)^2-m2/4-m3/(1-r)^2+m5/t^2 == 0)
# print "M5: ", test_relation_maxima(L*(1+t-c)-m1/(2+t+s)^2 - m2/(t+2)^2 - m3/(1+t-r)^2 - m4/t^2 == 0)
P = region_plot(f13/g13 - f43/g43 > 0, (t,0,6), (s,0,3))
P = P + text('s',(-0.5,1.75))
P = P + text('t',(2.5,-0.4))
show(P)
#f13/g13 - f43/g43
</script>
</div>
<h2><a href="E14">E<sub>14</sub> Boundary 3D Graph</a></h2>
<div class="sage">
<script type="text/x-sage">
# https://sagecell.sagemath.org/?z=eJyFVl1v2jAUfR4S_8FqHxoncRInzqZWYpq0h75Qqb-giIVA0Qggx9q6_frd668YCKwVikOOzzn3wzes5aEj_XLTZv2f7sdht20y2e6WanvYk213PEhFVNurhfty0S0_tt1yOrmXZEaKTEwn08mvpYweeqLInHTVA51OVvAs6pOSvpVxpMy1h496K0lCRAzP7LfRGYxZGJCs-Rxo1AgI9msiuOV0IKOEjXIDEvCU5isAcCRuLoj7gf3UIZJet5uI-JS6RM8cFeGjJTQidMWTniZaIWRCGRWaQVg8RCDi82hBFAV9LCehA0USnURwmkCRON-J9l2j70tdm1C_BgSCGwfuTzJ2BhPI6RXP7Gt9NpIa3Oj5h9wEJQgIsPvw_9t2r1q5bBT0Xrsmi0jO-t121cqoSIvs8fExJb1qj4t--7edFVlRcEqfphMCf7gdrxteYelyMCFNTJDVfAV6Qa2cC0AxqXObkLMCewJ22bUeArlgBhSk0QEdA7XGysqUGM2IMTaeKCZ9o1_SyQQTe1rOQMw6uQzE7cMO4njbQwmxtNqWqIYuGHYYSNjNV41qgSDOGgm9QW_MZnjIaxhk6MrQ3JdZ5dbN3lmEYRK4SwgcVC8PjDa70LJxYEAyh67nt6qIDPKTlcTZN_A1V_ia__FZ182Z6yZ0zX2coNns82blk9BxPHsYNm7jTdzQGJfQ5XFXUXNCDbI0yHJuBDyyvEAKgxQaKQak0EgIWMRqKI3bVZtd9dwG7nbVbpcaOyMukrU-lkEclnatT0Xg2sJ1VwYWHVw3V2AiaBfhZ8D9Ztl1yzocW4Mfw1PNg5YyjUSHWTEcL2PB8MWga0eVIXnxRwenjs71KAmeMU8CCaNmWmuSZzOugMvzvmCI1Vynt6vy5-EJ-f7eNj_97VHCvCR3L_yJ3KWjL_gIGp41jDN4V3Vljq8TpPQjIelEHpV2WeNSmREzg18F9EymvC3DWUNZx3PzUkcVP4JBReDFKFzjr27yS0sfDqfODx8diD-YKORn85iUuCnlIrH1YyAjmA5IM2Km1Bkxccz1bWaYnJbbp9qE4V-RJnHKTXOIK9BypX-FFpHtBsmPu4OCJq5y7EGGJyfHfvtKihTaPy3SzxQWPSwqSofdr9io7YeCX3wPacSKrE559qUehyiAlIAAmHCI_v3wO3p18V8Y-AdknnbG&lang=sage
#from sage.symbolic.relation import test_relation_maxima
#r = 0.4
var('s t L m3')
d = (s+2)^2*(t+2)^2*s^2*t^2 + 4*(s+t+2)^2*((s+2)^2*(t+2)^2-s^2*t^2)
f1L = t^2*((s+2)^2*(t+2)^2*(s^2 + (t+1)*(s+t+2)^2) - 4*(s+t+2)^2*((s+2)^2 + s^2))/d - 1
f1c = t^2*((s+2)^2*(s^2*(t+2)^2 + 4*(s+t+2)^2) - (s+t+2)^2*((s+2)^2*(t+2)^2+4*s^2))/d - 1
f2L = 1 + s + t^2*(4*s^2*(s+t+2)^2*(1+s)+4*(s+2)^2*(t+2)^2 - (t+2)^2*(s^2*(1+s)*(s+2)^2 + 4*(t+1)*(s+t+2)^2))/d
f2c = t^2*(4*(s+t+2)^2*(s^2+(t+2)^2) - (s+2)^2*(t+2)^2*(4+s^2))/d + 1
f5L = s^2*(1+s)*(s+2)^2 - 4*(s+2)^2 - 4*s^2
f5c = s^2*(s+2)^2 + 4*(s+2)^2 - 4*s^2
f4L = (t+2)^2*((t+1)*(s+t+2)^2+s^2)-s^2*(s+t+2)^2*(1+s)
f4c = s^2*(t+2)^2 - (s+t+2)^2*((t+2)^2+s^2)
var('r')
g13 = 1/(1+r)^2 + t^2/d*((4*(s+2)^2*(s+t+2)^2/(1-r)^2) + 4*s^2*(s+t+2)^2/(1+r)^2 -(s+2)^2*(t+2)^2*(s+t+2)^2/(t+1-r)^2 -s^2*(s+2)^2*(t+2)^2/(1+r)^2)
g23 = t^2/d*(4*(t+2)^2*(s+t+2)^2/(1+t-r)^2 + s^2*(s+2)^2*(t+2)^2/(r+1+s)^2 - 4*(s+2)^2*(t+2)^2/(1-r)^2 - 4*s^2*(s+t+2)^2/(r+1+s)^2) - 1/(r+s+1)^2
g43 = s^2*(s+t+2)^2/(r+s+1)^2 - (t+2)^2*((s+t+2)^2/(1+t-r)^2 + s^2/(r+1)^2)
g53 = 4*(s+2)^2/(1-r)^2 + 4*s^2/(1+r)^2 - s^2*(s+2)^2/(r+s+1)^2
cn = s^2*(f1L/(r+s+1)^2 + f2L/(r+1)^2) - t^2/d*(f4L*4*(s+2)^2/(r-1)^2 + f5L*(t+2)^2*(s+t+2)^2/(t+1-r)^2) - r
cd = t^2/d*(f4c*4*(s+2)^2/(r-1)^2 + f5c*(t+2)^2*(s+t+2)^2/(t+1-r)^2) - s^2*(f1c/(r+s+1)^2 + f2c/(r+1)^2) - 1
c = cn/cd
f13 = f1L + f1c*c
#f23 = f2L + f2c*c
f43 = f4L + f4c*c
#f53 = f5L + f5c*c
implicit_plot3d(f13/g13 - f43/g43, (s,0,3), (t,0,6), (r,0,1))
</script>
</div>
<h2><a href="E25">E<sub>25</sub></a></h2>
<div class="sage">
<script type="text/x-sage">
# https://sagecell.sagemath.org/?z=eJyFVtuO2jAQfS4S_2AtD8QJJjhxWu1KVJX6sC-stF-wKA2BRSWAHKvd9us7Y8cXQqC7QhhyfM6Z8cyYCdnKU0PaclfP2z_Nj9NhX81lfSjV_nQk--Z8koqoulVr--W6KT_2TTkeTSRZksVcjEfj0a9SRtOWKLIiTT6l49EGnkVtktG3LI6UeW_hpd4ykhARw7Pu26gHYx0MSLZ8BTRqAAT7NRF85NSTUcIGuQEJeErTDQA4EldXxK1nv3SIpLftJiK-pM7QM0dFeGkJjQhd8aSliVYImVBGhWYQFvsIRNyPFkRR0MVyETpQJNFFBJcJFIn1nWjfBfq-1u0S6taAQHBlwe1FxnowgZxOsWdf67OB1OBGx-9zExxBQIDVh__f9kdVy7JSUHv1lqwjuWwP-00tI7aYPz4-zhYz0qr6vG73f-vlYr5YcEqfxiMCf7gf33c8x7NLwYU0QUFa0w0IBodlbQCKSZ3chPRO2BGw67J1EEgGM6AgjxZoGWhnLMvNGaMZMcTGE8Wkq_RrOplgZi_PMxDrnFwHYvdhCXH82MIZ4tlqWyL3ZeB3GEhYzjeNaoEgzgIJnUFnrMuwz2sYZOjK0EyyeW7X1dFahGkSuEsIdKqTB8Yuu1CzcWBAMosuVvdOERnkp04Sh5_nq27wVf_j61xXPddV6Jq7OEGzOqbVxiWh4dh8GDZu41Vc0RiXUOVxk1PTogaZGWS2MgIOmV0hhUEKjRQeKTQSAhax8kdjdxVmV7HqAre7CrtLDfWIi2Sr-zIIxHy_1V0RuLZwXZaBxw6uiyswEZSLcDNgsiubpizCueX9GPp8FZSUKSTqZ4VvL-PA8MWg280qQ_LiWgenjs71IAn2mCOBhFEzrjXJsxlXwOV4XzDEfKXT2-Tps39Cvr_X1U_38SxhYJKHF_5EHmaDN3wEBc8qxhlcVk2W4n2ClG4kJI1Io6xbFrhUZsQs4WcB7clk92U4qyhreGpudVRxIxhUBL4ZhVv8-V1-2dGHw6lxw0cH4hoThdxsHpISd6VsJN35MZARTAekGTFTqkdMLHNxnxkmZ8ftUm3CcHekSZyy0xziCrTs0b9Cich6h-Tnw0lBW-YpDl6GDZXiVfOVwGUZtXBlfqawULDIKfW7X7FQ6w8VTdV0hvdrMePzL8UwpAVIBgiACYto30-_o1fq2zvFJgADAlYi_wc1x3c6&lang=sage
# from sage.symbolic.relation import test_relation_maxima
#r = 0.4
var('s t L m3')
d = (s+2)^2*(t+2)^2*s^2*t^2 + 4*(s+t+2)^2*((s+2)^2*(t+2)^2-s^2*t^2)
f1L = t^2*((s+2)^2*(t+2)^2*(s^2 + (t+1)*(s+t+2)^2) - 4*(s+t+2)^2*((s+2)^2 + s^2))/d - 1
f1c = t^2*((s+2)^2*(s^2*(t+2)^2 + 4*(s+t+2)^2) - (s+t+2)^2*((s+2)^2*(t+2)^2+4*s^2))/d - 1
f2L = 1 + s + t^2*(4*s^2*(s+t+2)^2*(1+s)+4*(s+2)^2*(t+2)^2 - (t+2)^2*(s^2*(1+s)*(s+2)^2 + 4*(t+1)*(s+t+2)^2))/d
f2c = t^2*(4*(s+t+2)^2*(s^2+(t+2)^2) - (s+2)^2*(t+2)^2*(4+s^2))/d + 1
f5L = s^2*(1+s)*(s+2)^2 - 4*(s+2)^2 - 4*s^2
f5c = s^2*(s+2)^2 + 4*(s+2)^2 - 4*s^2
f4L = (t+2)^2*((t+1)*(s+t+2)^2+s^2)-s^2*(s+t+2)^2*(1+s)
f4c = s^2*(t+2)^2 - (s+t+2)^2*((t+2)^2+s^2)
@interact
def _(r=slider(-0.999,0, step_size=0.001)):
g13 = 1/(1+r)^2 + t^2/d*((4*(s+2)^2*(s+t+2)^2/(1-r)^2) + 4*s^2*(s+t+2)^2/(1+r)^2 -(s+2)^2*(t+2)^2*(s+t+2)^2/(t+1-r)^2 -s^2*(s+2)^2*(t+2)^2/(1+r)^2)
g23 = t^2/d*(4*(t+2)^2*(s+t+2)^2/(1+t-r)^2 + s^2*(s+2)^2*(t+2)^2/(r+1+s)^2 - 4*(s+2)^2*(t+2)^2/(1-r)^2 - 4*s^2*(s+t+2)^2/(r+1+s)^2) - 1/(r+s+1)^2
g43 = s^2*(s+t+2)^2/(r+s+1)^2 - (t+2)^2*((s+t+2)^2/(1+t-r)^2 + s^2/(r+1)^2)
g53 = 4*(s+2)^2/(1-r)^2 + 4*s^2/(1+r)^2 - s^2*(s+2)^2/(r+s+1)^2
#2.3
cn = s^2*(f1L/(r+s+1)^2 + f2L/(r+1)^2) - t^2/d*(f4L*4*(s+2)^2/(r-1)^2 + f5L*(t+2)^2*(s+t+2)^2/(t+1-r)^2) - r
cd = t^2/d*(f4c*4*(s+2)^2/(r-1)^2 + f5c*(t+2)^2*(s+t+2)^2/(t+1-r)^2) - s^2*(f1c/(r+s+1)^2 + f2c/(r+1)^2) - 1
c = cn/cd
#m1 = ((f1L + f1c*c)*L + g13*m3)*s^2
#m2 = ((f2L + f2c*c)*L + g23*m3)*s^2
#m4 = ((f4L + f4c*c)*L + g43*m3)/d*4*t^2*(s+2)^2
#m5 = ((f5L + f5c*c)*L + g53*m3)/d*t^2*(t+2)^2*(s+t+2)^2
#f13 = f1L + f1c*c
f23 = f2L + f2c*c
#f43 = f4L + f4c*c
f53 = f5L + f5c*c
#2.4
#gamma5 = (t+2)^2*(s+t+2)^2
#f3L = s^2*(f1L + f2L) + t^2/d*(4*(s+2)^2*f4L + gamma5*f5L)
#fM3 = s^2*(g13 + g23) + t^2/d*(4*(s+2)^2*g43 + gamma5*g53) + 1
#G3 = 1/fM3
#M = f3L*L + m3/G3
# Check
#print "M1: ", test_relation_maxima(L*(-c-1-s)+m2/s^2+ m3/(r+1+s)^2+m4/(2+s)^2+m5/(2+t+s)^2 == 0)
#print "M2: ", test_relation_maxima(L*(-1-c)-m1/s^2 + m3/(1+r)^2 + m4/4 + m5/(2+t)^2 == 0)
#print "M3: ", test_relation_maxima(L*(r-c)-m1/(r+s+1)^2 - m2/(r+1)^2+m4/(r-1)^2 + m5/(t+1-r)^2 == 0)
#print "M4: ", test_relation_maxima(L*(1-c)-m1/(s+2)^2-m2/4-m3/(1-r)^2+m5/t^2 == 0)
# print "M5: ", test_relation_maxima(L*(1+t-c)-m1/(2+t+s)^2 - m2/(t+2)^2 - m3/(1+t-r)^2 - m4/t^2 == 0)
P = region_plot(f53/g53 - f23/g23 > 0, (s,0,6), (t,0,3))
P = P + text('t',(-0.5,1.75))
P = P + text('s',(2.5,-0.4))
show(P)
#f13/g13 - f43/g43
</script>
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<h2><a href="E25">E<sub>25</sub> Boundary 3D Graph</a></h2>
<div class="sage">
<script type="text/x-sage">
# https://sagecell.sagemath.org/?z=eJyFVl1v2jAUfR4S_8FqHxoncRInzqZWYpq0h75Qqb-giIVA0Qggx9q6_frd668YCKwVikOOzzn3wzes5aEj_XLTZv2f7sdht20y2e6WanvYk213PEhFVNurhfty0S0_tt1yOrmXZEaKTEwn08mvpYweeqLInHTVA51OVvAs6pOSvpVxpMy1h496K0lCRAzP7LfRGYxZGJCs-Rxo1AgI9msiuOV0IKOEjXIDEvCU5isAcCRuLoj7gf3UIZJet5uI-JS6RM8cFeGjJTQidMWTniZaIWRCGRWaQVg8RCDi82hBFAV9LCehA0USnURwmkCRON-J9l2j70tdm1C_BgSCGwfuTzJ2BhPI6RXP7Gt9NpIa3Oj5h9wEJQgIsPvw_9t2r1q5bBT0Xrsmi0jO-t121cqoSIvs8fExJb1qj4t--7edFVlRcEqfphMCf7gdrxteYelyMCFNTJDVfAV6Qa2cC0AxqXObkLMCewJ22bUeArlgBhSk0QEdA7XGysqUGM2IMTaeKCZ9o1_SyQQTe1rOQMw6uQzE7cMO4njbQwmxtNqWqIYuGHYYSNjNV41qgSDOGgm9QW_MZnjIaxhk6MrQ3JdZ5dbN3lmEYRK4SwgcVC8PjDa70LJxYEAyh67nt6qIDPKTlcTZN_A1V_ia__FZ182Z6yZ0zX2coNns82blk9BxPHsYNm7jTdzQGJfQ5XFXUXNCDbI0yHJuBDyyvEAKgxQaKQak0EgIWMRqKI3bVZtd9dwG7nbVbpcaOyMukrU-lkEclnatT0Xg2sJ1VwYWHVw3V2AiaBfhZ8D9Ztl1yzocW4Mfw1PNg5YyjUSHWTEcL2PB8MWga0eVIXnxRwenjs71KAmeMU8CCaNmWmuSZzOugMvzvmCI1Vynt6vy5-EJ-f7eNj_97VHCvCR3L_yJ3KWjL_gIGp41jDN4V3Vljq8TpPQjIelEHpV2WeNSmREzg18F9EymvC3DWUNZx3PzUkcVP4JBReDFKFzjr27yS0sfDqfODx8diD-YKORn85iUuCnlIrH1YyAjmA5IM2Km1Bkxccz1bWaYnJbbp9qE4V-RJnHKTXOIK9BypX-FFpHtBsmPu4OCJq5y7EGGJyfHfvtKihTaPy3SzxQWPSwqSofdr9io7YeCX3wPacSKrE559qUehyiAlIAAmHCI_v3wO3p18V8Y-AdknnbG&lang=sage
#from sage.symbolic.relation import test_relation_maxima
#r = 0.4
var('s t L m3')
d = (s+2)^2*(t+2)^2*s^2*t^2 + 4*(s+t+2)^2*((s+2)^2*(t+2)^2-s^2*t^2)
f1L = t^2*((s+2)^2*(t+2)^2*(s^2 + (t+1)*(s+t+2)^2) - 4*(s+t+2)^2*((s+2)^2 + s^2))/d - 1
f1c = t^2*((s+2)^2*(s^2*(t+2)^2 + 4*(s+t+2)^2) - (s+t+2)^2*((s+2)^2*(t+2)^2+4*s^2))/d - 1
f2L = 1 + s + t^2*(4*s^2*(s+t+2)^2*(1+s)+4*(s+2)^2*(t+2)^2 - (t+2)^2*(s^2*(1+s)*(s+2)^2 + 4*(t+1)*(s+t+2)^2))/d
f2c = t^2*(4*(s+t+2)^2*(s^2+(t+2)^2) - (s+2)^2*(t+2)^2*(4+s^2))/d + 1
f5L = s^2*(1+s)*(s+2)^2 - 4*(s+2)^2 - 4*s^2
f5c = s^2*(s+2)^2 + 4*(s+2)^2 - 4*s^2
f4L = (t+2)^2*((t+1)*(s+t+2)^2+s^2)-s^2*(s+t+2)^2*(1+s)
f4c = s^2*(t+2)^2 - (s+t+2)^2*((t+2)^2+s^2)
var('r')
g13 = 1/(1+r)^2 + t^2/d*((4*(s+2)^2*(s+t+2)^2/(1-r)^2) + 4*s^2*(s+t+2)^2/(1+r)^2 -(s+2)^2*(t+2)^2*(s+t+2)^2/(t+1-r)^2 -s^2*(s+2)^2*(t+2)^2/(1+r)^2)
g23 = t^2/d*(4*(t+2)^2*(s+t+2)^2/(1+t-r)^2 + s^2*(s+2)^2*(t+2)^2/(r+1+s)^2 - 4*(s+2)^2*(t+2)^2/(1-r)^2 - 4*s^2*(s+t+2)^2/(r+1+s)^2) - 1/(r+s+1)^2
g43 = s^2*(s+t+2)^2/(r+s+1)^2 - (t+2)^2*((s+t+2)^2/(1+t-r)^2 + s^2/(r+1)^2)
g53 = 4*(s+2)^2/(1-r)^2 + 4*s^2/(1+r)^2 - s^2*(s+2)^2/(r+s+1)^2
cn = s^2*(f1L/(r+s+1)^2 + f2L/(r+1)^2) - t^2/d*(f4L*4*(s+2)^2/(r-1)^2 + f5L*(t+2)^2*(s+t+2)^2/(t+1-r)^2) - r
cd = t^2/d*(f4c*4*(s+2)^2/(r-1)^2 + f5c*(t+2)^2*(s+t+2)^2/(t+1-r)^2) - s^2*(f1c/(r+s+1)^2 + f2c/(r+1)^2) - 1
c = cn/cd
#f13 = f1L + f1c*c
f23 = f2L + f2c*c
#f43 = f4L + f4c*c
f53 = f5L + f5c*c
implicit_plot3d(f53/g53 - f23/g23, (s,0,6), (t,0,3), (r,-1,0))
</script>
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