///////////////////////////////////
// Modifications for Theta Graph //
///////////////////////////////////

// The following contains an example for Cell5 combinatorial type of the modified tropical theta graphs in R^3. The curve can be visualized by means of projections. 

// We use the parameters b5, b4, b34, b2, where b3 = b34+b4, a5 = -b5^2, a4 = b4^2, a3 = b3^2, a2 = b2^2. we use this strategy since the modifications involve square roots.

// The parameters w5,w3, w34, w2 satisfy: 
// PiecesTypeIICone3[1][1].Hrepresentation()
// (An equation (1, -2, 1, 0) x + 0 == 0,
//  An inequality (1, 0, -1, 0) x + 0 >= 0,
//  An inequality (0, 0, 1, -1) x + 0 >= 0)

// We want cancellation, namely in(b4)^2 = in(b5*b34). We aim for a subdivision containing the edge xy---x^3. For this, we multiply our original vector by 3 (so we replace t by t^3) and add a tail to b34 of the form t^(6+e) for e=1, since we need ht(x3)>w3 + (w5+w3+w2)/3, and we want integer heights.

////////////////////////////////////////////////////////
// Case 5 (with cancellation): 2w3 = w34+w5, w34 > w2 //
////////////////////////////////////////////////////////

LIB "all.lib";
LIB "poly.lib";
LIB "tropical.lib";
LIB "elim.lib";

ring rr = (0,t), (b2,b34,b4,b5, x,y,z),dp;
poly f=y^2-x*(x-b2^2)*(x-(b4+b34)^2)*(x-b4^2)*(x+b5^2);
// f;
// b2^2*b34^2*b4^2*b5^2*x+2*b2^2*b34*b4^3*b5^2*x+b2^2*b4^4*b5^2*x+b2^2*b34^2*b4^2*x^2+2*b2^2*b34*b4^3*x^2+b2^2*b4^4*x^2-b2^2*b34^2*b5^2*x^2-2*b2^2*b34*b4*b5^2*x^2-2*b2^2*b4^2*b5^2*x^2-b34^2*b4^2*b5^2*x^2-2*b34*b4^3*b5^2*x^2-b4^4*b5^2*x^2-b2^2*b34^2*x^3-2*b2^2*b34*b4*x^3-2*b2^2*b4^2*x^3-b34^2*b4^2*x^3-2*b34*b4^3*x^3-b4^4*x^3+b2^2*b5^2*x^3+b34^2*b5^2*x^3+2*b34*b4*b5^2*x^3+2*b4^2*b5^2*x^3+b2^2*x^4+b34^2*x^4+2*b34*b4*x^4+2*b4^2*x^4-b5^2*x^4-x^5+y^2


setring(rr);
poly B5 = (1);
poly B4 = (2*t^3);
poly B34 = (4*t^6-t^7);
poly B2 = (17*t^9+13*t^12);

map P2 = rr, B2, B34, B4, B5, x, y, z;
poly ff = P2(f);

ring r = (0,t),(x,y),dp;
map newP2 = rr, 0,0,0,0,x,y,0;
poly f2 = newP2(ff);
// f2;
// -x5+(169t24+442t21+289t18+t14-8t13+16t12-4t10+16t9+8t6-1)*x4+(-169t38+1352t37-2704t36-442t35+4212t34-9776t33-289t32+4080t31-13048t30+1156t28-8160t27-2143t24+442t21-4t20+32t19+225t18+16t16-64t15+t14-8t13-4t10+16t9+8t6)*x3+(676t44-5408t43+10816t42+1768t41-16848t40+39104t39+987t38-14968t37+46784t36-442t35-412t34+15792t33-289t32+4080t31-8424t30+1156t28-8160t27-2312t24-4t20+32t19-64t18+16t16-64t15-16t12)*x2+y2+(676t44-5408t43+10816t42+1768t41-16848t40+39104t39+1156t38-16320t37+49488t36-4624t34+25568t33+4624t30)*x

drawTropicalCurve(f2,"max");

///////////////////
// XZ-projection //
///////////////////

poly B5 = newP2(B5);
poly B4 = newP2(B4);
poly B34 = newP2(B34);
poly B2 = newP2(B2);
poly B3 = B34 + B4;

poly g2 = substitute(f2, y, y+B3*B4*B5*x-B5*x^2);
// g2;
// -x5+(169t24+442t21+289t18+t14-8t13+16t12-4t10+16t9+8t6)*x4+(-169t38+1352t37-2704t36-442t35+4212t34-9776t33-289t32+4080t31-13048t30+1156t28-8160t27-2143t24+442t21-4t20+32t19+225t18+16t16-64t15+t14-8t13)*x3-2*x2y+(676t44-5408t43+10816t42+1768t41-16848t40+39104t39+987t38-14968t37+46784t36-442t35-412t34+15792t33-289t32+4080t31-8424t30+1156t28-8160t27-2312t24)*x2+(-4t10+16t9+8t6)*xy+y2+(676t44-5408t43+10816t42+1768t41-16848t40+39104t39+1156t38-16320t37+49488t36-4624t34+25568t33+4624t30)*x

drawTropicalCurve(g2,"max");

///////////////////
// ZY-projection //
///////////////////

ring s = (0,t),(x,y,z),dp;
map P = r, x,y;
poly ff= P(f2);

poly B5 = P(B5);
poly B4 = P(B4);
poly B3 = P(B3);
poly B34 = P(B34);

ideal I = (ff,z-y+B3*B4*B5*x-B5*x^2);

poly fnox= eliminate(I,x)[1];

// Replace z by y and y by x so that the pictures are not flipped.
ring r2 = (0,t),(x,y),dp;
setring(r2);
map PP = s,0,x,y;
poly newfnox = PP(fnox);
// newfnox;
// x5-5*x4y+10*x3y2-10*x2y3+5*xy4-y5+(28561t48+149396t45+293046t42+255476t39+83521t36+338t34-1352t33+884t31-4212t30+579t28-4096t27+96t26-256t25-568t24+72t23-288t22+1268t21+12t20-96t19+770t18+2t14-16t13+32t12)*x4+(-114244t48-597584t45-1172184t42-1021904t39-334084t36-1352t34+5408t33-3536t31+16848t30-2316t28+16384t27-384t26+1024t25+2948t24-288t23+1152t22-3304t21-48t20+384t19-1924t18-4t14+32t13-64t12+4t10-16t9-8t6)*x3y+(171366t48+896376t45+1758276t42+1532856t39+501126t36+2028t34-8112t33+5304t31-25272t30+3474t28-24576t27+576t26-1536t25-5098t24+432t23-1728t22+3188t21+72t20-576t19+1730t18+2t14-16t13+32t12-10t10+40t9+20t6+4)*x2y2+(-114244t48-597584t45-1172184t42-1021904t39-334084t36-1352t34+5408t33-3536t31+16848t30-2316t28+16384t27-384t26+1024t25+3624t24-288t23+1152t22-1536t21-48t20+384t19-768t18+8t10-32t9-16t6-4)*xy3+(28561t48+149396t45+293046t42+255476t39+83521t36+338t34-1352t33+884t31-4212t30+579t28-4096t27+96t26-256t25-906t24+72t23-288t22+384t21+12t20-96t19+192t18-2t10+8t9+4t6+1)*y4+(28561t76-456976t75+2741856t74-7162220t73+4749914t72+16398408t71-46177898t70+43627688t69+39231660t68-120532204t67+132025660t66+47417760t65-164057599t64+203746360t63+29929178t62-123097640t61+175301946t60+8863788t59-48224610t58+81190536t57+177486t56-6511696t55+15961748t54-1146176t53+2348932t52+115680t51-706608t50+1756056t49+586051t48-175512t47+619870t46+674448t45+19312t44-27296t43+596084t42+19224t41-98624t40+404788t39-1960t38-1800t37+82417t36+1012t35-400t34-15520t33+514t32-4112t31+3616t30-192t29+769t28-1040t27+80t26-128t25)*x3+(-85683t76+1370928t75-8225568t74+21486660t73-14249742t72-49195224t71+138533694t70-130883064t69-117694980t68+361596612t67-396076980t66-142253280t65+492172797t64-611239080t63-89787534t62+369292920t61-525905838t60-26591364t59+144673830t58-243571608t57-532458t56+19535088t55-47885244t54+3438528t53-7046796t52-347040t51+2119824t50-5268168t49-1760181t48+550872t47-1956954t46-1898856t45+9768t44-205152t43-1387368t42-5448t41+44544t40-822684t39+13492t38-55496t37-125459t36-1268t35-13652t34+78064t33-3458t32+9512t31+8008t30-960t29+2681t28+3088t27-368t26+1408t25+276t24-144t23+576t22-768t21-24t20+192t19-384t18)*x2y+(85683t76-1370928t75+8225568t74-21486660t73+14249742t72+49195224t71-138533694t70+130883064t69+117694980t68-361596612t67+396076980t66+142253280t65-492172797t64+611239080t63+89787534t62-369292920t61+525905838t60+26591364t59-144673830t58+243571608t57+532458t56-19535088t55+47885244t54-3438528t53+7046796t52+347040t51-2119824t50+5268168t49+1761195t48-563040t47+2005626t46+1836612t45-43620t44+348672t43+1186926t42-20664t41+81120t40+626844t39-17298t38+85944t37+64563t36+384t35+21078t34-93816t33+4416t32-8100t31-17436t30+1728t29-5175t28-3072t27+432t26-1920t25-414t24+216t23-864t22+1152t21+36t20-288t19+576t18)*xy2+(-28561t76+456976t75-2741856t74+7162220t73-4749914t72-16398408t71+46177898t70-43627688t69-39231660t68+120532204t67-132025660t66-47417760t65+164057599t64-203746360t63-29929178t62+123097640t61-175301946t60-8863788t59+48224610t58-81190536t57-177486t56+6511696t55-15961748t54+1146176t53-2348932t52-115680t51+706608t50-1756056t49-587065t48+187680t47-668542t46-612204t45+14540t44-116224t43-395642t42+6888t41-27040t40-208948t39+5766t38-28648t37-21521t36-128t35-7026t34+31272t33-1472t32+2700t31+5812t30-576t29+1725t28+1024t27-144t26+640t25+138t24-72t23+288t22-384t21-12t20+96t19-192t18)*y3+(228488t92-4569760t91+36558080t90-145037152t89+267190352t88-20809984t87-894229362t86+1827856680t85-1091372256t84-2183411464t83+4846552464t82-3842104448t81-2597146532t80+6409109840t79-6020138720t78-1331564840t77+4005038817t76-4556309936t75+179389710t74+129851004t73-786479674t72+619964904t71-1590957978t70+1503649536t69+381007484t68-1233981996t67+1480031880t66+136863968t65-528926223t64+731282744t63+25972794t62-151899784t61+251179962t60-7730132t59-11829172t58+51490880t57-7664178t56+17248240t55-11138496t54-2793216t53+9748332t52-12339264t51-979136t50+3515904t49-4628416t48-250240t47+1114112t46-1636352t45-18496t44+147968t43-295936t42)*x2+(-456976t92+9139520t91-73116160t90+290074304t89-534380704t88+41619968t87+1788458724t86-3655713360t85+2182744512t84+4366822928t83-9693104928t82+7684208896t81+5194293064t80-12818219680t79+12040277440t78+2663129680t77-8010077634t76+9112619872t75-358779420t74-259702008t73+1572959348t72-1239929808t71+3181915956t70-3007299072t69-762012264t68+2467909912t67-2959631120t66-275451424t65+1061155902t64-1463943280t63-58024548t62+316870160t61-512303796t60+8038056t59+41968064t58-120105600t57+11495140t56-22804912t55+9168512t54+4458880t53-15210464t52+18980928t51+1646816t50-5613312t49+7180704t48+539200t47-1998080t46+2704384t45+101456t44-418240t43+636928t42+22528t41-69536t40+80384t39+9728t38-28672t37+32576t36+2304t35-9216t34+12288t33+128t32-1024t31+2048t30)*xy+(228488t92-4569760t91+36558080t90-145037152t89+267190352t88-20809984t87-894229362t86+1827856680t85-1091372256t84-2183411464t83+4846552464t82-3842104448t81-2597146532t80+6409109840t79-6020138720t78-1331564840t77+4005038817t76-4556309936t75+179389710t74+129851004t73-786479674t72+619964904t71-1590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drawTropicalCurve(newfnox,"max");