//////////////////////////////////////////////////
// Modifications for Theta Graph yz-projections //
//////////////////////////////////////////////////

// The following contains an example for Cell2_0 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: 
// PiecesTypeIICone[1][0].Hrepresentation()
// (An inequality (1, -1, 0, 0) x + 0 >= 0,
// An inequality (0, 0, -1, 1) x + 0 >= 0,
// An inequality (-1, 2, 0, -1) x + 0 >= 0)

///////////////////////////////
// Case 2_0: [0, -3, -6, -5] //
///////////////////////////////

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+t^2);
poly B4 = (11*t^3);
poly B34 = (5*t^6);
poly B2 = (3*t^5);

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+(25t12+9t10+110t9+242t6-t4-2t2-1)*x4+(-225t22-990t19-3025t18-2153t16-13310t15+59t14+110t13-14598t12+220t11+251t10+110t9+484t8+242t6)*x3+(27225t28-225t26+119790t25-450t24-990t23+128519t22-1980t21-8228t20-14300t19-7381t18-26620t17-16819t16-13310t15-29282t14-14641t12)*x2+y2+(27225t32+54450t30+119790t29+27225t28+239580t27+131769t26+119790t25+263538t24+131769t22)*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+(25t12+9t10+110t9+242t6)*x4+(-225t22-990t19-3025t18-2153t16-13310t15+59t14-14598t12+9t10)*x3+(-2t2-2)*x2y+(27225t28-225t26+119790t25-450t24-990t23+131544t22-1980t21-2178t20-990t19-4356t18-2178t16)*x2+(110t11+110t9+242t8+242t6)*xy+y2+(27225t32+54450t30+119790t29+27225t28+239580t27+131769t26+119790t25+263538t24+131769t22)*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+(625t26+625t24+4125t23+81t22+3630t21+9156t20-495t19+8036t18-921t16+204t14+104t12+18t10)*x4+(-2500t26-2500t24-16500t23-324t22-14520t21-36624t20+1980t19-32044t18+4020t16-110t15-408t14-330t13-450t12-330t11-762t10-110t9-726t8-242t6)*x3y+(3750t26+3750t24+24750t23+486t22+21780t21+54936t20-2970t19+47966t18-6366t16+275t15+204t14+825t13+709t12+825t11+1837t10+275t9+1835t8+645t6+40t4+20t2+4)*x2y2+(-2500t26-2500t24-16500t23-324t22-14520t21-36624t20+1980t19-31944t18+4356t16-220t15-660t13-484t12-660t11-1456t10-220t9-1472t8-524t6-40t4-20t2-4)*xy3+(625t26+625t24+4125t23+81t22+3630t21+9156t20-495t19+7986t18-1089t16+55t15+165t13+121t12+165t11+364t10+55t9+368t8+131t6+10t4+5t2+1)*y4+(50625t48-309375t47+101250t46-284625t45-2671875t44-3800500t43-4709925t42-16934500t41-28704225t40-21953250t39-64051350t38-69019170t37-47162150t36-120742930t35-53939899t34-61635310t33-88566015t32-2420715t31-45588552t30-2045285t29-2609982t28-811910t27-2232793t26-119790t25-888561t24-130833t22+81t20)*x3+(-151875t48+928125t47-303750t46+853875t45+8015625t44+11401500t43+14129775t42+50803500t41+86112675t40+65934000t39+191549050t38+207354510t37+139229800t36+360012290t35+158843997t34+174546680t33+261335345t32-8684115t31+125096606t30-4629955t29-9664626t28-376420t27-5260103t26+255420t25-587447t24-34650t23+164709t22-2310t21-76473t20+990t19-5082t18+2178t16)*x2y+(151875t48-928125t47+303750t46-853875t45-8015625t44-11401500t43-14129775t42-50803500t41-86112675t40-65971125t39-191246550t38-207503010t37-138101475t36-358904040t35-157356147t34-169367055t33-259153995t32+16657245t31-119262081t30+10012860t29+18411912t28+1782495t27+11239344t26-203445t25+2214012t24+51975t23-50814t22+3465t21+114588t20-1485t19+7623t18-3267t16)*xy2+(-50625t48+309375t47-101250t46+284625t45+2671875t44+3800500t43+4709925t42+16934500t41+28704225t40+21990375t39+63748850t38+69167670t37+46033825t36+119634680t35+52452049t34+56455685t33+86384665t32-5552415t31+39754027t30-3337620t29-6137304t28-594165t27-3746448t26+67815t25-738004t24-17325t23+16938t22-1155t21-38196t20+495t19-2541t18+1089t16)*y3+(-336909375t65+2058890625t64-1007943750t63+3936054375t62+17121431250t61-554709375t60+48929116500t59+55130675625t58+38664301500t57+172564138350t56+89305053750t55+163501753500t54+254571842250t53+87081369975t52+251875626750t51+129653592525t50+66047082750t49+132140758725t48-21907388250t47+31475530800t46-11285356500t45-16177507200t44-2591424000t43-8441601300t42-251224875t41-2053854000t40-83853000t39-275979825t38-11979000t37-92238300t36-13176900t34)*x2+(673818750t65-4117781250t64+2015887500t63-7872108750t62-34242862500t61+1109418750t60-97933101750t59-109803820000t58-77702946750t57-343334754200t56-175332520000t55-324900224500t54-490839772500t53-161251586200t52-469246076750t51-196298213750t50-83157288500t49-137083498150t48+150797229000t47+73227015050t46+216960268250t45+146249513250t44+201469078250t43+135664891450t42+102098345250t41+113271106850t40+23848858000t39+57487880450t38+2554521750t37+14011437000t36+845517750t35+1879904400t34+120788250t33+620046350t32+88578050t30)*xy+(-336909375t65+2058890625t64-1007943750t63+3936054375t62+17121431250t61-554709375t60+48966550875t59+54901910000t58+38851473375t57+171667377100t56+87666260000t55+162450112250t54+245419886250t53+80625793100t52+234623038375t51+98149106875t50+41578644250t49+68541749075t48-75398614500t47-36613507525t46-108480134125t45-73124756625t44-100734539125t43-67832445725t42-51049172625t41-56635553425t40-11924429000t39-28743940225t38-1277260875t37-7005718500t36-422758875t35-939952200t34-60394125t33-310023175t32-44289025t30)*y2+(-18530015625t80+113238984375t79-111180093750t78+516369768750t77+967678593750t76+719863031250t75+6567119893125t74+5296889193750t73+15192059788125t72+30541357427625t71+28827268841250t70+75830778997875t69+79928542140975t68+110619782014500t67+178157278299825t66+146791037349000t65+240100541156025t64+221684414675625t63+209890343482875t62+269139374410500t61+157698253458900t60+203306399084250t59+130603499900775t58+85727165460750t57+91276659720675t56+18153205698375t55+38769549548400t54+2450491621875t53+8567933331375t52+701538156000t51+1346873539275t50+87692269500t49+385845985800t48+48230748225t46)*x+(18530015625t80-113238984375t79+111180093750t78-516369768750t77-967678593750t76-719863031250t75-6567119893125t74-5296889193750t73-15192059788125t72-30541357427625t71-28827268841250t70-75830778997875t69-79928542140975t68-110619782014500t67-178157278299825t66-146791037349000t65-240100541156025t64-221684414675625t63-209890343482875t62-269139374410500t61-157698253458900t60-203306399084250t59-130603499900775t58-85727165460750t57-91276659720675t56-18153205698375t55-38769549548400t54-2450491621875t53-8567933331375t52-701538156000t51-1346873539275t50-87692269500t49-385845985800t48-48230748225t46)*y

drawTropicalCurve(newfnox,"max");