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

// The following contains an example for Cell7_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 7_0: [0, -2, -5, -4] //
///////////////////////////////

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^2);
poly B34 = (5*t^5);
poly B2 = (3*t^4);

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+(25t10+9t8+110t7+241t4-2t2-1)*x4+(-225t18-990t15-3000t14-2119t12-13200t11+43t10+220t9-14390t8+110t7+484t6+242t4)*x3+(27000t22-450t20+118800t19-3250t18-1980t17+123541t16-14300t15-7381t14-26620t13-16819t12-13310t11-29282t10-14641t8)*x2+y2+(27225t26+54450t24+119790t23+27225t22+239580t21+131769t20+119790t19+263538t18+131769t16)*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+(25t10+9t8+110t7+242t4)*x4+(-225t18-990t15-3000t14-2119t12-13310t11+43t10-14632t8)*x3+(-2t2-2)*x2y+(27000t22-450t20+118800t19-225t18-1980t17+129591t16-990t15-4356t14-2178t12)*x2+(110t9+110t7+242t6+242t4)*xy+y2+(27225t26+54450t24+119790t23+27225t22+239580t21+131769t20+119790t19+263538t18+131769t16)*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+(625t22+625t20+4125t19+81t18+3630t17+9206t16-495t15+8154t14-885t12+104t10+18t8)*x4+(-2500t22-2500t20-16500t19-324t18-14520t17-36724t16+1980t15-32280t14-110t13+3948t12-330t11-450t10-330t9-762t8-110t7-726t6-242t4)*x3y+(3750t22+3750t20+24750t19+486t18+21780t17+54986t16-2970t15+48084t14+275t13-6330t12+825t11+713t10+825t9+1853t8+275t7+1855t6+645t4+20t2+4)*x2y2+(-2500t22-2500t20-16500t19-324t18-14520t17-36624t16+1980t15-31944t14-220t13+4356t12-660t11-488t10-660t9-1472t8-220t7-1492t6-524t4-20t2-4)*xy3+(625t22+625t20+4125t19+81t18+3630t17+9156t16-495t15+7986t14+55t13-1089t12+165t11+122t10+165t9+368t8+55t7+373t6+131t4+5t2+1)*y4+(50625t40-309375t39+101250t38-284625t37-2671875t36-3766125t35-4709925t34-16794250t33-28703600t32-21768670t31-63992700t30-69333055t29-46977468t28-122029435t27-54152085t26-62814840t25-89877723t24-2045285t23-46906477t22-811910t21-2240373t20-119790t19-892165t18-131688t16)*x3+(-151875t40+928125t39-303750t38+853875t37+8015625t36+11298375t35+14129775t34+50457000t33+85505800t32+65603010t31+189722350t30+205774415t29+137961204t28+355688795t27+158083505t26+172428960t25+257880047t24-4662295t23+123067899t22-330220t21-5316611t20+324720t19-479495t18-2310t17+318834t16+990t15-5082t14+2178t12)*x2y+(151875t40-928125t39+303750t38-853875t37-8015625t36-11298375t35-14129775t34-50494125t33-85203300t32-65751510t31-188594475t30-204662040t29-136475604t28-350489040t27-155897130t26-164421180t25-252003486t24+10061370t23-114242133t22+1713195t21+11335476t20-307395t19+2057490t18+3465t17-280719t16-1485t15+7623t14-3267t12)*xy2+(-50625t40+309375t39-101250t38+284625t37+2671875t36+3766125t35+4709925t34+16831375t33+28401100t32+21917170t31+62864825t30+68220680t29+45491868t28+116829680t27+51965710t26+54807060t25+84001162t24-3353790t23+38080711t22-571065t21-3778492t20+102465t19-685830t18-1155t17+93573t16+495t15-2541t14+1089t12)*y3+(-334125000t53+2041875000t52-996806250t51+3892545000t50+16910437500t49-541423125t48+48109718250t47+54180174375t46+37846932750t45+168433998750t44+87510060000t43+158387755950t42+246916506375t41+85756219425t40+241268247000t39+128483868600t38+65616828750t37+127799890650t36-11288178000t35+31906066500t34-2425413375t33-8431473600t32-83853000t31-1870384725t30-11979000t29-92238300t28-13176900t26)*x2+(668250000t53-4083750000t52+1993612500t51-7785090000t50-33895743750t49+1540377500t48-96593780250t47-106568187500t46-72407959250t45-334772882500t44-156672285000t43-303828511900t42-459247310500t41-108199691000t40-433446219250t39-129040596100t38-23610328500t37-118637654900t36+218540484250t35+50540653350t34+202795483000t33+136124243750t32+101797542000t31+113469208050t30+22041692750t29+56883213200t28+845517750t27+12597848450t26+120788250t25+620046350t24+88578050t22)*xy+(-334125000t53+2041875000t52-996806250t51+3892545000t50+16947871875t49-770188750t48+48296890125t47+53284093750t46+36203979625t45+167386441250t44+78336142500t43+151914255950t42+229623655250t41+54099845500t40+216723109625t39+64520298050t38+11805164250t37+59318827450t36-109270242125t35-25270326675t34-101397741500t33-68062121875t32-50898771000t31-56734604025t30-11020846375t29-28441606600t28-422758875t27-6298924225t26-60394125t25-310023175t24-44289025t22)*y2+(-18530015625t66+113238984375t65-111180093750t64+516032859375t63+969737484375t62+717504665625t61+6579308525625t60+5308269243750t59+15217672387500t58+30653859703500t57+28900235925000t56+76154183547750t55+80310049632450t54+111121663182750t53+179268015202650t52+147530998147500t51+241724630317500t50+223122855990375t49+210985656824250t48+271158292609875t47+157805641899675t46+204309036892125t45+130343565782925t44+84470484507750t43+90858757631400t42+15713947383750t41+37747536007500t40+701538156000t39+7182794074500t38+87692269500t37+385845985800t36+48230748225t34)*x+(18530015625t66-113238984375t65+111180093750t64-516032859375t63-969737484375t62-717504665625t61-6579308525625t60-5308269243750t59-15217672387500t58-30653859703500t57-28900235925000t56-76154183547750t55-80310049632450t54-111121663182750t53-179268015202650t52-147530998147500t51-241724630317500t50-223122855990375t49-210985656824250t48-271158292609875t47-157805641899675t46-204309036892125t45-130343565782925t44-84470484507750t43-90858757631400t42-15713947383750t41-37747536007500t40-701538156000t39-7182794074500t38-87692269500t37-385845985800t36-48230748225t34)*y

drawTropicalCurve(newfnox,"max");