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

// The following contains an example for Cell2_1 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_1: [0, -2, -5, -3] //
///////////////////////////////

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^3);

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+110t7+9t6+241t4-2t2-1)*x4+(-225t16-3000t14-990t13+50t12-13200t11-2144t10+220t9-14381t8+110t7+493t6+242t4)*x3+(27000t20-3475t18+118800t17-6275t16-15290t15+126566t14-27610t13-18997t12-13310t11-31460t10-14641t8)*x2+y2+(27225t24+54450t22+119790t21+27225t20+239580t19+131769t18+119790t17+263538t16+131769t14)*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+110t7+9t6+242t4)*x4+(-225t16-3000t14-990t13+50t12-13310t11-2144t10-14623t8+9t6)*x3+(-2t2-2)*x2y+(27000t20-450t18+118800t17-225t16-1980t15+129591t14-990t13-4356t12-2178t10)*x2+(110t9+110t7+242t6+242t4)*xy+y2+(27225t24+54450t22+119790t21+27225t20+239580t19+131769t18+119790t17+263538t16+131769t14)*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+4125t17+9125t16-495t15+9306t14-495t13-840t12-985t10+54t8+18t6)*x4+(-2500t22-2500t20-16500t19-16500t17-36400t16+1980t15-36924t14+1870t13+3696t12-330t11+3906t10-330t9-834t8-110t7-762t6-242t4)*x3y+(3750t22+3750t20+24750t19+24750t17+54500t16-2970t15+55086t14-2695t13-5880t12+825t11-5821t10+825t9+1889t8+275t7+1873t6+645t4+20t2+4)*x2y2+(-2500t22-2500t20-16500t19-16500t17-36300t16+1980t15-36624t14+1760t13+4032t12-660t11+3868t10-660t9-1472t8-220t7-1492t6-524t4-20t2-4)*xy3+(625t22+625t20+4125t19+4125t17+9075t16-495t15+9156t14-440t13-1008t12+165t11-967t10+165t9+368t8+55t7+373t6+131t4+5t2+1)*y4+(-309375t37+50625t36-4743750t35-2621250t34-8156500t33-32845625t32-12232000t31-56887425t30-78075250t29-35800125t28-130841920t27-63452325t26-62344095t25-99572118t24-1746305t23-46346704t22-1033670t21-1869330t20-474705t19-1113115t18-119790t17-515610t16-131283t14+81t12)*x3+(928125t37-151875t36+14231250t35+7863750t34+24469500t33+97931875t32+36770250t31+168242275t30+231852500t29+103934625t28+382273760t27+185649425t26+171234525t25+287283254t24-5493895t23+121872240t22+330110t21-6277892t20+1394415t19+176965t18+365970t17+1482390t16+5940t15+408369t14+990t13+12825t12+2178t10)*x2y+(-928125t37+151875t36-14231250t35-7863750t34-24469500t33-97629375t32-36807375t31-167032275t30-230665875t29-102201750t28-377147760t27-183295650t26-163335645t25-281566704t24+10860300t23-113288304t22+1055340t21+12220833t20-1379565t19+1404225t18-369270t17-1450170t16-8910t15-415629t14-1485t13-19359t12-3267t10)*xy2+(309375t37-50625t36+4743750t35+2621250t34+8156500t33+32543125t32+12269125t31+55677425t30+76888625t29+34067250t28+125715920t27+61098550t26+54445215t25+93855568t24-3620100t23+37762768t22-351780t21-4073611t20+459855t19-468075t18+123090t17+483390t16+2970t15+138543t14+495t13+6453t12+1089t10)*y3+(2041875000t50-334125000t49+6091593750t48+16896515625t47+3807466875t46+52246631250t45+54056953125t44+46982355750t43+167483818125t42+87112228500t41+163101301875t40+237130764750t39+85399926750t38+236780614125t37+121323966750t36+65944939500t35+124657503750t34-11202642000t33+32272184475t32-2423408625t31-8332238475t30-83518875t29-1865974275t28-11979000t27-91503225t26-13176900t24)*x2+(-4083750000t50+668250000t49-11725656250t48-33867900000t47-5327277500t46-100833012500t45-104034088750t44-74523792750t43-319530910000t42-135628784500t41-262067007500t40-419951405500t39-42115040650t38-365928915000t37-103603149050t36+58638778000t35-135383174100t34+215917710250t33+50722747850t32+105305706000t31+123110506200t30+22148835500t29+59541244400t28+863486250t27+12714970400t26+120788250t25+639811700t24+88578050t22)*xy+(2041875000t50-334125000t49+5862828125t48+16933950000t47+2663638750t46+50416506250t45+52017044375t44+37261896375t43+159765455000t42+67814392250t41+131033503750t40+209975702750t39+21057520325t38+182964457500t37+51801574525t36-29319389000t35+67691587050t34-107958855125t33-25361373925t32-52652853000t31-61555253100t30-11074417750t29-29770622200t28-431743125t27-6357485200t26-60394125t25-319905850t24-44289025t22)*y2+(113238984375t63-18530015625t62+679433906250t61+1136507625000t60+1535183718750t59+7212293859375t58+6782584668750t57+17831145791250t56+32272736203125t55+33620942705625t54+79323975187875t53+83407882820625t52+116850281526000t51+178785552891825t50+150789370000500t49+240828933834225t48+219075388475625t47+210503668881825t46+263984373477000t45+155102898292650t44+200225573979750t43+126371086209225t42+83583415599750t41+88585702850700t40+15679703914875t39+37241711052975t38+696646231875t37+7157682197325t36+87692269500t35+382258574775t34+48230748225t32)*x+(-113238984375t63+18530015625t62-679433906250t61-1136507625000t60-1535183718750t59-7212293859375t58-6782584668750t57-17831145791250t56-32272736203125t55-33620942705625t54-79323975187875t53-83407882820625t52-116850281526000t51-178785552891825t50-150789370000500t49-240828933834225t48-219075388475625t47-210503668881825t46-263984373477000t45-155102898292650t44-200225573979750t43-126371086209225t42-83583415599750t41-88585702850700t40-15679703914875t39-37241711052975t38-696646231875t37-7157682197325t36-87692269500t35-382258574775t34-48230748225t32)*y

drawTropicalCurve(newfnox,"max");