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

// The following contains an example for Cell0_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 0_1: [0, -1, -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^1);
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+110t6-t4+240t2-1)*x4+(-225t18-965t14-2966t12-2025t10-13081t8+352t6-14157t4+242t2)*x3+(-225t22+26775t20-1215t18+114785t16-9218t14+111078t12-28798t10-27951t8-29282t6-14641t4)*x2+y2+(27225t24+54450t22+147015t20+239580t18+251559t16+263538t14+131769t12)*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+110t6+242t2)*x4+(-225t18-965t14-2966t12-2135t10-13301t8-14641t4)*x3+(-2t2-2)*x2y+(-225t22+26775t20-1215t18+117810t16-3168t14+127413t12-2178t10)*x2+(110t8+110t6+242t4+242t2)*xy+y2+(27225t24+54450t22+147015t20+239580t18+251559t16+263538t14+131769t12)*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+4206t18+3761t16+8748t14+8190t12-985t10+18t8)*x4+(-2500t22-2500t20-16824t18-14944t16-34656t14-32462t12+3818t10-608t8-836t6-726t4-242t2)*x3y+(3750t22+3750t20+25236t18+22316t16+51648t14+48395t12-5601t10+1468t8+2130t6+1855t4+625t2+4)*x2y2+(-2500t22-2500t20-16824t18-14844t16-34320t14-32164t12+3692t10-1164t8-1712t6-1492t4-504t2-4)*xy3+(625t22+625t20+4206t18+3711t16+8580t14+8041t12-923t10+291t8+428t6+373t4+126t2+1)*y4+(50625t40-208125t38-234000t36-2329250t34-8359925t32-18191945t30-48399999t28-73646483t26-108621701t24-133219429t22-108230565t20-92102449t18-46663448t16-893101t14-131769t12)*x3+(-151875t40+624375t38+702000t36+6987750t34+25154025t32+54268735t30+143385997t28+215562839t26+313733493t24+380563115t22+302136513t20+255712067t18+128046544t16-324137t14+390225t12+2178t10)*x2y+(151875t40-624375t38-702000t36-6987750t34-25191150t32-54115185t30-142478997t28-212874534t26-307667688t24-371015529t22-290858922t20-245414427t18-122074644t16+1825857t14-387684t12-3267t10)*xy2+(-50625t40+208125t38+234000t36+2329250t34+8397050t32+18038395t30+47492999t28+70958178t26+102555896t24+123671843t22+96952974t20+81804809t18+40691548t16-608619t14+129228t12+1089t10)*y3+(2784375t52-339952500t50+1015475625t48+2697167250t46+16328022750t44+42076083825t42+90572997600t40+166317985800t38+230547399525t36+283830440850t34+276676302375t32+204947458650t30+126029697750t28+39970029825t26-1686643200t24-13176900t22)*x2+(-5568750t52+679905000t50-2030951250t48-5469203250t46-32574219250t44-83108542650t42-175740108450t40-314192032350t38-415213892000t36-473365291300t34-393633748200t32-184725950750t30+16641087650t28+184149772950t26+212836949050t24+130632126350t22+55570647550t20+11337990400t18+88578050t16)*xy+(2784375t52-339952500t50+1015475625t48+2734601625t46+16287109625t44+41554271325t42+87870054225t40+157096016175t38+207606946000t36+236682645650t34+196816874100t32+92362975375t30-8320543825t28-92074886475t26-106418474525t24-65316063175t22-27785323775t20-5668995200t18-44289025t16)*y2+(-18530015625t62+1721981250t60+237783150000t58+1600042516875t56+6515640140625t54+19432158607125t52+47679913375875t50+97893454430475t48+171585839078775t46+260543555165025t44+341437022826075t42+386116405436025t40+374271512137875t38+303928542138375t36+201727695359025t34+104201136435825t32+36453676431150t30+6221766521025t28+48230748225t26)*x+(18530015625t62-1721981250t60-237783150000t58-1600042516875t56-6515640140625t54-19432158607125t52-47679913375875t50-97893454430475t48-171585839078775t46-260543555165025t44-341437022826075t42-386116405436025t40-374271512137875t38-303928542138375t36-201727695359025t34-104201136435825t32-36453676431150t30-6221766521025t28-48230748225t26)*y

drawTropicalCurve(newfnox,"max");