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

// The following contains an example for Cell7_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 7_1: [0, -1, -4, -2] //
///////////////////////////////

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

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+(25t8+110t5+8t4+240t2-1)*x4+(-200t12-2975t10-880t9+34t8-13090t7-1918t6+110t5-14148t4+242t2)*x3+(-225t16+23750t14-990t13-6275t12+104500t11-5203t10-27610t9+112772t8-13310t7-31460t6-14641t4)*x2+y2+(27225t18+54450t16+119790t15+27225t14+239580t13+131769t12+119790t11+263538t10+131769t8)*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+(25t8+110t5+9t4+242t2)*x4+(-200t12-2975t10-990t9+34t8-13310t7-2160t6-14632t4)*x3+(-2t2-2)*x2y+(-225t16+26775t14-990t13-225t12+117810t11-2178t10-990t9+127413t8-2178t6)*x2+(110t7+110t5+242t4+242t2)*xy+y2+(27225t18+54450t16+119790t15+27225t14+239580t13+131769t12+119790t11+263538t10+131769t8)*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+(625t18+625t16+4125t15+50t14+4125t13+9225t12-495t11+9324t10-495t9-904t8-1035t6+18t4)*x4+(-2500t18-2500t16-16500t15-100t14-16500t13-36600t12+1870t11-36960t10+1650t9+3582t8-330t7+3522t6-110t5-762t4-242t2)*x3y+(3750t18+3750t16+24750t15+50t14+24750t13+54600t12-2695t11+55108t10-2145t9-5319t8+825t7-4625t6+275t5+1873t4+625t2+4)*x2y2+(-2500t18-2500t16-16500t15-16500t13-36300t12+1760t11-36628t10+1320t9+3528t8-660t7+2864t6-220t5-1492t4-504t2-4)*xy3+(625t18+625t16+4125t15+4125t13+9075t12-440t11+9157t10-330t9-882t8+165t7-716t6+55t5+373t4+126t2+1)*y4+(-275000t29+51250t28-4640625t27-2617500t26-8050625t25-32836425t24-12522125t23-56821525t22-79097920t21-35996419t20-131722470t19-64433663t18-62140430t17-100523050t16-1033670t15-46162625t14-474705t13-1115770t12-119790t11-516789t10-131688t8)*x3+(825000t29-153750t28+13921875t27+7247500t26+24217875t25+96090175t24+35151875t23+166985175t22+226968500t21+103199057t20+379333350t19+181711217t18+175738640t17+284316918t16+409310t15+126704245t14+1430715t13+354670t12+365310t11+1564887t10+990t9+408132t8+2178t6)*x2y+(-825000t29+153750t28-13921875t27-6945000t26-24250875t25-94880625t24-33944625t23-165245475t22-221805870t21-100803957t20-371416320t19-175916331t18-170397315t17-275690802t16+936540t15-120812430t14-1434015t13+1141650t12-368280t11-1572147t10-1485t9-414666t8-3267t6)*xy2+(275000t29-51250t28+4640625t27+2315000t26+8083625t25+31626875t24+11314875t23+55081825t22+73935290t21+33601319t20+123805440t19+58638777t18+56799105t17+91896934t16-312180t15+40270810t14+478005t13-380550t12+122760t11+524049t10+495t9+138222t8+1089t6)*y3+(-17015625t40+2475000t39+1973863125t38-549759375t37+6027969375t36+16006159125t35+2812044375t34+51186155625t33+49916141325t32+45036176625t31+162185098950t30+79705766250t29+161631287100t28+226833391125t27+84525403500t26+236434732875t25+117345545100t24+76652965125t23+125156346975t22-2257707375t21+40076098425t20-11979000t19-1685908125t18-13176900t16)*x2+(34031250t40-4950000t39-3490195000t38+1032968750t37-9769643750t36-28302155750t35-1497535000t34-82820919500t33-84085191400t32-51275367000t31-259491595100t30-104460532000t29-193760061500t28-344543749000t27-29547735050t26-280766235750t25-120360997400t24+40240436500t23-134565969450t22+103358802250t21+26058009450t20+20350657250t19+57978353950t18+120788250t17+11357755750t16+88578050t14)*xy+(-17015625t40+2475000t39+1745097500t38-516484375t37+4884821875t36+14151077875t35+748767500t34+41410459750t33+42042595700t32+25637683500t31+129745797550t30+52230266000t29+96880030750t28+172271874500t27+14773867525t26+140383117875t25+60180498700t24-20120218250t23+67282984725t22-51679401125t21-13029004725t20-10175328625t19-28989176975t18-60394125t17-5678877875t16-44289025t14)*y2+(113238984375t49-16471125000t48+679096996875t47+1150919859375t46+1550943590625t45+7251248070000t44+6902674143750t43+17933019699375t42+32620775566500t41+34023826423125t40+79880017907250t39+84509013763350t38+117624660986250t37+180394017544125t36+152162019652500t35+241921816037550t34+220961712621375t33+210578622383250t32+264920545803375t31+154771423490475t30+199008365329125t29+125916301976850t28+81207752298750t27+87594612199875t26+13960101993750t25+35903208139425t24+87692269500t23+6218179110000t22+48230748225t20)*x+(-113238984375t49+16471125000t48-679096996875t47-1150919859375t46-1550943590625t45-7251248070000t44-6902674143750t43-17933019699375t42-32620775566500t41-34023826423125t40-79880017907250t39-84509013763350t38-117624660986250t37-180394017544125t36-152162019652500t35-241921816037550t34-220961712621375t33-210578622383250t32-264920545803375t31-154771423490475t30-199008365329125t29-125916301976850t28-81207752298750t27-87594612199875t26-13960101993750t25-35903208139425t24-87692269500t23-6218179110000t22-48230748225t20)*y

drawTropicalCurve(newfnox,"max");
w