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

// The following contains an example for Cell0_3 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_3: [0, -1, -4, -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^1);
poly B34 = (5*t^4);
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+(25t8+9t6+110t5-t4+240t2-1)*x4+(-225t14+25t12-990t11-2966t10+110t9-2135t8-13090t7+251t6+110t5-14157t4+242t2)*x3+(-225t18+26775t16-990t15-3250t14+117810t13-8228t12-14300t11+124388t10-26620t9-16819t8-13310t7-29282t6-14641t4)*x2+y2+(27225t20+54450t18+119790t17+27225t16+239580t15+131769t14+119790t13+263538t12+131769t10)*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+9t6+110t5+242t2)*x4+(-225t14+25t12-990t11-2966t10-2135t8-13310t7+9t6-14641t4)*x3+(-2t2-2)*x2y+(-225t18+26775t16-990t15-225t14+117810t13-2178t12-990t11+127413t10-2178t8)*x2+(110t7+110t5+242t4+242t2)*xy+y2+(27225t20+54450t18+119790t17+27225t16+239580t15+131769t14+119790t13+263538t12+131769t10)*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+131t14+3630t13+9324t12-495t11+8190t10-985t8+18t6)*x4+(-2500t18-2500t16-16500t15-424t14-14520t13-36960t12+1870t11-32352t10-330t9+3906t8-330t7-762t6-110t5-726t4-242t2)*x3y+(3750t18+3750t16+24750t15+536t14+21780t13+55104t12-2695t11+48124t10+825t9-5805t8+825t7+1873t6+275t5+1855t4+625t2+4)*x2y2+(-2500t18-2500t16-16500t15-324t14-14520t13-36624t12+1760t11-31948t10-660t9+3852t8-660t7-1492t6-220t5-1492t4-504t2-4)*xy3+(625t18+625t16+4125t15+81t14+3630t13+9156t12-440t11+7987t10+165t9-963t8+165t7+373t6+55t5+373t4+126t2+1)*y4+(50625t32-309375t31+101250t30-250250t29-2671250t28-3625875t27-4705725t26-16609670t25-28637619t24-22082555t23-63804013t22-70619560t21-47197756t20-123208965t19-55478843t18-62439410t17-91203082t16-811910t15-46533002t14-119790t13-893020t12-131769t10)*x3+(-151875t32+928125t31-303750t30+750750t29+8013750t28+10951875t27+13513075t26+50117760t25+83661607t24+63982655t23+188430789t22+201389870t21+137136446t20+353578995t19+154526447t18+176566280t17+255878466t16-260920t15+127658416t14+357060t13-325370t12+990t11+390225t10+2178t8)*x2y+(151875t32-928125t31+303750t30-750750t29-8013750t28-10989000t27-13211025t26-50262135t25-82535982t24-62850150t23-186940164t22-196155465t21-134908035t20-345555045t19-148571406t18-171190305t17-247013076t16+1609245t15-121688121t14-355905t13+1827585t12-1485t11-387684t10-3267t8)*xy2+(-50625t32+309375t31-101250t30+250250t29+2671250t28+3663000t27+4403675t26+16754045t25+27511994t24+20950050t23+62313388t22+65385155t21+44969345t20+115185015t19+49523802t18+57063435t17+82337692t16-536415t15+40562707t14+118635t13-609195t12+495t11+129228t10+1089t8)*y3+(2784375t43-16965000t42-323296875t41+1998669375t40-1209322125t39+3903086250t38+16089022125t37-1511203050t36+47282239125t35+50009495850t34+35987069250t33+163287737775t32+79702981875t31+157036359600t30+236208270375t29+84500597250t28+240998546250t27+124065305100t26+76570102125t25+128377690650t24-2258041500t23+39982743900t22-11979000t21-1686643200t20-13176900t18)*x2+(-5568750t43+33930000t42+646593750t41-3997338750t40+2343775500t39-7350002500t38-32544069250t37+4806399850t36-91234649000t35-97888030450t34-53545778000t33-313324699000t32-124666863750t31-250030849550t30-422685389500t29-40290003650t28-372799190500t27-110710196300t26+44482350000t25-141922987250t24+202494679750t23+39860848500t22+99990376750t21+112864540800t20+20332688750t19+55469624650t18+120788250t17+11337990400t16+88578050t14)*xy+(2784375t43-16965000t42-323296875t41+1998669375t40-1171887750t39+3675001250t38+16272034625t37-2403199925t36+45617324500t35+48944015225t34+26772889000t33+156662349500t32+62333431875t31+125015424775t30+211342694750t29+20145001825t28+186399595250t27+55355098150t26-22241175000t25+70961493625t24-101247339875t23-19930424250t22-49995188375t21-56432270400t20-10166344375t19-27734812325t18-60394125t17-5668995200t16-44289025t14)*y2+(-18530015625t52+112902075000t51-109121203125t50+513674493750t49+981926116875t48+728884715625t47+6604921125000t46+5420771519625t45+15290639471250t44+30977264253375t43+29281743416475t42+76656064716000t41+81420786535275t40+111861623981250t39+180892104364125t38+148969439462250t37+242819943658875t36+225141774189750t35+211093045265025t34+272160930417750t33+157545707781825t32+203052355939125t31+129925663693650t30+82031226193125t29+89836744090500t28+13964993917875t27+36362396750625t26+87692269500t25+6221766521025t24+48230748225t22)*x+(18530015625t52-112902075000t51+109121203125t50-513674493750t49-981926116875t48-728884715625t47-6604921125000t46-5420771519625t45-15290639471250t44-30977264253375t43-29281743416475t42-76656064716000t41-81420786535275t40-111861623981250t39-180892104364125t38-148969439462250t37-242819943658875t36-225141774189750t35-211093045265025t34-272160930417750t33-157545707781825t32-203052355939125t31-129925663693650t30-82031226193125t29-89836744090500t28-13964993917875t27-36362396750625t26-87692269500t25-6221766521025t24-48230748225t22)*y

drawTropicalCurve(newfnox,"max");