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

// The following contains an example for Cell0_2 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_2: [0, -1, -6, -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^6);
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+(25t12+110t7+9t6-t4+240t2-1)*x4+(-225t18+25t16-2975t14-990t13+25t12+110t11+9t10-13090t9-2160t8+110t7+251t6-14157t4+242t2)*x3+(-225t22+26775t20-3250t18-990t17-6050t16+117810t15-3025t14-14300t13-2178t12-26620t11+127413t10-13310t9-16819t8-29282t6-14641t4)*x2+y2+(27225t24+54450t22+27225t20+119790t19+239580t17+119790t15+131769t14+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+(25t12+110t7+9t6+242t2)*x4+(-225t18+25t16-2975t14-990t13+25t12+9t10-13310t9-2160t8+9t6-14641t4)*x3+(-2t2-2)*x2y+(-225t22+26775t20-225t18-990t17+117810t15-990t13-2178t12+127413t10-2178t8)*x2+(110t9+110t7+242t4+242t2)*xy+y2+(27225t24+54450t22+27225t20+119790t19+239580t17+119790t15+131769t14+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+(625t26+625t24+4125t21+4125t19+50t18+9225t16-495t15+9306t14-495t13+149t12-1035t10-1035t8+18t6)*x4+(-2500t26-2500t24-16500t21-16500t19-100t18-36600t16+1980t15-36924t14+1870t13-460t12-330t11+4248t10-330t9+4006t8-110t7-762t6-726t4-242t2)*x3y+(3750t26+3750t24+24750t21+24750t19+50t18+54600t16-2970t15+55086t14-2695t13+554t12+825t11-6476t10+825t9-5855t8+275t7+1873t6+1855t4+625t2+4)*x2y2+(-2500t26-2500t24-16500t21-16500t19-36300t16+1980t15-36624t14+1760t13-324t12-660t11+4352t10-660t9+3852t8-220t7-1492t6-1492t4-504t2-4)*xy3+(625t26+625t24+4125t21+4125t19+9075t16-495t15+9156t14-440t13+81t12+165t11-1088t10+165t9-963t8+55t7+373t6+373t4+126t2+1)*y4+(-309375t41+50625t40-584375t39+101250t38-4331250t37-2671250t36-7778375t35-5441250t34-3316500t33-30165000t32-8467250t31-54155725t30-18078500t29-25915200t28-70556420t27-8919300t26-122729805t25-19794969t24-61818185t23-55400963t22-794090t21-90676631t20-714285t19-45824018t18-479160t17-852682t16-119790t15-780327t14-526995t12-131769t10)*x3+(928125t41-151875t40+1753125t39-303750t38+12993750t37+8013750t36+23335125t35+15718750t34+10023750t33+88075000t32+25690500t31+158838075t30+51969500t29+75494350t28+201194510t27+26801650t26+352127655t25+57345907t24+174688745t23+154884739t22-314380t21+254268621t20+2154405t19+125500972t18+1452330t17-446384t16+365310t15+2366391t14+990t13+1613655t12+408375t10+2178t8)*x2y+(-928125t41+151875t40-1753125t39+303750t38-12993750t37-8013750t36-23335125t35-15416250t34-10060875t33-86865000t32-25834875t31-157023525t30-50836500t29-74368725t28-195957135t27-26823525t26-344096775t25-56326407t24-169305840t23-149225664t22+1662705t21-245387985t20-2160180t19-119515431t18-1459755t17+1948599t16-368280t15-2379096t14-1485t13-1629990t12-414909t10-3267t8)*xy2+(309375t41-50625t40+584375t39-101250t38+4331250t37+2671250t36+7778375t35+5138750t34+3353625t33+28955000t32+8611625t31+52341175t30+16945500t29+24789575t28+65319045t27+8941175t26+114698925t25+18775469t24+56435280t23+49741888t22-554235t21+81795995t20+720060t19+39838477t18+486585t17-649533t16+122760t15+793032t14+495t13+543330t12+138303t10+1089t8)*y3+(-17015625t54+2784375t53+1973812500t52-323296875t51+6006566250t50-1209656250t49+6031372500t48+16005825000t47-129616875t46+51775168500t45-7535469375t44+52183803375t43+47941593750t42+12496114125t41+166445919450t40-16340308875t39+168664157100t38+62713963125t37+51225222150t36+236435042250t35-12628152900t34+240998855625t33+30131541000t32+76570102125t31+125159750100t30-2258041500t29+128381093775t28-11979000t27+39982743900t26-1686643200t24-13176900t22)*x2+(34031250t54-5568750t53-3947625000t52+646593750t51-11555601250t50+2344443750t49-9775088750t48-32377675000t47+4833185000t46-100214518250t45+19142588750t44-84810225500t43-96039882500t42+15181344750t41-323752255150t40+72030634500t39-274730779200t38-110562320000t37+29832418500t36-459533464500t35+158698027950t34-393097559250t33+3286281350t32+40397843750t31-233513096300t30+202486361000t29-209460044750t28+99990376750t27+26189814750t26+20332688750t25+112809637050t24+120788250t23+55469624650t22+11337990400t20+88578050t18)*xy+(-17015625t54+2784375t53+1973812500t52-323296875t51+5777800625t50-1172221875t49+4887544375t48+16188837500t47-2416592500t46+50107259125t45-9571294375t44+42405112750t43+48019941250t42-7590672375t41+161876127575t40-36015317250t39+137365389600t38+55281160000t37-14916209250t36+229766732250t35-79349013975t34+196548779625t33-1643140675t32-20198921875t31+116756548150t30-101243180500t29+104730022375t28-49995188375t27-13094907375t26-10166344375t25-56404818525t24-60394125t23-27734812325t22-5668995200t20-44289025t18)*y2+(113238984375t65-18530015625t64+679433906250t63-109121203125t62+1698247856250t61+984149718750t60+2099020275000t59+7160821593750t58+728884715625t57+18519556460625t56+3840317662500t55+24389618343750t54+30220284290625t53+15550059690000t52+80756403044625t51+11871233921250t50+109067429394000t49+64364462229375t48+78275700384750t47+176015831102100t46+35244164587500t45+241585347190275t44+71534787662250t43+181000978500375t42+191768317624125t41+73527744373875t40+266535217674000t39+39969339235650t38+203034237701625t37+83587474084950t36+82031226193125t35+117389655167400t34+13964993917875t33+89776953906750t32+87692269500t31+36362396750625t30+6221766521025t28+48230748225t26)*x+(-113238984375t65+18530015625t64-679433906250t63+109121203125t62-1698247856250t61-984149718750t60-2099020275000t59-7160821593750t58-728884715625t57-18519556460625t56-3840317662500t55-24389618343750t54-30220284290625t53-15550059690000t52-80756403044625t51-11871233921250t50-109067429394000t49-64364462229375t48-78275700384750t47-176015831102100t46-35244164587500t45-241585347190275t44-71534787662250t43-181000978500375t42-191768317624125t41-73527744373875t40-266535217674000t39-39969339235650t38-203034237701625t37-83587474084950t36-82031226193125t35-117389655167400t34-13964993917875t33-89776953906750t32-87692269500t31-36362396750625t30-6221766521025t28-48230748225t26)*y

drawTropicalCurve(newfnox,"max");