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

// The following contains an example for Cell7_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)

// We want a cancellation, so we need in(b4^2) = in(b2*b5). 

///////////////////////////////
// Case 7_2: [0, -1, -3, -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 = (2*t^1);
poly B34 = (t^3);
poly B2 = (4*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+(t6+19t4+6t2-1)*x4+(-15t10-46t8-95t6+20t4+8t2)*x3+(-16t14-36t12-40t10-116t8-176t6-16t4)*x2+y2+(64t16+384t14+832t12+768t10+256t8)*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+(t6+20t4+8t2)*x4+(-15t10-50t8-111t6)*x3+(-2t2-2)*x2y+(-16t14-32t12-16t10-64t8-128t6)*x2+(4t6+12t4+8t2)*xy+y2+(64t16+384t14+832t12+768t10+256t8)*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+(t14+9t12+280t10+274t8+34t6+32t4)*x4+(-4t14-32t12-1048t10-912t8+24t6-92t4-8t2)*x3y+(6t14+44t12+1504t10+1204t8-156t6+142t4+40t2+4)*x2y2+(-4t14-28t12-980t10-748t8+144t6-96t4-36t2-4)*xy3+(t14+7t12+245t10+187t8-36t6+24t4+9t2+1)*y4+(227t24+1862t22+6507t20+10788t18+11841t16+10694t14+6065t12+1296t10)*x3+(-681t24-5374t22-18273t20-29360t18-31451t16-28166t14-14979t12-1804t10+808t8+128t6)*x2y+(681t24+5268t22+17649t20+27858t18+29415t16+26208t14+13371t12+762t10-1212t8-192t6)*xy2+(-227t24-1756t22-5883t20-9286t18-9805t16-8736t14-4457t12-254t10+404t8+64t6)*y3+(672t34+6560t32+29728t30+80864t28+141984t26+159072t24+100832t22+18720t20-16192t18-9216t16-1024t14)*x2+(-1344t34-13456t32-63056t30-178256t28-326320t26-384016t24-264432t22-71600t20+24816t18+20000t16+3136t14+128t12)*xy+(672t34+6728t32+31528t30+89128t28+163160t26+192008t24+132216t22+35800t20-12408t18-10000t16-1568t14-64t12)*y2+(-5376t42-73728t40-448000t38-1586176t36-3615744t34-5521408t32-5667328t30-3769344t28-1431296t26-149504t24+104448t22+40960t20+4096t18)*x+(5376t42+73728t40+448000t38+1586176t36+3615744t34+5521408t32+5667328t30+3769344t28+1431296t26+149504t24-104448t22-40960t20-4096t18)*y

drawTropicalCurve(newfnox,"max");