///////////////////////////////////
// Modifications for Theta Graph //
///////////////////////////////////


// The following contains an example for Cell7_0 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_0: [0, -2, -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 = (2*t^2+t^5);
poly B34 = (t^5+t^10);
poly B2 = (4*t^4+t^16);

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+(t32+9t20+4t15+4t12+5t10+16t8+12t7+7t4-2t2-1)*x4+(-t52-4t47-4t44-5t42-8t40-12t39-7t36-32t35+2t34-31t32-41t30-16t28-100t27-4t25-59t24-64t23-2t22-59t20-28t19-80t18-16t17-12t16-188t15-39t14-98t12-36t11+37t10+24t9+8t8+12t7+16t6+8t4)*x3+(t62+4t59+4t57+3t56+18t54+3t52+28t51+8t50+16t49+12t48+28t47+39t46+32t45+10t44+36t43+139t42-24t41+32t40+212t39+128t37+88t36+32t35+311t34+64t33-66t32+284t31+247t30-204t29+108t28+340t27-156t26+252t25+80t24-96t23+596t22-88t21-244t20+496t19-164t18-408t17+8t16-240t15-308t14-96t13-144t12-48t11-32t10-16t8)*x2+y2+(t66+2t64+4t63+t62+12t61+4t60+12t59+28t58+4t57+48t56+32t55+36t54+88t53+36t52+112t51+92t50+120t49+152t48+144t47+276t46+128t45+400t44+304t43+272t42+704t41+208t40+704t39+688t38+384t37+1024t36+576t35+864t34+832t33+896t32+896t31+704t30+1408t29+448t28+1280t27+1344t26+384t25+1920t24+768t23+832t22+1536t21+256t20+768t19+512t18+256t16)*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+(t32+9t20+4t15+4t12+5t10+16t8+12t7+8t4)*x4+(-t52-4t47-4t44-5t42-8t40-12t39-7t36-32t35+2t34-31t32-41t30-16t28-100t27-4t25-59t24-64t23-2t22-59t20-30t19-80t18-20t17-16t16-190t15-51t14-110t12-48t11+33t10)*x3+(-2t2-2)*x2y+(t62+4t59+4t57+3t56+18t54+3t52+28t51+8t50+16t49+12t48+28t47+39t46+32t45+10t44+36t43+139t42-24t41+32t40+212t39+128t37+88t36+32t35+312t34+64t33-64t32+288t31+248t30-192t29+112t28+352t27-128t26+256t25+128t24-64t23+624t22-224t20+576t19-80t18-384t17+128t16-192t15-256t14-128t12)*x2+(2t17+2t15+4t14+8t12+4t10+12t9+12t7+8t6+8t4)*xy+y2+(t66+2t64+4t63+t62+12t61+4t60+12t59+28t58+4t57+48t56+32t55+36t54+88t53+36t52+112t51+92t50+120t49+152t48+144t47+276t46+128t45+400t44+304t43+272t42+704t41+208t40+704t39+688t38+384t37+1024t36+576t35+864t34+832t33+896t32+896t31+704t30+1408t29+448t28+1280t27+1344t26+384t25+1920t24+768t23+832t22+1536t21+256t20+768t19+512t18+256t16)*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;
// 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drawTropicalCurve(newfnox,"max");