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

// The following contains an example for Cell0_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)

///////////////////////////////
// Case 0_0: [0, -2, -7, -5] //
///////////////////////////////

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^2);
poly B34 = (5*t^7);
poly B2 = (3*t^5);

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+(25t14+9t10+110t9+241t4-2t2-1)*x4+(-225t24-990t19-3000t18+50t16-2144t14-13200t13+18t12+220t11+9t10+110t9-14399t8+484t6+242t4)*x3+(27000t28-450t26-225t24+118800t23-3025t22-1980t21-6050t20-990t19+126566t18-13310t17-4356t16-26620t15-2178t14-13310t13-14641t12-29282t10-14641t8)*x2+y2+(27225t32+54450t30+27225t28+119790t27+239580t25+119790t23+131769t22+263538t20+131769t18)*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+(25t14+9t10+110t9+242t4)*x4+(-225t24-990t19-3000t18+50t16-2144t14-13310t13+18t12+9t10-14641t8)*x3+(-2t2-2)*x2y+(27000t28-450t26-225t24+118800t23-1980t21-990t19+129591t18-4356t16-2178t14)*x2+(110t11+110t9+242t6+242t4)*xy+y2+(27225t32+54450t30+27225t28+119790t27+239580t25+119790t23+131769t22+263538t20+131769t18)*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+(625t30+625t28+4125t25+4125t23+81t22-495t21+9206t20-495t19+9225t18-921t16-985t14+54t12+18t10)*x4+(-2500t30-2500t28-16500t25-16500t23-324t22+1980t21-36724t20+1980t19-36600t18+4020t16-110t15+4148t14-330t13-108t12-330t11-278t10-110t9-726t8-726t6-242t4)*x3y+(3750t30+3750t28+24750t25+24750t23+486t22-2970t21+54986t20-2970t19+54600t18-6366t16+275t15-6430t14+825t13+54t12+825t11+627t10+275t9+1835t8+1855t6+645t4+20t2+4)*x2y2+(-2500t30-2500t28-16500t25-16500t23-324t22+1980t21-36624t20+1980t19-36300t18+4356t16-220t15+4356t14-660t13-660t11-488t10-220t9-1472t8-1492t6-524t4-20t2-4)*xy3+(625t30+625t28+4125t25+4125t23+81t22-495t21+9156t20-495t19+9075t18-1089t16+55t15-1089t14+165t13+165t11+122t10+55t9+368t8+373t6+131t4+5t2+1)*y4+(50625t52-309375t51+101250t50-618750t49+50625t48+24750t47-2722500t46-3456750t45-5445000t44-7847125t43-1987425t42-12900250t41-25981100t40-17682500t39-54164925t38-8722670t37-37270275t36-60560555t35-19532350t34-122055395t33-9529968t32-62492595t31-44384810t30-1805705t29-89920304t28-1051490t27-46573420t26-479160t25-1979710t24-119790t23-1156153t22-526995t20-131769t18)*x3+(-151875t52+928125t51-303750t50+1856250t49-151875t48-74250t47+8167500t46+10370250t45+16335000t44+23541375t43+5962275t42+38775000t41+77338300t40+53344500t39+160074775t38+26613510t37+108345075t36+179308415t35+56835850t34+355542935t33+28978504t32+171383025t31+130879480t30-5389945t29+258116062t28+383570t27+121886388t26+1407780t25-6122702t24+365970t23+300679t22+5940t21+1515645t20+990t19+409827t18+13068t16+2178t14)*x2y+(151875t52-928125t51+303750t50-1856250t49+151875t48+74250t47-8167500t46-10370250t45-16335000t44-23541375t43-5962275t42-38812125t41-77035800t40-53493000t39-158864775t38-26836260t37-106612200t36-178121790t35-55955250t34-350231310t33-29172804t32-163335645t31-129742005t30+10793475t29-252293637t28+1001880t27-112969452t26-1392930t25+12153618t24-369270t23+1283211t22-8910t21-1482975t20-1485t19-417087t18-19602t16-3267t14)*xy2+(-50625t52+309375t51-101250t50+618750t49-50625t48-24750t47+2722500t46+3456750t45+5445000t44+7847125t43+1987425t42+12937375t41+25678600t40+17831000t39+52954925t38+8945420t37+35537400t36+59373930t35+18651750t34+116743770t33+9724268t32+54445215t31+43247335t30-3597825t29+84097879t28-333960t27+37656484t26+464310t25-4051206t24+123090t23-427737t22+2970t21+494325t20+495t19+139029t18+6534t16+1089t14)*y3+(-334125000t67+2041875000t66-996806250t65+6091593750t64-982884375t63+3807466875t62+17584256250t61-4659204375t60+53243437500t59-6509795625t58+47294205750t57+56722336875t56+1592204625t55+174117898125t54-14994032625t53+165120688125t52+81203351625t51+38652830325t50+252535833000t49-15329530125t48+241170930000t47+42808835025t46+65533458375t45+137074608000t44-11289873375t43+128572458300t42-2425747500t41+31830653250t40-83853000t39-8432480925t38-11979000t37-1871119800t36-92238300t34-13176900t32)*x2+(668250000t67-4083750000t66+1993612500t65-12183187500t64+1965768750t63-7614933750t62-35243381250t61+9775940000t60-106861218750t59+15307247500t58-95337099000t57-109364856250t56+101497000t55-346140681250t54+49803327750t53-332923643750t52-123137212000t51-68495499400t50-469995157500t49+94958611500t48-472916050750t47+47075872800t46-118338563750t45-143402169900t44+115274247000t43-196901491050t42+201055203250t41-46617718400t40+199225078250t39+67242446700t38+101681412250t37+111901163000t36+22023724250t35+110418029700t34+845517750t33+56709717350t32+120788250t31+12578083100t30+620046350t28+88578050t26)*xy+(-334125000t67+2041875000t66-996806250t65+6091593750t64-982884375t63+3807466875t62+17621690625t61-4887970000t60+53430609375t59-7653623750t58+47668549500t57+54682428125t56-50748500t55+173070340625t54-24901663875t53+166461821875t52+61568606000t51+34247749700t50+234997578750t49-47479305750t48+236458025375t47-23537936400t46+59169281875t45+71701084950t44-57637123500t43+98450745525t42-100527601625t41+23308859200t40-99612539125t39-33621223350t38-50840706125t37-55950581500t36-11011862125t35-55209014850t34-422758875t33-28354858675t32-60394125t31-6289041550t30-310023175t28-44289025t26)*y2+(-18530015625t82+113238984375t81-111180093750t80+679433906250t79-277950234375t78+1535183718750t77+877087406250t76+1283699587500t75+7212293859375t74-756810759375t73+18090566010000t72+2898169312500t71+21762885684375t70+30673240228125t69+10694619106875t68+80927439206625t67+8819793303750t66+105682942629000t65+65900394635625t64+71501653748250t63+178765046341200t62+31191477223500t61+241270518107025t60+74089459149750t59+178421231372175t58+196542835570125t57+72693870611175t56+269271796266000t55+42370911616275t54+205051159900125t53+86562235027125t52+84486609739125t51+118944598546125t50+15718839307875t49+90886261115925t48+701538156000t47+37712857700925t46+87692269500t45+7186381485525t44+385845985800t42+48230748225t40)*x+(18530015625t82-113238984375t81+111180093750t80-679433906250t79+277950234375t78-1535183718750t77-877087406250t76-1283699587500t75-7212293859375t74+756810759375t73-18090566010000t72-2898169312500t71-21762885684375t70-30673240228125t69-10694619106875t68-80927439206625t67-8819793303750t66-105682942629000t65-65900394635625t64-71501653748250t63-178765046341200t62-31191477223500t61-241270518107025t60-74089459149750t59-178421231372175t58-196542835570125t57-72693870611175t56-269271796266000t55-42370911616275t54-205051159900125t53-86562235027125t52-84486609739125t51-118944598546125t50-15718839307875t49-90886261115925t48-701538156000t47-37712857700925t46-87692269500t45-7186381485525t44-385845985800t42-48230748225t40)*y

drawTropicalCurve(newfnox,"max");