deriv2[F_,x_]:={-D[F,x[[2]]],D[F,x[[1]]]}
antipH[x_]:={Conjugate[x[[2]]],-Conjugate[x[[1]]]}
antip[x_]:=-1/x//Conjugate//ce

sProj[p_] := p/eucNorm[p]

trf3pt[{a1_, a2_, a3_}, {b1_, b2_, b3_}] :=
Inverse[{(b1 - b3)*{1, -b2}, (b1 - b2)*{1, -b3}}].{(a1 - a3)*{1, -a2}, (a1 - a2)*{1, -a3}}

vS = {{0, 0, -1}, {0, 0, 1}, {-(2/Sqrt[5]), 0, -(1/Sqrt[5])}, {2/Sqrt[5], 0, 1/Sqrt[5]}, {1/10 (-5 + Sqrt[5]),
 -Sqrt[1/10 (5 + Sqrt[5])], -(1/Sqrt[5])}, {1/10 (5 - Sqrt[5]), Sqrt[1/10 (5 + Sqrt[5])], 1/Sqrt[5]},
 {1/10 (5 + Sqrt[5]), -(1/2) (-1 + Sqrt[5]) Sqrt[1/10 (5 + Sqrt[5])], -(1/Sqrt[5])}, {1/10 (-5 - Sqrt[5]),
 1/2 (-1 + Sqrt[5]) Sqrt[1/10 (5 + Sqrt[5])], 1/Sqrt[5]}, {1/10 (5 + Sqrt[5]),
 -(1/10) (-5 + Sqrt[5]) Sqrt[1/2 (5 + Sqrt[5])], -(1/Sqrt[5])}, {1/10 (-5 - Sqrt[5]),
    1/10 (-5 + Sqrt[5]) Sqrt[1/2 (5 + Sqrt[5])], 1/Sqrt[5]}, {-((-1 + Sqrt[5])/(2 Sqrt[5])),
 Sqrt[1/10 (5 + Sqrt[5])], -(1/Sqrt[5])}, {(-1 + Sqrt[5])/(2 Sqrt[5]), -Sqrt[1/10 (5 + Sqrt[5])], 1/Sqrt[5]}};

fSBase =
  {
   Sum[{vS[[1]], vS[[3]], vS[[5]]}[[i]], {i, 3}],
   Sum[{vS[[1]], vS[[7]], vS[[5]]}[[i]], {i, 3}],
   Sum[{vS[[1]], vS[[7]], vS[[9]]}[[i]], {i, 3}],
   Sum[{vS[[1]], vS[[9]], vS[[11]]}[[i]], {i, 3}],
   Sum[{vS[[1]], vS[[3]], vS[[11]]}[[i]], {i, 3}],
   
   Sum[{vS[[2]], vS[[4]], vS[[6]]}[[i]], {i, 3}],
   Sum[{vS[[2]], vS[[6]], vS[[8]]}[[i]], {i, 3}],
   Sum[{vS[[2]], vS[[8]], vS[[10]]}[[i]], {i, 3}],
   Sum[{vS[[2]], vS[[10]], vS[[12]]}[[i]], {i, 3}],
   Sum[{vS[[2]], vS[[4]], vS[[12]]}[[i]], {i, 3}],
   
   Sum[{vS[[3]], vS[[5]], vS[[10]]}[[i]], {i, 3}],
   Sum[{vS[[3]], vS[[8]], vS[[10]]}[[i]], {i, 3}],
   Sum[{vS[[3]], vS[[8]], vS[[11]]}[[i]], {i, 3}],
   
   Sum[{vS[[4]], vS[[7]], vS[[12]]}[[i]], {i, 3}],
   Sum[{vS[[4]], vS[[7]], vS[[9]]}[[i]], {i, 3}],
   Sum[{vS[[4]], vS[[6]], vS[[9]]}[[i]], {i, 3}],
   
   Sum[{vS[[5]], vS[[10]], vS[[12]]}[[i]], {i, 3}],
   Sum[{vS[[5]], vS[[7]], vS[[12]]}[[i]], {i, 3}],
   
   Sum[{vS[[6]], vS[[8]], vS[[11]]}[[i]], {i, 3}],
   Sum[{vS[[6]], vS[[9]], vS[[11]]}[[i]], {i, 3}]
   };

fS =
  Table[
   fSBase[[i]]/Sqrt[Sum[fSBase[[i,j]]^2,{j,3}]]
   , {i,20}];

vC = Complement[Table[s2C[vS[[i]]],{i,12}]//simp//Quiet,{Indeterminate}];
fC = Table[s2C[fS[[i]]],{i,20}]//simp//Quiet;

Fz = Product[z-vC[[i]],{i,11}]//simp;
Fh = y^12*Fz/.z->x/y//simp;

Hh = (hess[Fh,{x,y}]//Det)/-11^2//simp;
Hz = Hh/.{x->z,y->1};

Th = (jacob[{Fh,Hh},{x,y}]//Det)/-20//simp;
Tz = Th/.{x->z,y->1};

phi = deriv2[Fh,{x,y}]//simp;
eta = deriv2[Hh,{x,y}]//simp;
tau = deriv2[Th,{x,y}]//simp;

tetV[1] = {fC[[3]], fC[[11]], fC[[19]], fC[[10]]};
tetV[2] = {fC[[4]], fC[[18]], fC[[12]], fC[[6]]};
tetV[3] = {fC[[5]], fC[[15]], fC[[17]], fC[[7]]};
tetV[4] = {fC[[1]], fC[[20]], fC[[14]], fC[[8]]};
tetV[5] = {fC[[2]], fC[[13]], fC[[16]], fC[[9]]};

P[1,2,3,4,5] = 
trf3pt[{tetV[1][[1]],tetV[2][[1]],tetV[3][[1]]},{tetV[2][[1]],tetV[3][[1]],tetV[4][[1]]}]//simp//fs;
P[1,5,4,3,2]=MatrixPower[P[1,2,3,4,5],4]//simp//fs;

Z0[{1,2},{3,4}] = 
trf3pt[{tetV[3][[1]],tetV[4][[1]],tetV[1][[2]]},{tetV[4][[1]],tetV[3][[1]],tetV[2][[1]]}]//simp;

Z0[{1,3},{2,4}] =
trf3pt[{tetV[1][[1]],tetV[2][[1]],tetV[3][[2]]},{tetV[3][[2]],tetV[4][[3]],tetV[1][[1]]}]//simp;

T0[1,2,4] =
trf3pt[{tetV[1][[1]],tetV[2][[2]],tetV[4][[1]]},{tetV[2][[2]],tetV[4][[1]],tetV[1][[1]]}]//simp;

PAff[1,2,3,4,5] = norm[P[1,2,3,4,5].{z,1},2][[1]];
ZAff[{1,2},{3,4}] = norm[Z[{1,2},{3,4}].{z,1},2][[1]]// simp;
ZAff[{1, 3}, {2, 4}] = norm[Z[{1,3},{2,4}].{z,1},2][[1]]//simp;
TAff[1, 2, 4] = norm[T[1,2,4].{z,1},2][[1]]//simp;