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Question: how to eliminate this the find K?

Question:how to eliminate this the find K?

ComputerUser 535 Maple 12
April 02 2018
0

1.

with(Groebner):
K := {r-x^4,u-(x^3)*y,v-x*y^3,w-y^4};
G := Basis(K, 'tord', degrevlex(r,u,v,w));
R1 := eliminate(G, {r,u,v,w}); # eliminate is the reverse of Basis
Ga := Basis({a*G[1],a*G[2],a*G[3],a*G[4],a*G[5],a*G[6],a*G[7],a*G[8],a*G[9],a*G[10],a*G[11],a*G[12],a*G[13],a*G[14], (1-a)*K[1], (1-a)*K[2], (1-a)*K[3], (1-a)*K[4]}, 'tord', deglex(a,r,u,v,w));
Ga := remove(has, Ga, [x,y,a]);
eliminate(Ga, [r,u,v,w]);

how to eliminate Ga to find back K ?

2.

A1A2 and A3A4 parallel
A(0,0), B(u1,0), D(u2,u3), C(x1,x2), E(x3,x4)
#BC = A1A2
xx1 := u1:
xx2 := x1:
yy1 := 0:
yy2 := x2:
#AD = A3A4
xx3 := 0:
xx4 := u2:
yy3 := 0:
yy4 := u3:
eq1 := (xx2 - xx1)*(yy4 - yy3) - (yy2 - yy1)*(xx4 - xx3);
(x1-u1)*u3-x2*u2;
#CD = A1A2
xx1 := x1:
xx2 := u2:
yy1 := x2:
yy2 := u3:
#AB = A3A4
xx3 := 0:
xx4 := u1:
yy3 := 0:
yy4 := 0:
eq2 := (xx2 - xx1)*(yy4 - yy3) - (yy2 - yy1)*(xx4 - xx3);
-(u3-x2)*u1;
with(LinearAlgebra):
#E is on the same line of AC
xx1 := x3:
yy1 := x4:
xx2 := 0:
yy2 := 0:
xx3 := x1:
yy3 := x2:
eq3 := Determinant(Matrix([[xx1,yy1,1],[xx2,yy2,1],[xx3,yy3,1]]));
-x3*x2+x1*x4;
#E is on the same line of BD
xx1 := x3:
yy1 := x4:
xx2 := u1:
yy2 := 0:
xx3 := u2:
yy3 := u3:
eq4 := Determinant(Matrix([[xx1,yy1,1],[xx2,yy2,1],[xx3,yy3,1]]));
-x3*u3+u1*u3-u1*x4+u2*x4;
 

sol := eliminate({eq1,eq2,eq3,eq4},[x1,x2,x3,x4]);

with(Groebner):
K := {(rhs(sol[1][1])-lhs(sol[1][1])),(rhs(sol[1][2])-lhs(sol[1][2])),(rhs(sol[1][3])-lhs(sol[1][3])),(rhs(sol[1][4])-lhs(sol[1][4]))};
G := Basis(K, 'tord', degrevlex(x1,x2,x3,x4));
R1 := eliminate(G, {x1,x2,x3,x4}); # eliminate is the reverse of Basis
Ga := Basis({a*G[1],a*G[2],a*G[3],a*G[4], (1-a)*K[1], (1-a)*K[2], (1-a)*K[3], (1-a)*K[4]}, 'tord', deglex(a,x1,x2,x3,x4));
Ga := remove(has, Ga, [u1,u2,u3,u4,a]);

From Question1, is it possible to find from sol to eq1, eq2, eq3 and eq4 ?
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