#### D.5.11.8 rncItProjEven

Procedure from library `paraplanecurves.lib` (see paraplanecurves_lib).

Usage:
  rncItProjEven(I); I ideal

Assume:
  I is a homogeneous ideal in the basering with n+1 variables defining a rational normal curve C in PP^n with n even.

Note:
  The procedure will fail or give a wrong output if I is not the ideal of a rational normal curve. It will test whether n is odd.

Return:
  ring with an ideal CONIC defining a conic C2 in PP^2.
  In addition, an ideal PHI in the basering defining an isomorphic projection of C to C2 will be exported.
  Note that the entries of PHI should be considered as
  representatives of elements in R/I, where R is the basering.

Theory:
  We iterate the procedure rncAntiCanonicalMap to obtain PHI.

Example:

    LIB "paraplanecurves.lib";
    ring R = 0,(x,y,z),dp;
    poly f = y^8-x^3*(z+x)^5;
    ideal adj = adjointIdeal(f);
    def Rn = mapToRatNormCurve(f,adj);
    ==> //'mapToRatNorm' created a ring together with an ideal RNC.
    ==> // Supposing you typed, say,  def RPn = mapToRatNorm(f,AI);
    ==> // you may access the ideal by typing
    ==> //      setring RPn; RNC;
    setring(Rn);
    RNC;
    ==> RNC[1]=y(5)*y(6)-y(4)*y(7)
    ==> RNC[2]=y(4)*y(6)-y(3)*y(7)
    ==> RNC[3]=y(2)*y(6)-y(1)*y(7)
    ==> RNC[4]=y(4)*y(5)-y(2)*y(7)
    ==> RNC[5]=y(3)*y(5)-y(1)*y(7)
    ==> RNC[6]=y(1)*y(5)-y(7)^2
    ==> RNC[7]=y(4)^2-y(1)*y(7)
    ==> RNC[8]=y(3)*y(4)-y(1)*y(6)
    ==> RNC[9]=y(2)*y(4)-y(1)*y(5)
    ==> RNC[10]=y(1)*y(4)-y(6)*y(7)
    ==> RNC[11]=y(2)*y(3)-y(6)*y(7)
    ==> RNC[12]=y(1)*y(3)-y(6)^2
    ==> RNC[13]=y(2)^2-y(5)*y(7)
    ==> RNC[14]=y(1)*y(2)-y(4)*y(7)
    ==> RNC[15]=y(1)^2-y(3)*y(7)
    ==> RNC[16]=y(1)*y(6)^2-y(3)^2*y(7)
    ==> RNC[17]=y(6)^4-y(3)^3*y(7)
    def Rc = rncItProjEven(RNC);
    PHI;
    ==> PHI[1]=-y(7)
    ==> PHI[2]=-y(2)
    ==> PHI[3]=-y(5)
    setring Rc;
    CONIC;
    ==> y(2)^2-y(1)*y(3)

See also: rncItProjOdd.

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