Singular

D.2.4.18 FamElemsAtEnvCompPoints

Usage:

FamElemsAtEnvCompPoints(poly F,ideal C,ideal E);
poly F must be the family of hyper-surfaces whose envelop is analyzed. ideal C must be the ideal of restrictions on the variables for defining the family. Must contain less polynomials than the dimension n of the space.
ideal E must be a component of the envelop of (F,C), previously computed by envelop. (x_1,..,x_n) are the variables of the hypersurfaces of F, that are considered as parameters of the parametric ring. (u_1,..,u_m) are the parameteres of the hyper-surfaces, that are considered as variables of the parametric ring. Having computed an envelop component E of a family of hyper-surfaces F, with constraints C, it returns the parameter values of the set of all hyper-surfaces of the family passing at one point of the envelop component E.
Calling sequence:
ring R=(0,x_1,..,x_n),(u_1,..,u_m),lp;
poly F=F(x_1,..,x_n,u_1,..,u_m);
ideal C=g_1(u_1,..u_m),..,g_s(u_1,..u_m);
poly E(x_1,..,x_n);
FamElemsAtEnvCompPoints(F,C,E[,options]);

Return:

list [lpp,basis,segment]. The basis determines the parameter values of the of hyper-surfaces that pass at a fixed point of the envelop component E. The lpp determines the dimension of the set. The segment is the component and is given in Prep.
Fixing the values of (x_1,..,x_n) inside E, the basis allows to detemine the values of the parameters (u_1,..u_m), of the hyper-surfaces passing at a point of E. See the not yet published book A. Montes. "Discussing Parametric Polynomial Systems: The Groebner Cover"

Note:

grobcov is called internally.
The basering R, must be of the form Q[a][x] (a=parameters, x=variables).

LIB "grobcov.lib";
if(defined(R)){kill R;}
ring R=(0,y0,x,y),(t),dp;
short=0;
poly F=(x-5*t)^2+y^2-3^2*t^2;
F;
==> 16*t^2+(-10*x)*t+(x^2+y^2)
ideal C;
C;
==> C[1]=0
def Env=envelop(F,C);
Env;
==> [1]:
==>    [1]:
==>       _[1]=(3*x+4*y)
==>    [2]:
==>       [1]:
==>          _[1]=1
==>    [3]:
==>       Normal
==>    [4]:
==>       1
==> [2]:
==>    [1]:
==>       _[1]=(3*x-4*y)
==>    [2]:
==>       [1]:
==>          _[1]=1
==>    [3]:
==>       Normal
==>    [4]:
==>       1
// E is a component of the envelop:
ideal E=Env[1][1];
E;
==> E[1]=(3*x+4*y)
def A=AssocTanToEnv(F,C,E);
A;
==> [1]:
==>    [1]:
==>       _[1]=t
==>    [2]:
==>       _[1]=12*t+(5*y)
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(3*x+4*y)
==>          [2]:
==>             [1]:
==>                _[1]=1
// The basis of the parameter values of the associated tangent component is
A[1][2][1];
==> 12*t+(5*y)
// Thus t=(5/12)*y0 the assocoated tangent family element at (x0,y0) is
subst(F,t,(5/12)*y0);
==> (50*y0^2-75*y0*x+18*x^2+18*y^2)/18
FamElemsAtEnvCompPoints(F,C,E);
==> [1]:
==>    [1]:
==>       _[1]=t^2
==>    [2]:
==>       _[1]=144*t^2+(120*y)*t+(25*y^2)
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(3*x+4*y)
==>          [2]:
==>             [1]:
==>                _[1]=1
// Thus (12*t^2-5*y0)^2=0 and the unique circle of the family passing at (x0,y0) in E
// is the associated   tangent circle:
subst(F,t,(5/12)*y0);
==> (50*y0^2-75*y0*x+18*x^2+18*y^2)/18