Singular

D.2.4.14 envelop

Usage:

envelop(poly F,ideal C[,options]);
poly F must represent the family of hyper-surfaces for which on want to compute its envelop. ideal C must be
the ideal of restrictions on the variables defining the family, and should contain less polynomials than the
number of variables. (x_1,..,x_n) are the variables of the hyper-surfaces 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.
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);
envelop(F,C[,options]); where s<m.

Return:

The output is a list of the components [C_1, .. , C_n] of the locus. Each component is given by
Ci=[pi,[pi1,..pi_s_i],tax] where
pi,[pi1,..pi_s_i] is the canonical P-representation of the component.
Concerning tax: (see help for locus)
For normal-point components is
tax=[d,taxonomy,anti-image], being
d=dimension of the anti-image
taxonomy="Normal" or "Special"
anti-image=values of the mover corresponding
to the component
For non-normal-point components is
tax=[d,taxonomy]
d=dimension of the component
taxonomy="Accumulation" or "Degenerate".

Options:

An option is a pair of arguments: string, integer.
To modify the default options,
pairs of arguments -option name, value- of valid options must be added to the call. The algorithm allows the
following option as pair of arguments:
"comments", c: by default it is 0, but it can be set to 1. "anti-image", a: by default a=1 and the anti-image is shown also for "Normal" components.
For a=0, it is not shown.

Note:

grobcov and locus are called internally.
The basering R, must be of the form Q[x][u]
(x=parameters, u=variables).
This routine uses the generalized definition of envelop introduced in the book
A. Montes. "The Groebner Cover" (Discussing Parametric
Polynomial Systems) not yet published.

LIB "grobcov.lib";
// Steiner Deltoid
// 1. Consider the circle x1^2+y1^2-1=0, and a mover point M(x1,y1) on it.
// 2. Consider the triangle A(0,1), B(-1,0), C(1,0).
// 3. Consider lines passing through M perpendicular to two sides of ABC triangle.
// 4. Obtain the envelop of the lines above.
if(defined(R)){kill R;}
ring R=(0,x,y),(x1,y1,x2,y2),lp;
short=0;
ideal C=(x1)^2+(y1)^2-1,
x2+y2-1,
x2-y2-x1+y1;
matrix M[3][3]=x,y,1,x2,y2,1,x1,0,1;
poly F=det(M);
// Curves Family F
F;
==> -x1*y2+(y)*x1+(-y)*x2+(x)*y2
// Conditions C=
C;
==> C[1]=x1^2+y1^2-1
==> C[2]=x2+y2-1
==> C[3]=-x1+y1+x2-y2
envelop(F,C);
==> [1]:
==>    [1]:
==>       _[1]=(x^4+2*x^2*y^2+10*x^2*y-x^2+y^4-6*y^3+12*y^2-8*y)
==>    [2]:
==>       [1]:
==>          _[1]=1
==>    [3]:
==>       [1]:
==>          1
==>       [2]:
==>          Normal
==>       [3]:
==>          _[1]=x2+y2-1
==> [2]:
==>    [1]:
==>       _[1]=(x+y-1)
==>    [2]:
==>       [1]:
==>          _[1]=1
==>    [3]:
==>       [1]:
==>          0
==>       [2]:
==>          Special
==>       [3]:
==>          _[1]=4*y2^3-4*y2^2-y2
==>          _[2]=x2+y2-1