#### D.2.4.12 envelop

Procedure from library `grobcov.lib` (see grobcov_lib).

Usage:
  envelop(F,C);
  The first argument F must be the family of curves of which on want to compute the envelop.
  The second argument C must be the ideal of conditions
  over the variables, and should contain as polynomials
  as the number of variables -1.

Return:
  The components of the envelop with its taxonomy:
  The taxonomy distinguishes 'Normal',
  'Special', 'Accumulation', 'Degenerate' components.
  In the case of 'Special' components, it also
  outputs the antiimage of the component
  and an integer (0-1). If the integer is 0
  the component is not a curve of the family and is
  not considered as 'Relevant' by the envelopdg routine applied to it, but is considered as 'Relevant' if the integer is 1.

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

Example:

    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.
    ring R=(0,x,y),(x1,y1,x2,y2),lp;
    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);
    short=0;
    // 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
    def Env=envelop(F,C);
    Env;
    ==> [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]:
    ==>       Normal
    ==>    [4]:
    ==>       1
    ==> [2]:
    ==>    [1]:
    ==>       _[1]=(x+y-1)
    ==>    [2]:
    ==>       [1]:
    ==>          _[1]=1
    ==>    [3]:
    ==>       [1]:
    ==>          Special
    ==>       [2]:
    ==>          y2,x2-1,y1,x1-1
    ==>       [3]:
    ==>          0
    ==>    [4]:
    ==>       1

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