A family of surfaces may or may not have an envelope. The shock cone is an
example of an envelope of a set of translating and expanding Huygen spheres. At
the point of contact, the envelope and the spherical wavefront represent a
multiple (nonunique) solution to the problem of supersonic flow. The key idea
is that of nonuniqueness. Nonuniqueness implies either a point of
intersection or a point of tangency. An envelope is built from the points of
tangency.
Cartan's methods can be applied to such problems, for consider a family of
implicit surfaces given by the zero set of some function F(x,y,z; t), with
parameter t. Then construct the 1form which is equal to the perfect
differential of the given function, and subtract from it those terms which
multiply the differential dt. The resulting 1form,
need not be exact, nor closed, nor integrable. If A^dA = 0, the Frobenious
integrability conditions for uniqueness are satisfied for the equation A = 0,
and no envelope exists.
However, when A^dA # 0, the
Torsion Vector is not zero,
and the conditions for uniqueness are not satisfied. The equation A = 0 then
does not have a unique solution, and an envelope can exist.
If Ftt = 0, and A^dA # 0, intersections of the family can
occur, but to obtain an envelope it is necessary that both Ftt # 0, and A^dA # 0.
The concept of an envelope also can be associated with the Jacobian neighborhood
of any vector field representing an evolutionary process. The family of
surfaces is given by the Hamilton characteristic polynomial, which is globally
zero for every Jacobian matrix. The parameter of the family is the eigenvalue
parameter. Of particular significance are those polynomials which are
homogeneous of degree 1, for then the surfaces can be put into correspondence
with the primitive equilibrium thermodynamic surfaces of Gibbs. Such a method
gives an explanation for the universal behavior of the Van der Waals gas, and
the chemical engineering law of corresponding states.
