The curve that at every point touches one of
the curves of the family such that the points of contact along the envelope pass from one curve
of the family to another. For example, the family of circles of the same radius
with centres on a straight line has an envelope consisting of two parallel lines. If
is the parameter of the family,
is the parameter along the envelope and
the value of
for one of the curves of the family touching the envelope at the point with parameter
,
then it is assumed that it is possible to choose
such that the function is not constant on any part of the range of
.
For the family of curves given by
,
where
and
,
a necessary condition for the existence of an envelope is that
,
,
satisfy the condition
The system
(1)
serves to determine the points of the envelope, but
other singular points of the family may also satisfy
(1).
A sufficient
condition for a point to belong to the envelope is that
and satisfies, in addition to
(1),
the condition
For a family of plane curves given by a
-function
where
is the parameter of the family and
the parameter along its curves, a necessary condition for
a point to be on the envelope is that
,
or, which is the same thing,
A sufficient condition is that
and that, in addition to
(3),
Violation of conditions
(2)
and
(4)
is most often
related to the appearance of cusps on the envelope.
The
envelope of a family of surfaces
in space depending on one parameter
is the surface that at each of its points with intrinsic parameters
touches the surface of the family with parameter
,
and is such that the function
is not constant on any domain in domain of definition of
.
For example, the envelope of spheres with the same radius and centres
on a straight line is a cylinder. For a family given by
,
where
and
,
a necessary condition for the envelope is satisfaction of the system of equations
and a sufficient condition is:
and, in addition to
(5),
the conditions
For the family
,
where
and
,
a necessary condition for the envelope is satisfaction of the equation
and a sufficient condition is that
and that, besides
(7),
the following conditions are satisfied:
Violation of the first of the conditions in
(6)
and
(8)
is most often
related to the appearance of a cuspidal edge on the envelope. The line of
contact of the envelope with one of the surfaces of the family is called a
characteristic.
A cuspidal edge on the envelope in turn is usually the envelope of the characteristics.
The envelope of a family of surfaces in space depending on two parameters
and
is the surface touching at each of its points
the surface of the family with parameters
,
and such that there is no function
on any domain of the range of
with
.
For the family given by the equation
,
where
and
,
a necessary condition for the envelope is satisfaction of the system of equations
and a sufficient condition is that
and that, as well as
(9),
the following conditions holds:
For a family
,
where
and
,
a necessary condition is
and a sufficient condition is that
and, as well as
(10),
satisfaction of
The more complicated concept of the
envelope of a family of
-dimensional submanifolds depending on
parameters
in an
-dimensional
manifold can be introduced (see
[1])
based on the theory of singularities of differentiable mappings
as a special form of singularity of a family.