find the envelope of the plane
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Applications of the Derivative
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Envelope of a Family of Curves
Consider a one-parameter family of plane curves defined by the equation
f
(
x
,
y
,
C
)
=
0
,
where
C
is a parameter.
The envelope of this family of curves is a curve such that at each point it touches tangentially one of the curves of the family (Figure
1
).
Envelope of a family of curves
Figure 1.
The parametric equations of the envelope are defined by the system of equations
{
f
(
x
,
y
,
C
)
=
0
f
′
C
(
x
,
y
,
C
)
=
0
,
that is, by the original equation of the family of curves and the equation obtained by differentiating the original equation with respect to the parameter
C
.
Eliminating the parameter
C
from these equations, we can get the equation of the envelope in explicit or implicit form.
The above system of equations is a necessary condition for the existence of an envelope. Besides the envelope curve, the solution of this system may comprise, for example, singular points of the curves of the family that do not belong to the envelope. The set of all solutions of the system is called the discriminant curve. Thus, in general, the envelope is a part of the discriminant curve.
To find the equation of the envelope uniquely, the sufficient conditions are used. They assume that the following inequalities are satisfied (in addition to the above system of equations):
∣
∣
∣
∣
∂
f
∂
x
∂
f
∂
y
∂
f
′
C
∂
x
∂
f
′
C
∂
y
∣
∣
∣
∣
≠
0
,
∂
2
f
∂
C
2
≠
0.
Note that not any one-parameter family of curves has an envelope. A classic counter-example is the family of concentric circles (Figure
2
), which is described by the equation
x
2
+
y
2
=
C
2
.