Graphics Reference
In-Depth Information
Figure 9.10.
An envelope of circles.
y = x + √2
y = x -
√
2
Equation (9.21) implies the constraint
cos u
+
sin u
= 0,
that is, u =-p/4 or 3p/4. The envelope therefore consists of a subset of the points
1
2
1
2
1
2
1
2
Ó
Ê
Ë
ˆ
¯
˛
»
Ó
Ê
Ë
ˆ
¯
˛
()
+
()
+-
tt
,
,
-
t
Œ
R
tt
,
,
t
Œ
R
,
which corresponds to the lines y = x +
2
and y = x -
2
. This is clearly a correct
answer.
Finally, we look at the problem of finding the envelope of a family of curves in the
xy-plane defined implicitly in the form
(
)
= 0
a xyt
,,
.
(9.23)
Think of Equation (9.23) as defining an equation in x and y for each fixed t.
9.5.3. Theorem.
If a family of curves is defined implicitly by equation (9.23), then
the envelope of that family is a subset of the set of points (x,y) satisfying (9.23) and
∂a
∂t
(
)
= 0
xyt
,,
.
(9.24)
Proof.
One can give an argument similar to the one for Theorem 9.5.1 by thinking
of y in equation (9.23) as a function of x and t (possibly after some change in coor-
dinates). Alternatively, one can look at the intersection of two of the curves f(x,y,t) and
f(x,y,t + h) and let h go to zero. See also [Brec92].
9.5.4. Example.
We redo Example 9.5.2 by expressing the family of circles implic-
itly as
