Graphics Reference
In-Depth Information
Figure 9.9.
Intersecting envelopes.
v
a
t+h
a
t
u
u
h
(
)
-
(
)
fu t h
,
+
fu t
,
h
h
0 =
.
h
See Figure 9.9. If we assume that the numbers u
h
approach a limit u(t) as h approaches
0, then we must have
∂
∂
f
t
(
()
)
=
(
()
)
=
Dfut t
,
ut t
,
0
.
(9.18)
2
The envelope p(t) will be defined by
()
=
(
()
(
()
)
)
pt
ut fuut
,
,
.
Next, consider the general case. The functions a
t
may not be the graphs of func-
tions of u, but, locally, we can find a coordinate system so that they will be graphs of
functions with respect to one of the new coordinate variables (simply rotate the stan-
dard coordinate system appropriately). Therefore, if
()
=
(
() ()
)
a
ut
,
a
ut
,
,
a
ut
,
,
1
2
we can find a function T(u,t) so that
(
()
)
=
a
1
Tut t
,,
u
.
(9.19)
Define
()
=
(
()
)
fut
,
a
2
Tut t
,
,
.
Note that the function b(u,t) =a(T(u,t),t) is then the graph of the function f because
()
=
(
()
)
=
(
(
()
)
(
(
)
)
)
=
(
()
)
b
ut
,
a
Tutt
,,
a
Tutt
,,,
a
Tutt
,,
ufut
, , .
1
2
Applying equation (9.18) to f(u,t) gives
