Graphics Reference
In-Depth Information
(
()
)
(
)
+
(
()
)
0
=
D
a
Tut tDTut
,,
,
D
a
Tut t
,,.
(9.20)
12
2
22
On the other hand, differentiating equation (9.19) gives
(
()
)
(
)
+
(
()
)
=
DTut t DTut
a
,,
,
D Tut t
a
,,
0
,
11
2
21
that is,
(
()
)
()
DTut t
DTut t
a
a
,,
,,
.
21
11
()
=-
DTut
,
2
(
)
After substituting into equation (9.20), we finally obtain
(
)
(
(
)
)
=
DD
aa aa
-
DD Tut t
,,
0
.
12 21
11 22
We conclude that the envelope in general should consist of the points a(u,t) where
(u,t) satisfies
(
()
)
= 0
det D
i
a
j
ut
,
.
(9.21)
Getting criterion (9.21) involved somewhat loose reasoning, but independent of
how we got equation (9.21), one can show that the envelope we want
must
be a subset
of the points defined by that equation.
9.5.1. Theorem.
Equation (9.21) is a necessary condition that points of an envelope
defined by a family of curves a(u,t) must satisfy. Alternatively, the condition can be
expressed as
∂a
∂
∂a
u
¥=0.
(9.22)
Proof.
See [Spiv75] or [Stok69].
Note:
Condition (9.21) is only a
necessary
condition for an envelope but not a
sufficient
condition! See [Stok69].
9.5.2. Example.
To determine the envelope of circles of radius 1 centered on the
line y = x in the plane. See Figure 9.10.
Solution.
If we parameterize the circles by angles, then we can apply Theorem 9.5.1
to the map a(u,t) defined by
()
=
()
+
(
)
a ut
,
tt
,
cos u, sin u
.
Since
D
a
=
cos u,
D
a
=
1
,
D
a
= -
sin u,
and
D
a
=
1
,
12
2
1
11
2
2
