Graphics Reference
In-Depth Information
Figure 8.23.
Parameterizing a handle.
(u
k+1
,...,u
n
) - axis
(u
1
,...,u
k
) - axis
p
e
k
f = c - e
f = c - e
f = c + e
f = c + e
f = c
f = c
Clearly, ∂
e
k
= M
c-e
«
e
k
.
Next, define functions
2
2
(
)
=+ +
x
uu
,
,...,
u
u
...
u
12
n
1
k
2
2
(
)
=
h
uu
,
,...
u
u
+
...
+
u
,
12
n
k
+
1
n
so that
f=-xh,
and choose a C
•
function m :
R
Æ
R
satisfying
(1) m(0) >e
(2) m(x) = 0 for x ≥ 2e, and
(3) -1 <m¢(x) £ 0 for all x.
Finally, using the local coordinates u
i
, define the function
F:
MR
Æ
in local coordinates by
(
)
=-+-
(
)
Ff
=-
mx
+
2
h
c
x
h
mx
+
2
h
on
U
,
=
f
outside
U
.
F
-1
([-•,c -e]) =
M
c+e
.
Claim 1.
Claim 1 is proved by considering the ellipsoid defined by x+2h£2e and noting
that the functions f and F agree outside the ellipsoid and inside it we have
