Graphics Reference
In-Depth Information
Figure 8.4.
Regular reparameter-
izations.
R
n
X
y
F
R
k
R
k
U
V
m
(cos q, sin q) = (t,
√
1- t
2
)
q
t
Y
F
Figure 8.5.
Reparameterizing the
half circle.
R
R
0
q
p
m
-1
0
t
1
[
]
Æ
1
[
]
Æ
1
F:
0
,
p
S
and
Y:
-
1 1
,
S
+
+
defined by
(
)
()
=
(
)
( )
=
2
F
q
cos
q
,
sin
q
and
Y
t
t
,
1
-
t
.
The map
p
[
]
Æ
[
]
()
=-
(
)
m
:
-
11
,
0
,
p
,
m
t
1
t
,
2
¢
()
=-
is an orientation-reversing reparameterization because
m
t
p
2
. See Figure 8.5.
Manifolds in R
n
8.3
Topological manifolds were defined in Section 5.3. Now we add a differential struc-
ture. In this section we restrict ourselves to subsets of Euclidean space because, by
