Graphics Reference
In-Depth Information
()
¢
()
(
)
=
()(
1.
DF t g t
DF t
Thus, if we understand tangent vectors of the curve corresponding to the x-axis, then
we get the tangent vector for an arbitrary parametric curve by mapping the tangent
vectors along the x-axis over to it.
Now let us move on to higher dimensions. It clearly makes sense to talk about the
tangent plane of a surface in
R
3
at a point. Higher-dimensional manifolds in
R
n
also
have tangent planes. There are a number of different rigorous definitions that can be
given for the tangent space of a submanifold of Euclidean space. The easiest way is
to take advantage of the fact that we understand tangent vectors to parametric curves.
Definition.
Let
M
be a manifold and let
p
Œ
M
. If
[
]
Æ>
g :
-
aa
,
M
,
a
0
,
is any parametric curve in
M
with g(0) =
p
, then g¢(0) is called a
tangent vector of
M
at
p
. The set of tangent vectors to
M
at
p
is called the
tangent space of
M
at
p
and is
denoted by T
p
or T
p
(
M
). The set
{
}
pvv
p
+Œ
|
T
is called the
tangent plane of
M
at
p
.
See Figure 8.12.
Caution.
Later on, such as in Sections 8.11, 8.12, and 9.16, we shall, for technical
reasons, want to “index” tangent vectors by the point at which they are defined. There-
fore the definition of “tangent vector” and “tangent space” will be changed slightly.
This will in no way alter any facts about these concepts but only involve a simple and
obvious translation of terminology. The later definitions are needed for the theory of
abstract manifolds. Right now we are quite happy to stick to subspaces of
R
n
and to
keep the notation as simple as possible.
Note that the tangent space and the tangent plane are just translations of each
other. The tangent space passes through the origin (since the constant curve g(t) =
p
,
produces the zero tangent vector) and the tangent plane at
p
passes through
p
. To
justify the terminology we need to show that the T
p
(
M
) is a vector space. We also need
a practical way to compute these spaces.
g ¢ (0)
p = g (0)
g (t)
M
Figure 8.12.
A tangent vector at
p
in a mani-
fold
M
.
