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In-Depth Information
3. GEOMETRY, TOPOLOGY, AND SHAPE REPRESENTATION
e
genus
of a connected, orientable surface is an integer representing the maximum number
of cuttings along non-intersecting closed simple curves without rendering the resultant manifold
disconnected, see also chapter
7
. It is equal to the number of handles on it.
3.9
TANGENT SPACE
Proceeding with the generalization of intuitive notions from the Euclidean space to more general
ones, we have to introduce the notion of tangent space. In the domain of geometry, the tangent
space is defined by tangency conditions between curves on the manifold space, as follows.
Let
M
be a
C
k
manifold,
k1
, and
p2M
. Fix a chart
'WU!E
n
, where
p2U
M
. Suppose that two curves
1
W.1;1/!M
and
2
W.1;1/!M
with
1
.0/D
2
.0/Dp
are given such that
'
1
and
'
2
are both differentiable at
0
. en
1
and
2
are called
tangent
at
0
if the ordinary derivatives of
'
1
and
'
2
coincide at
0
:
.'
1
/
0
.0/D.'
2
/
.0/
. If the functions
'
i
W.1;1/!E
n
,
iD1;2
, are given by
n
real-valued component
functions
.'
i
/
1
.t/;:::;.'
i
/
n
.t/
, the condition above means that their Jacobian matrices
0
d.'
i
/
1
.t/
dt
;:::;
d.'
i
/
n
.t/
coincide at
0
. is is an equivalence relation, and the equivalence
dt
0
class
.0/
of the curve
is called a
tangent vector
of
M
at
p
. e tangent space
T
p
.M/
of
M
at
a point
p
is defined as the set of all tangent vectors at
p
. Figure
3.8
represents the tangent plane
of a point on a surface, which reflects the intuition of tangency that we have in Euclidean spaces.
e dual of the tangent space is called the
cotangent
space.
Figure 3.8:
Tangent plane in
p
. e vectors
Eu
and
Ev
are tangent vectors [
217
].
3.10 RIEMANNIAN MANIFOLD
We eventually finish the list of basic notions with the Riemannian manifold, where many of the
concepts introduced so far are integrated. is space, very rich in terms of geometric structure,
allows us to handle properly well-behaved spaces in shape analysis and to define a set of geometric
measures which have well-grounded mathematical definitions.
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