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In-Depth Information
3. GEOMETRY, TOPOLOGY, AND SHAPE REPRESENTATION
this term originated a century ago in the work by M. Frèchet (1906) and F. Hausdorff (1914),
who introduced the theory of
metric spaces
. Readers interested in more details on notations and
definitions presented in this section are referred to the topic [
61
].
Let
X
be a set. A function
dWX!X
is called a
metric
on
X
if, for all
x
,
y
,
z2X
, the
following holds:
1.
d.x;y/0
(
non-negativity
)
2.
d.x;y/D0
if and only if
xDy
(
identity
)
3.
d.x;y/Dd.y;x/
(
symmetry
)
4.
d.x;y/d.x;z/Cd.z;y/
(
triangle inequality
)
A
metric space
.X;d/
is a set
X
equipped with a metric
d
.
e Euclidean 3D world we live in is an example of a metric space, where the metric is given
by the well-known Euclidean distance, that is, the distance between two points is the length of
the straight line that joins them. In Figure
3.1
, on the left side, the dashed segment measures the
Euclidean distance between the yellow points. But imagine the shape is embedded in the empty
space and the distance has to be measured by walking along the shape boundary: the distance
would change considerably (see the dashed line on the right of the figure).
Figure 3.1:
On the left side, the red segment between the yellow points measures the Euclidean
distance, while on the right side it measures the distance on the object boundary.
We will now introduce formally the mathematical notion of
geodesic distance
, which gener-
alizes the notion of distances in a flatland to distances in curved spaces. e term
geodesic
comes
indeed from the science of measuring the Earth, namely geodesy, and stems from the fact that
measures on the curved surface of the Earth require some more considerations.
3.2 GEODESIC DISTANCE
To define the geodesic distance, we need another intuitive geometric notion:
curve
. In mathe-
matical terms, curves are continuous functions, which map an interval of the real numbers onto
a subset of the target space, that is, the space where measures are necessary. e intuition behind
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