Geometry, Topology, and Shape Representation - Mathematical Tools for Shape Analysis and Description - page 16

3. GEOMETRY, TOPOLOGY, AND SHAPE REPRESENTATION
Shortest path Now, is there a way to move from one point on X to another one by moving
along a path as short as possible? e answer is yes, and the way to do it is to follow the so-called
shortest path between the two points. e definition is the following:
e curve which satisfies l./Dd..a/;.b// is called the shortest path between .a/
and .b/ .
Not all paths between two points are shortest paths: in Figure 3.3 , left side, we see two paths
over a sphere between two points P and Q but only the blue one represents a geodesic segment.
Note that there may not be a shortest path, for instance when the shape is not connected by arcs.
When existing, the shortest path may not be necessarily unique: in Figure 3.3 , the picture on the
right side shows that there are two shortest paths between P and Q as the distances to travel
around the Gaussian bell are the same.
Figure 3.3: Left: the blue line represents the geodesic segment between the points P and Q while the
yellow one is a possible path whose length is not minimal. Right: the shortest path is not necessarily
unique.
Geodesic distance With these premises, the geodesic distance between two points is defined as
the length of a shortest path between the two points. e geodesic segment and the geodesic
curve are concepts which relate to quite a considerable mathematical machinery. ere are subtle
differences between the two, which largely depend on the structure of the space S . ere are
indeed metric spaces where the limit process referred to in the definition of curve length may
yield unexpected results. For readers interested in additional mathematical details, we suggest the
following topics [ 35 , 62 ].
As we will see in the following chapters, the notion of geodesic distance is quite important
for shape analysis and the existence of metrics defined by the geodesic distance characterizes
strongly the nature of the space where they are defined.
Intrinsic metric Given a metric space .X;d/ in which every pair of points are joined by a recti-
fiable curve, the internal metric on X is defined as the infimum of the lengths of all rectifiable
curves connecting two given points. e metric d is called the intrinsic metric if it coincides with
the internal metric . A metric space with the intrinsic metric is called an intrinsic or inner metric
space .