Geometry, Topology, and Shape Representation - Mathematical Tools for Shape Analysis and Description - page 15

this is that we can travel on the curve by moving on a one-dimensional interval of the real num-
bers. On a curved space, without the straightness constraint, there are many paths to travel from
one point to another. In order to quantify the distance between two points on curved spaces, it is
necessary to introduce the concept of shortest (curved) path which will be the one along which
the measure will be computed. is is stated formally as follows:

Curve Let .X;d/ be a metric space. A curve WI!X is a continuous function defined on an
interval IR . is called regular of class C r if it is r -times continuously differentiable, and it is
called smooth if rD1 . A curve is a path if it does not cross itself, that is, is injective.
For a regular curve Wa;b!X let us define the parameter sDs.t/Dl.j a;t / , with
l.j a;t / the length of the curve portion corresponding to a;t . A curve D.s/ parametrized
according to s is called parametrized by arc length , and sDs.t/ is also known as the curvilinear
abscissa of the curve (see Fig. 3.2 ).

Figure 3.2: A curve over a surface X and its parametrization by arc length.

P

n
iD1 d..t i1 ;.t i ///
with the supremum taken over all finite decompositions aDt 0 < t 1 < ::: < t n Db , n2N , of
a;b . A curve with finite length is called rectifiable .
e expression above for the curve length states that we can measure the length of the
curve using the distance defined on the metric space X ; we have to cut the interval a;b into
sub-intervals, and measure the distance of these sub-interval extrema under the mapping of
on X . erefore, the length of a curve Wa;b!X is at least the distance between its end
points, that is, l./d..a/;.b// , because the definition above mentions a subdivision process
and a supremum selection over all decompositions, while the value l./Dd..a/;.b// can be
obtained with the trivial decomposition t 0 Da;t 1 Db . In case the curve is expressed by its arc
length parametrization, the curve length takes the form of an integral measure.

Curve length e length l./ of a curve Wa;b!X is defined as sup