  This paper contains a comparative study of two families of simple curves
drawn in the plane. On the one hand, we have the fractal curves on the unit
interval, with self-similar structure, which have associated a Hausdorff
dimension. On the other hand, we have the opposite: a class of locally
rectifiable unbounded curves, which have another "fractional dimension" defined
by M. Mendes France. We propose a geometrical constructive process that will
allow us to obtain - as the limit of a sequence of polygonal curves - one curve
of the first family, by contractive transformations; and another of the second
family, by expansive transformations. Thanks to this process of linking curves
from both families, we are able to compare their dimensions - our aim in this
work -, and to obtain interesting results such as the equality of the latter in
the case of strict self-similarity.
