  In this article, we functorially associate definable sets to $k$-analytic
curves, and definable maps to analytic morphisms between them, for a large
class of $k$-analytic curves. Given a $k$-analytic curve $X$, our association
allows us to have definable versions of several usual notions of Berkovich
analytic geometry such as the branch emanating from a point and the residue
curve at a point of type 2. We also characterize the definable subsets of the
definable counterpart of $X$ and show that they satisfy a bijective relation
with the radial subsets of $X$. As an application, we recover (and slightly
extend) results of Temkin concerning the radiality of the set of points with a
given prescribed multiplicity with respect to a morphism of $k$-analytic
curves.
  In the case of the analytification of an algebraic curve, our construction
can also be seen as an explicit version of Hrushovski and Loeser's theorem on
iso-definability of curves. However, our approach can also be applied to
strictly $k$-affinoid curves and arbitrary morphisms between them, which are
currently not in the scope of their setting.
