  Euler's elastica is defined by a critical point of the total squared
curvature under the fixed length constraint, and its $L^p$-counterpart is
called $p$-elastica. In this paper we completely classify all $p$-elasticae in
the plane and obtain their explicit formulae as well as optimal regularity. To
this end we introduce new types of $p$-elliptic functions which streamline the
whole argument and result. As an application we also classify all closed planar
$p$-elasticae.
