  Expansions of Lie algebras are the opposite process of contractions. Starting
from a Lie algebra, the expansion process goes to another one, non-isomorphic
and less abelian. We propose an expansion method based in the Casimir
invariants of the initial and expanded algebras and where the free parameters
involved in the expansion are the curvatures of their associated homogeneous
spaces. This method is applied for expansions within the family of Lie algebras
of 3d spaces and (2+1)d kinematical algebras. We show that these expansions are
classed in two types. The first type makes different from zero the curvature of
space or space-time (i.e., it introduces a space or universe radius), while the
other has a similar interpretation for the curvature of the space of
worldlines, which is non-positive and equal to $-1/c^2$ in the kinematical
algebras. We get expansions which go from Galilei to either Newton--Hooke or
Poincar\'e algebras, and from these ones to de Sitter algebras, as well as some
other examples.
