  We are interested in geometric approximation by parameterization of
two-dimensional multiple-component shapes, in particular when the number of
components is a priori unknown. Starting a standard method based on successive
shape deformations with a one-component initial shape in order to approximate a
multiple-component target shape usually leads the deformation flow to make the
boundary evolve until it surrounds all the components of the target shape. This
classical phenomenon tends to create double points on the boundary of the
approximated shape. In order to improve the approximation of multiple-component
shapes (without any knowledge on the number of components in advance), we use
in this paper a piecewise B\'ezier parameterization and we consider two
procedures called intersecting control polygons detection and flip procedure.
The first one allows to prevent potential collisions between two parts of the
boundary of the approximated shape, and the second one permits to change its
topology by dividing a one-component shape into a two-component shape. For an
experimental purpose, we include these two processes in a basic geometrical
shape optimization algorithm and test it on the classical inverse obstacle
problem. This new approach allows to obtain a numerical approximation of the
unknown inclusion, detecting both the topology (i.e. the number of connected
components) and the shape of the obstacle. Several numerical simulations are
performed.
