  In this work, we present a translation of the complete pipeline for
variational shape approximation (VSA) to the setting of point sets. First, we
describe an explicit example for the theoretically known non-convergence of the
currently available VSA approaches. The example motivates us to introduce an
alternate version of VSA based on a switch operation for which we prove
convergence. Second, we discuss how two operations - split and merge - can be
included in a fully automatic pipeline that is in turn independent of the
placement and number of initial seeds. Third and finally, we present two
approaches how to obtain a simplified mesh from the output of the VSA
procedure. This simplification is either based on simple plane intersection or
based on a variational optimization problem. Several qualitative and
quantitative results prove the relevance of our approach.
