  We characterize mutation-finite cluster algebras of rank at least 3 using
positive semi-definite quadratic forms. In particular, we associate with every
unpunctured bordered surface a positive semi-definite quadratic space $V$, and
with every triangulation a basis in $V$, such that any mutation of a cluster
(i.e., a flip of a triangulation) transforms the corresponding bases into each
other by partial reflections. Furthermore, every triangulation gives rise to an
extended affine Weyl group of type $A$, which is invariant under flips. The
construction is also extended to exceptional skew-symmetric mutation-finite
cluster algebras of types $E$.
