  We discuss generalizations of some results on lattice polygons to certain
piecewise linear loops which may have a self-intersection but have vertices in
the lattice $\mathbb{Z}^2$. We first prove a formula on the rotation number of
a unimodular sequence in $\mathbb{Z}^2$. This formula implies the generalized
twelve-point theorem in [12]. We then introduce the notion of lattice
multi-polygons which is a generalization of lattice polygons, state the
generalized Pick's formula and discuss the classification of Ehrhart
polynomials of lattice multi-polygons and also of several natural subfamilies
of lattice multi-polygons.
