  We consider a model consisting of a self-avoiding polygon occupying a
variable density of the sites of a square lattice. A fixed energy is associated
with each $90^\circ$-bend of the polygon. We use a grand canonical ensemble,
introducing parameters $\mu$ and $\beta$ to control average density and average
(total) energy of the polygon, and show by Monte Carlo simulation that the
model has a first order, nematic phase transition across a curve in the
$\beta$-$\mu$ plane.
