  The distinction between the true total area and the projected area is
elucidated with soluble models which represent the membrane as a self-avoiding
string on a plane. Constraining the total area to a predetermined value changes
the averages very significantly. The latter are calculated exactly from the
generating functions of self-avoiding walks and are shown as functions of
activities $q$ and $r$ related to temperature $T=\pm 1/\log (q)$ and lateral
force $f=-\log (r)$. The constraint makes the partition functions and averages
valid for all $q,r >0$ and reduces the ratio of $A_{tot}$ to the projected area
$L$. High temperature divergences are supressed. Possible applications to
simulated bilayers/membranes are discussed.
