  We describe how to generate a longitudinal brightness map for a tidally
locked exoplanet from its phase function light curve. We operate under a number
of simplifying assumptions, neglecting limb darkening/brightening, star spots,
detector ramps, as well as time-variability over a single planetary rotation.
We develop the transformation from a planetary brightness map to a phase
function light curve and simplify the expression for the case of an edge-on
system. We introduce two models--composed of longitudinal slices of uniform
brightness, and sinusoidally varying maps, respectively--which greatly simplify
the transformation from map to light curve. We discuss numerical approaches to
extracting a longitudinal map from a phase function light curve, explaining how
to estimate the uncertainty in a computed map and how to choose an appropriate
number of fit parameters. We demonstrate these techniques on a simulated map
and discuss the uses and limitations of longitudinal maps. The sinusoidal model
provides a better fit to the planet's underlying brightness map, although the
slice model is more appropriate for light curves which only span a fraction of
the planet's orbit. Regardless of which model is used, we find that there is a
maximum of ~5 free parameters which can be meaningfully fit based on a full
phase function light curve, due to the insensitivity of the latter to certain
modes of the map.
