  The pentagram map was invented by Richard Schwartz in his search for a
projective-geometric analogue of the midpoint map. It turns out that the
dynamical behavior of the pentagram map is totally different from that of the
midpoint map. Recently, Schwartz has constructed a related map, the projective
heat map, which empirically exhibits similar dynamics as the midpoint map. In
this paper, we will demonstrate that there is a one-parameter family of maps
which behaves a lot like Schwartz's projective heat map.
