  We study flips in hypertriangulations of planar points sets. Here a level-$k$
hypertriangulation of $n$ points in the planes is a subdivision induced by the
projection of a $k$-hypersimplex, which is the convex hull of the barycenters
of the $(k-1)$-dimensional faces of the standard $(n-1)$-simplex. In
particular, we introduce four types of flips and prove that the level-2
hypertriangulations are connected by these flips.
