  A convex combination mapping of a planar graph is a plane mapping in which
the external vertices are mapped to the corners of a convex polygon and every
internal vertex is a proper weighted average of its neighbours. If a planar
graph is nodally 3-connected or triangulated then every such mapping is an
embedding (Tutte, Floater).
  We give a simple characterisation of nodally 3-connected planar graphs, and
generalise the above result to any planar graph which admits any convex
embedding.
