  We characterize the triples of interior angles that are possible in
non-self-crossing triangles with circular-arc sides, and we prove that a given
cyclic sequence of angles can be realized by a non-self-crossing polygon with
circular-arc sides whenever all angles are at most pi. As a consequence of
these results, we prove that every cactus has a planar Lombardi drawing (a
drawing with edges depicted as circular arcs, meeting at equal angles at each
vertex) for its natural embedding in which every cycle of the cactus is a face
of the drawing. However, there exist planar embeddings of cacti that do not
have planar Lombardi drawings.
