Menagerie of polygons
I think(?) I [Joe O'Rourke] was the first to introduce monotone mountains, but perhaps I copied
that class from someone else.  It is a subclass of monotone polygons, with one of the
two chains that constitutes a monotone polygon a single segment.
I used it in the paper,
"Vertex pi-lights for monotone mountains,"
9th Canad. Conf. Comput. Geom., Aug. 1997, pp. 1-5.

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Joe Mitchell subsequently used this class of polygons in his abstract,
"Triangulating Monotone Mountains,"
AMS 345/CSE 355 (PDF link).