For strictly convex C with m singular points, C-polygons have n to 2(n-1)+m singular boundary points and translative C-polygons have n to n+m.
McMullen, On the upper-bound conjecture for convex polytopes, J
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Vertex Classification of Planar C-polygons
For strictly convex C with m singular points, C-polygons have n to 2(n-1)+m singular boundary points and translative C-polygons have n to n+m.