Missing sets in rational parametrizations of surfaces of revolution

• We give a simple description of the missing area of ruled surface parametrization.
• We provide algorithms to compute, in each case, the missing area.
• We analyze the real and the complex case.

Parametric representations do not cover, in general, the whole geometric object that they parametrize. This can be a problem in practical applications. In this paper we analyze the question for surfaces of revolution generated by real rational profile curves, and we describe a simple small superset of the real zone of the surface not covered by the parametrization. This superset consists, in the worst case, of the union of a circle and the mirror curve of the profile curve.