Representations of the real projective plane are called Steiner surfaces. A classification based on real parameters and transformations results in 10 types [1, 2].

[1] A. Coffman, A. J. Schwartz, and C. Stanton, "The Algebra and Geometry of Steiner and Other Quadratically Parametrizable Surfaces," Computer Aided Geometric Design, 13, 1996 pp. 257–286.