  This is a case study of the algebraic boundary of convex hulls of varieties.
We focus on surfaces in fourspace to showcase new geometric phenomena that
neither curves nor hypersurfaces do. Our method is a detailed analysis of a
general purpose formula by Ranestad and Sturmfels in the case of smooth real
algebraic surfaces of low degree (that are rational over the complex numbers).
We study both the complex and the real features of the algebraic boundary of
Veronese, Del Pezzo and Bordiga surfaces.
