In a previous work we introduced a new general representation of algebraic surfaces, that we called semi-implicit, which encapsulates both usual and less known surfaces. Here we specialize this notion in order to apply it in Solid Modeling: we view a surface in the real space as a one-parameter (algebraic) family of algebraic low-degree curves. The paper mainly addresses the topic of performing the usual CAD operations with semi-implicit representation of surfaces. We derive formulae for computing the normal and the curvatures at a regular point. We provide exact algorithms for computing self-intersections of a surface and more generally its singular locus. We also present some surface/surface intersection algorithms relying on generalized resultant calculations.