Fitting C²-continuous or superior surfaces to a set S of points sampled on a 2-manifold is central to reverse engineering, computer aided geometric modeling, entertaining, modeling of art heritage, etc. This article addresses the fitting of analytic (ellipsoid, cones, cylinders) surfaces in general position in . Currently, the state of the art presents limitations in 1) automatically finding an initial guess for the analytic surface F sought, and 2) economically estimating the geometric distance between a point of S and the analytic surface F. These issues are central in estimating an analytic surface which minimizes its accumulated distances to the point set. In response to this situation, this article presents and tests novel user-independent strategies for addressing aspects 1) and 2) above, for cylinders, cones and ellipsoids. A conjecture for the calculation of the distance point-ellipsoid is also proposed. Our strategies produce good initial guesses for F and fast fitting error estimation for F, leading to an agile and robust optimization algorithm. Ongoing work addresses the fitting of free-form parametric surfaces to S.