A theory and low computational cost linear algorithm is presented for estimating algebraic surfaces of second degree for representing an object in 3D, based on fitting in the dual space (space of tangent planes) computed from images taken by a calibrated camera in a number of positions. The approach and algorithm are designed to handle implicit quadric surfaces, which are regular or singular, in a uniform way without distinguishing the two cases. A significance of these quadric surface estimation results is, as illustrated in the paper, the estimation of complex 3D free form shapes in a computationally simple way in terms of quadric patches. The paper explains how singular quadrics cause instabilities in the 3D surface fitting and representation, and presents regularization, based on this understanding, to produce accurate stable surface representations.