Generally, the analysis of corneal topography involves fitting the raw data to a parametric geometric model that includes a regular basis surface, plus some sort of polynomial expansion to adjust the more irregular residual component. So far, these parametric models have been used in their canonical form, ignoring that the observation (keratometric) coordinate system is different from corneal axes of symmetry. Here we propose, instead, to use the canonical form when the topography is referenced to the intrinsic corneal system of coordinates, defined by its principal axes of symmetry. This idea is implemented using the general expression of an ellipsoid to fit the raw data given by the instrument. Then, the position and orientation of the three orthogonal semiaxes of the ellipsoid, which define the intrinsic Cartesian system of coordinates for normal corneas, can be identified by passing to the canonical form, by standard linear algebra. This model has been first validated experimentally obtaining significantly lower values for rms fitting error as compared with previous standard models: spherical, conical, and biconical. The fitting residual was then adjusted by a Zernike polynomial expansion. The topographies of 123 corneas were analyzed obtaining their radii of curvature, conic constants, Zernike coefficients, and the direction and position of the optical axis of the ellipsoid. The results were compared with those obtained using the standard models. The general ellipsoid model provides more negative values for the conic constants and lower apex radii (more prolate shapes) than the standard models applied to the same data. If the data are analyzed using standard models, the resulting mean shape of the cornea is consistent with previous studies, but when using the ellipsoid model we find new interesting features: The mean cornea is a more prolate ellipsoid (apical power 50 D), the direction of the optical axis is about 2.3 degrees nasal, and the residual term shows three Zernike coefficients significantly higher than zero (third-order trefoil and fourth- and sixth-order spherical). These three nonzero Zernike coefficients are responsible for most of the higher-order aberrations of the average cornea. Finally, we propose and implement a simple method for three-dimensional registration of corneal topographies, passing from the general to the canonical form of the ellipsoid.