with the ellipticals FP. The approach of this work is the use of analytic toy models, which cannot capture the intrinsic complexity of galaxies due to their hierarchical mass assembly, but resort to absorb it into a single parameter, $\Gamma_e$, the mass-to-light ratio inside the half-light radius $r_e$, and to fit this parameter assuming that galaxies lie on the manifold. Catinella et al. (2012) make use instead of dynamical indicators such as rotational velocity and dispersion to obtain a generalised baryonic FJ relation that holds for all galaxy types. Courteau et al. (2007) investigate galaxy size, and analyse the spiral size-luminosity and luminosity-velocity relations and their dependence on morphology and stellar population content.

In this work we consider two questions: 1) is there a third dynamical parameter that characterises the morphology-dependence of the Tully-Fisher relation, and can be used to expand such relation into a three-dimensional manifold, describing the structure of spiral galaxies?; and 2) can we use this parameter to classify galaxy morphology, when visualisation is not available (for instance at high redshift) or impractical (for instance for large surveys)?

The novelty of our analysis is that we take advantage of a full semi-analytic hierarchical model (based from Croton 