A Symmetry-induced Model of Elliptical Galaxy Patterns
  S\'ersic (1968) generalized the de Vaucouleurs law which follows the projected (observed) one dimensional radial profile of elliptical galaxies closely and Dehnen (1993) proposed an analytical formula of the 3-dimensional light distributions whose projected line profile resembles the de Vaucouleurs law. This paper is involved to recover the Dehnen model and generalize the model to account for galaxy elliptical shapes by means of curvilinear coordinate systems and employing a symmetry principle. The symmetry principle maps an orthogonal coordinate system to a light distribution pattern. The coordinate system for elliptical galaxy patterns turns out to be the one which is formed by the complex-plane reciprocal transformation $Z=1/W$. The resulting spatial (3-dimensional) light distribution is spherically symmetric and has infinite gradient at its centre, which is called spherical-nucleus solution and is used to model galaxy central area. We can make changes of the coordinate system by cutting out some column areas of its definition domain, the areas containing the galaxy centre. The resulting spatial (3-dimensional) light distributions are axisymmetric or triaxial and have zero gradient at the centre, which are called elliptical-shape solutions and are used to model global elliptical patterns. The two types of logarithmic light distributions are added together to model full elliptical galaxy patterns. The model is a generalization of the Dehnen model. One of the elliptical-shape solutions permits realistic numerical calculation and is fitted to all R-band elliptical images from the Frei {\it et al.}(1996)'s galaxy sample. The fitting is satisfactory. This suggests that elliptical galaxy patterns can be represented in terms of a few basic parameters. 
Jin He