  This work introduces a new fitting formalism for isophotes which enables more
accurate modelling of galaxies with non-elliptical shapes, such as disk
galaxies viewed edge-on or galaxies with X-shaped/peanut bulges. Within this
scheme, the angular parameter which defines quasi-elliptical isophotes is
transformed from the commonly used, but inappropriate, polar co-ordinate to the
`eccentric anomaly'. This provides a superior description of deviations from
ellipticity, better capturing the true isophotal shape. Furthermore, this makes
it possible to accurately recover both the surface brightness profile, using
the correct azimuthally-averaged isophote, and the two-dimensional model of any
galaxy: the hitherto ubiquitous, but artificial, cross-like features in
residual images are completely removed. The formalism has been implemented into
the IRAF tasks $Ellipse$ and $Bmodel$ to create the new tasks `$Isofit$', and
`$Cmodel$'. The new tools are demonstrated here with application to five
galaxies, chosen to be representative case-studies for several areas where this
technique makes it possible to gain new scientific insight. Specifically:
properly quantifying boxy/disky isophotes via the fourth harmonic order in
edge-on galaxies, quantifying X-shaped/peanut bulges, higher-order Fourier
moments for modelling bars in disks, and complex isophote shapes. Higher order
(n > 4) harmonics now become meaningful and may correlate with structural
properties, as boxyness/diskyness is known to do. This work also illustrates
how the accurate construction, and subtraction, of a model from a galaxy image
facilitates the identification and recovery of over-lapping sources such as
globular clusters and the optical counterparts of X-ray sources.
