  A large international effort is under way to assess the presence of a shadow
in the radio emission from the compact source at the centre of our Galaxy,
Sagittarius A$^*$ (Sgr A$^*$). If detected, this shadow would provide the first
direct evidence of the existence of black holes and that Sgr A$^*$ is a
supermassive black hole. In addition, the shape of the shadow could be used to
learn about extreme gravity near the event horizon and to determine which
theory of gravity better describes the observations. The mathematical
description of the shadow has so far used a number of simplifying assumptions
that are unlikely to be met by the real observational data. We here provide a
general formalism to describe the shadow as an arbitrary polar curve expressed
in terms of a Legendre expansion. Our formalism does not presume any knowledge
of the properties of the shadow, e.g. the location of its centre, and offers a
number of routes to characterize the distortions of the curve with respect to
reference circles. These distortions can be implemented in a coordinate
independent manner by different teams analysing the same data. We show that the
new formalism provides an accurate and robust description of noisy
observational data, with smaller error variances when compared to previous
approaches for the measurement of the distortion.

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena;Solar and Stellar Astrophysics