When the observation of a newly discovered asteroid are not enough to compute a meaningful orbit we can represent the data with an attributable (containing two angles and their time derivatives). The undetermined variables range and range rate span an admissible region of solar system orbits, which can be represented by a set of Virtual Asteroids (VAs) selected by means of an optimal triangulation (see the presentation by Z. Knězević).

However, the attributable 4 coordinates are themselves the result of a fit (whenever there are more than 2 observations) and they have an uncertainty, represented by a covariance matrix. Thus the predictions of future observations can be described by a quasi-product structure (admissible region times confidence ellipsoid), which can be approximated by a triangulation with each node surrounded by a confidence ellipsoid.

The problem of computing a preliminary orbit starting from two short arcs of observations, represented by two attributables, can be solved as follows. For each VA (selected in the admissible region of the first arc) we consider prediction at the time of the second arc and the corresponding covariance matrix, and we compare them with the attributable of the second arc with its own covariance. By using the identification penalty (as in the algorithms for orbit identification) we can select the VAs which allow to fit together both arcs at the attributable level, that is in the 8-dimensional space. This provides us with a preliminary orbit, formed with the identified attributable and the range, range rate at the time of the second arc.

Even two attributables may not be enough to compute an orbit with a convergent differential corrections algorithm. The preliminary orbit obtained as above is used in a constrained differential correction, providing solutions along the Line Of Variations (LOV) which can be used as second generation VAs to further predict the observations at the time of a third arc. In general the identification with a third arc will ensure a well determined orbit. This algorithm is well defined and mathematically rigorous, and has been tested in a few significant cases. However, a full scale test, based upon a realistic survey data set, is necessary to assess how effective it is; this will be our next goal.