With the substantial improvements in observational techniques we have to deal with very big databases, consisting of a few positions of an object over a short time span; this is often not enough to compute a preliminary orbit with traditional tools. In this paper we first review a classical method by C.F. Gauss to compute a preliminary orbit for asteroids. This method, followed by a least squares fit to improve the orbit, still today gives successful results when we have at least three separate observations. Then we introduce the basics of a very recent orbit determination theory, that has been thought just to be used with modern sets of data. These data allow us in many cases to know the angular position and velocity of an asteroid at a given time, even though the radial distance and velocity $(r,\dot r)$, needed to compute its full orbit, are unknown. The variables $(r,\dot r)$ can be constrained to a compact set, that we call the admissible region(AR), whose definition requires that the body belongs to the Solar System, that it is not a satellite of the Earth, and that it is not a “shooting star” (i.e. very close and very small). We provide a mathematical description of the AR: its topological properties are surprisingly simple, in fact it turns out that the AR cannot have more than two connected components. A sampling of the AR can be performed by means of a Delaunay triangulation; a finite number of six-parameter sets of initial conditions are thus defined, with each node of the triangulation representing a possible orbit (a virtual asteroid).
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