We want to have a reasonable ``safety margin'' with respect to the VIR, a margin that will be reflected in a slightly larger sky region to be observed; in these matters safety is obviously worth some extra effort. We therefore consider a rectangular region in the MTP, centered on the VI point; the exact definition is detailed in Sec. 2.5, but the idea is to ensure that points outside it are either passing at Earth radii during that encounter, or belong to orbits for which , that is well outside the confidence region.
We then compute the preimage of the MTP rectangle back into the confidence region in element space, thus obtaining the set of orbits of interest, which is somewhat larger than the VIR because of the safety margins. The details of this computation are described in Sec. 2.5, but a qualitative description could be as follows: this region is similar to the Cartesian product of the MTP rectangle, pulled back in the orbital element space, and of a 4-dimensional region; this is due to the fact that the orbital element space is 6-dimensional, thus each point on the 2-dimensional MTP has a preimage which is 4-dimensional.
If we apply the procedure described in [Milani 1999] to obtain the sky coordinates at a given epoch of the observation confidence region in which a lost asteroid should be located, we typically obtain a thin strip which can be extremely long, even more than a full tour around the sky (because sometimes the string of virtual asteroids extends over more than a full orbit).
When this observation confidence region is too large to be completely scanned, we need to identify the subset of observations corresponding to the orbits which are both compatible with the existing observations and capable of having a dangerously close approach, that is to the VIR, slightly augmented for safety. At a given time, the footprint in the sky of this region is the skyprint corresponding to a given VI.
If the skyprint of a given VI is, at some time, small enough to be observed at once with a single plate/CCD frame (at most with a few frames), and the corresponding virtual asteroid is bright enough at that time to be detected with certainty (with the given telescope, exposure, and detection technique), then a negative observation is possible with small resources. Again, some safety margins are required to allow us to rely on the negative observation, as discussed in Sec. 4.
The main results of Sec. 2 can now be described as follows: we have a reliable procedure to find virtual impactors, and an efficient procedure to compute the skyprint associated with them at any given time. These tools can be used to plan a campaign to search and destroy VIs. One such campaign to hunt those of 1998 OX4 is proposed in Sec. 3.