Once the VIR, with safety margins, *X*_{R} has been computed, or at
least approximated in the
space, we need to be able to
compute the image of it onto the sky at any given time in which
the observations might take place. Given the approximate
representation of *X*_{R} as a product, we can separately map the two
factors, which can be done in a very efficient way.
One observation at time *t* is mathematically described by a smooth
*observation function*
,
where
is the celestial sphere. A point on the
celestial sphere is represented by two angular coordinates
.
At time *t* the virtual impactor has an ephemeris
.

We first map the four corners *R*_{i} of *R* to the four initial
conditions
according
to (4). For these four initial conditions, we compute four
displacements with respect to the VI ephemerides
.
They mark on the sky a quadrilateral
*F*_{s}(*H*_{T}(*R*)) which is almost flattened to a segment in the cases of a
low probability impact at a very late epoch. (See in the Figures of
the next section.)

The space of changes in orbital elements is a product ; we need to map the 4-dimensional ellipsoid onto the sky plane. To do this we use the same formalism used in [Milani 1999, Sec. 3] to map the confidence ellipsoid onto the sky: the same formulas are used, but they refer to a 4-dimensional space rather than to a 6-dimensional space.

To this purpose, let us consider the differential (linearized map)
*DF*_{S} of *F*_{S}, computed at the VI initial conditions. If we
restrict the map *F*_{S} to the 4-dimensional subspace
,
we
can describe *DF*_{S} by means of a
matrix of partial
derivatives, with rows
and
.
Let
be the 2-dimensional
subspace of
spanned by these two gradients, and
the orthogonal space (also 2-dimensional), so that
.

Then *DF*_{S} restricted to
is a composition of an
orthogonal projection
onto
,
followed
by an invertible map
:

It is possible to perform a change of coordinates in , by means of an orthogonal matrix

with and ; then the normal matrix

This allows to select a representative in the
space for
every point on the sky near the ephemerides of the virtual impactor:

where

The last step in the procedure is to represent the image, on the sky,
of the Cartesian product
by the Cartesian
product of the displacements on the sky, that is

The left hand side is, by definition, the

Please note that these computations contain a number of approximations
which are accurate only provided the skyprint is small. If this is the
case, as in the Figures of the next section, then the product
structure is clearly visible, since
*F*_{S}(*H*_{t}(*R*)) is almost a segment
and
is almost an ellipse, thus the skyprint looks
very much like the drawing of an ordinary 3-dimensional cylinder.
If the skyprint was large, the linear, semilinear, and Cartesian
product approximations used in this section would break down.

On the other hand, if the skyprint was big, to the point of being not well approximated with the linear and semilinear formalisms, then the negative observation campaign would not be worthwhile. We do not have a simple way to predict in which cases the skyprint will be small, in which cases it will be too large to be usable; this depends upon the size of the confidence region (in turn depending upon the number of observations and the length of the observed arc), the date of the possible impact, its probability, the date and the geometry of the observation. Thus we need to apply the computational procedure outlined above case by case; to this purpose we have developed rather efficient software.