We use Figure 5 to explain the lighthouse model. It
shows a simplified top and side view of the lighthouse. Each view
shows the two mirror's rotation axes and the corresponding reflected
rotating laser beams. Note that in general the angle enclosed by the
mirror rotation axis and the mirror will not be exactly
(i.e.,
) due to manufacturing
limitations. Therefore, the rotating reflected laser beams will form
two cones as depicted in Figure 5. Moreover, the two
mirror's rotation axes will not be perfectly aligned. Instead, the
dashed vertical line (connecting the apexes of the two cones formed by
the rotating laser beams) and the mirror rotation axes will enclose
angles
in the side view and angles
in the top
view that are different from
. Additionally, the figure shows
the rotation axis of the lighthouse platform and its distances
and
to the apexes of the two light cones. The lighthouse
center is defined as the intersection point of the lighthouse
platform rotation axis and the dashed vertical line in Figure
5. Note that the idealistic lighthouse described in
Section 4.1 is a special case of this more complex
model with
and
.
Now let us consider an observer (black square) located at distance
from the main lighthouse platform rotation axis and at height
over
the lighthouse center. We are interested in the width
of the
virtual wide beam as seen by the observer. Let us assume for this that we
can build a lighthouse with
and
, i.e., we do our best to approximate the
perfect lighthouse described in Section 4.1. Then we
can express
approximately as follows:
The inaccuracy results from the last two terms, which are linear
approximations of rather complex non-linear expressions. For
, however, expression 6 becomes an
equation. We will allow these factors to be built into the error term.
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With
,
,
, and
we can rewrite expression 6 as
Note that
, and
are fixed
lighthouse parameters. We will show below how they can be determined
using a simple calibration procedure. We can express
also in terms
of the angle
obtained using Equation 2:
Combining expressions 7 and 8 we obtain the
following expression which defines the possible locations of
the observer given a measured angle
and the lighthouse
calibration values
:
Note that for given and
the points in space whose
and
values are solutions of Equation 9 form a
rotational hyperboloid centered at the rotation axis of the
lighthouse. In the special case
and
this hyperboloid becomes a cylinder as in the
idealistic model described in Section 4.1.