The Lighthouse Model

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 $45^\circ$ (i.e., $\beta_i \ne 0^\circ$) 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 $\gamma_i$ in the side view and angles $\delta_i$ in the top view that are different from $0^\circ$. Additionally, the figure shows the rotation axis of the lighthouse platform and its distances $b_1$ and $b_2$ 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 $\beta_i = \gamma_i = \delta_i = 0^\circ$ and $b_1 = b_2$.

Now let us consider an observer (black square) located at distance $d$ from the main lighthouse platform rotation axis and at height $h$ over the lighthouse center. We are interested in the width $b$ of the virtual wide beam as seen by the observer. Let us assume for this that we can build a lighthouse with $b_1 \approx b_2$ and $\beta_i, \gamma_i, \delta_i \approx 0^\circ$, i.e., we do our best to approximate the perfect lighthouse described in Section 4.1. Then we can express $b$ approximately as follows:

$$\begin{array}{rcl} b & \approx & b_1 + b_2 + \sqrt{d^2 + h^2... ...\tan \gamma_2) + d (\sin \delta_1 + \sin \delta_2) \end{array}$$ (6)

The inaccuracy results from the last two terms, which are linear approximations of rather complex non-linear expressions. For $\beta_1 = \beta_2 = 0^\circ$, however, expression 6 becomes an equation. We will allow these factors to be built into the error term.

Figure 5: Model of a realistic lighthouse based on rotating mirrors. The zoom-ins show detail for one rotating mirror in the top and side views. The other rotating mirror has respective parameters $\beta _2, \gamma _2$, and $\delta _2$.

With $C^b := b_1 + b_2$, $C^\beta := \sin \beta_1 + \sin \beta_2$, $C^\gamma := \tan \gamma_1 + \tan \gamma_2$, and $C^\delta := \sin \delta_1 + \sin \delta_2$ we can rewrite expression 6 as

$$b \approx C^b + \sqrt{d^2 + h^2}C^\beta + h C^\gamma + d C^\delta$$ (7)

Note that $C^b, C^\beta, C^\gamma$, and $C^\delta$ are fixed lighthouse parameters. We will show below how they can be determined using a simple calibration procedure. We can express $b$ also in terms of the angle $\alpha$ obtained using Equation 2:

$$b = 2 d \sin \frac{\alpha}{2}$$ (8)

Combining expressions 7 and 8 we obtain the following expression which defines the possible $(d,h)$ locations of the observer given a measured angle $\alpha$ and the lighthouse calibration values $C^\ast$:

$$2 d \sin \frac{\alpha}{2} \approx C^b + \sqrt{d^2 + h^2}C^\beta + h C^\gamma + d C^\delta$$ (9)

Note that for given $C^\ast$ and $\alpha$ the points in space whose $d$ and $h$ values are solutions of Equation 9 form a rotational hyperboloid centered at the rotation axis of the lighthouse. In the special case $\beta_i = \gamma_i = \delta_i = 0^\circ$ and $b_1 = b_2$ this hyperboloid becomes a cylinder as in the idealistic model described in Section 4.1.