Stroboscope-and-telegraph problem

Question

Narrative:

Consider, in a suitably flat region, two straight lines which don't necessarily intersect. Let vector $\mathbf{x}$ point along one line, and vector $\mathbf{y}$ point along the other. Let $X$ and $Y$ be the two points closest to each other on these two lines, or otherwise let $P$ be their intersection point.

The real number $\phi := \text{ArcCos}\bigl( \frac{\mathbf{x} \, \cdot \, \mathbf{y}}{|\mathbf{x}| \, |\mathbf{y}|}\bigr)$ is supposed to be known as part of the description of how the two lines are related to each other.

Now consider a stroboscope "$B_x$" moving uniformely along one of these lines (in direction of vector $\mathbf{x}$, passing point $X$), and a telegraph "$A_y$" moving uniformely along one of these lines (in direction of vector $\mathbf{y}$, passing point $Y$). The case of intersecting lines may be dealt with later on.

Stroboscope $B_x$ is supposed to blink ceaselessly (therefore its name: "$B_x$") with equal durations $\tau_B$ from any one "blink" ($b_j$) to the next one ($b_{j+1}$); and telegraph $A_y$ is supposed to re-shuffle its "arms" all the time (... cmp. "$A_y$") with equal durations $\tau_A$ from settling into any one signal state ($a_k$) to settling to the next ($a_{k+1}$). The real "ratio" number $r := \tau_B / \tau_A$ is supposed to be known, too.

Let $a_0$ be the state of telegraph $A_y$ while passing point $Y$, and $b_0$ be the state of stroboscope $B_x$ while passing point $X$. Let integer $p_{BA}$ be given as the number of "blinks" stated by stroboscope $B_x$ after having passed point $X$ (i.e. after having been in state $b_0$) and before observing telegraph $A_y$ passing point $Y$ (i.e. before observing that telegraph $A_y$ had been in state $a_0$). If the described order of observations is inverted (i.e. if stroboscope $B_x$ first observed that telegraph $A_y$ had passed point $Y$ and only later passed point $X$) then integer $p_{BA}$ is obtained as negative value (or as value $0$).

Finally suppose that stroboscope $B_x$ is (partially) surrounded by a shiny case such that telegraph $A_y$ can see its (own) reflection and recognize its states in the mirror image; and suppose that telegraph $A_y$ is (perhaps rather diffusely) reflective as well such that stroboscope $B_x$ would be able to see reflections of its " blinks".

Now: For any particular signal state $a_k$ telegraph $A_y$ can count how many subsequent times it went on to re-shuffle its arms before recognizing the reflection (from $B_x$) corresponding to state $a_k$; lets call this number $\alpha_{Ak}$.

Along with that, again for any particular signal state $a_k$, telegraph $A_y$ can count how many subsequent times it saw stroboscope $B_x$ blink subsequently before recognizing the reflection (from $B_x$) corresponding to state $a_k$; lets call this number $\alpha_{Bk}$.

Correspondingly, stroboscope $B_x$ may determine the numbers $\beta_{Bj}$ and $\beta_{Aj}$ ...
(No!, too bad: stroboscope $B_x$ may not readily determine these numbers because it cannot easily tell the reflection of any one blink from that of any other. But this may lead to yet another question ... In order to fix the narrative let's here just assume that "stroboscope $B_x$" is in fact a "telegraph" as well, with distinct states $b_j$ whose reflections from $A_y$ can be individually recognized, too.)

Question (1):

Do the numbers $\alpha_{Ak}$ have a minimum, $\alpha_{\text{min}}$ ?,
in particular, if the two lines don't intersect.
(And similarly: Do the numbers $\beta_{Bj}$ have a minimum, $\beta_{\text{min}}$ ?)

And if so:

Question (2):

Can the "ratio" number $r$ be calculated from the (given) numbers $\phi$, $p$, $\alpha_{Ak}$, $\alpha_{Bk} $, $\beta_{Bj}$ and $\beta_{Aj}$ ?, or perhaps at least approximated for sufficiently large numbers $\alpha_{\text{min}}$ and/or $\beta_{\text{min}}$ ?

Answer 1

While I find it too difficult, at present, to attempt answering the stated problem in general, I'd like to present initial results for some special cases (for now without derivations, which are generally lengthy):

• Special cases of the two lines not intersecting´and
  either $\text{Cos}( \phi ) = 0$, or $\text{Cos}( \phi ) = 1$, or $\text{Cos}( \phi ) = -1$:

  In all these cases there exist minima $\alpha_{\text{min}}$ and $\beta_{\text{min}}$, and
  if these numbers are suitably large then the "ratio number $r := \tau_B / \tau_A$" is obtained (or suitably approximated) as
  $r = \alpha_{\text{min}} / \beta_{\text{min}}$.

• Special case "sufficiently late (after states $a_0$ and $b_0$)", for any value $\phi$
  (also including the case of intersecting lines):

  For any suitably large $j_{\text{late}}$ and $k_{\text{late}}$, "ratio number $r$" is obtained (or correspondingly approximated) as
  $r = \sqrt{ ( \alpha_{Ak_{\text{late}}} / \alpha_{Bk_{\text{late}}} ) ( \beta_{Aj_{\text{late}}} / \beta_{Bj_{\text{late}}} ) }.$

(Of course, in the derivations of these results suitably defined auxiliary quantities were used, such as "$u$", "$v$", and, not least as a suitable abbreviation, "$w$"$ := \sqrt{ 1 - \frac{(1 - u^2) (1 - v^2)}{(1 - u~ v~ \text{Cos}(\phi))^2}}$.
Obviously the stated initial results don't explicitly involve or require those quantities.

However, the "auxiliary quantity $w$" is in fact readily evaluated (or approximated) as
$w = \frac{ 1 - ( \alpha_{Bk_{\text{late}}} / \alpha_{Ak_{\text{late}}} ) ( \beta_{Aj_{\text{late}}} / \beta_{Bj_{\text{late}}} )}{ 1 + ( \alpha_{Bk_{\text{late}}} / \alpha_{Ak_{\text{late}}} ) ( \beta_{Aj_{\text{late}}} / \beta_{Bj_{\text{late}}} ) }.$

It might be interesting, whether the values of "$u$", "$v$", and perhaps even the value of $\phi$, can be evaluated from given values $\alpha_{Ak}$, $\alpha_{Bk}$, $\beta_{Bj}$ und $\beta_{Aj}$, too.)

Answer 2

For addressing the above questions it is convenient to introduce coordinates$\,^{(1)}$ and suitable parameters$\,^{(2)}$.

The ping duration $T_{Ak} \approx \alpha_{Ak} \frac{\tau_A}{\sqrt{1 - u^2}}$, for any particular state $a_k$ of the telegraph (and the corresponding coordinate value $t_k$), is obtained from the following system of two equations

$H_{XY}^2 + (u \, t_k)^2 + (v (t_{Rk} - T_{AB}))^2 - 2 (u \, t_k) (v (t_{Rk} - T_{AB})) \text{Cos}(\phi) = (t_{Rk} - t_k)^2$,

$H_{XY}^2 + (u (t_k + T_{Ak}))^2 + (v (t_{Rk} - T_{AB}))^2 - 2 (u (t_k + T_{Ak})) (v (t_{Rk} - T_{AB})) \text{Cos}(\phi) = (t_{Rk} - t_k + T_{Ak})^2$

by eliminating the "reflection coordinate $t_{Rk}$":

$T_{Ak}( t_k ) :=$
$2 t_k \bigl( \frac{(1 - u \, v \, \text{Cos}(\phi))^2}{(1 - u^2) (1 - v^2)} - 1 \bigr) +$
$2 T_{AB} \bigl( \frac{1}{(1 - u^2)} - \frac{1 - u \, v \, \text{Cos}(\phi)}{(1 - u^2) (1 - v^2)} \bigr) +$ $2 \frac{1 - u \, v \, \text{Cos}(\phi)}{(1 - u^2) (1 - v^2)} {\sqrt{ (t_k (1 - u \, v \, \text{Cos}(\phi)) - T_{AB})^2 + (1 - v^2) (H_{XY}^2 + (u \, t_k)^2 - (t_k - T_{AB})^2) }}$.

The value $t_{A{\text{min}}}$ for which the ping duration $T_{Ak}$ attains its minimal value, $T_{A{\text{min}}}$, is consequently obtained (or at least well approximated, provided $\alpha_{{\text{min}}} \gg 1$) as solution of

$d/dt_k( T_{Ak} ) = 0$.

This results in

$t_{A{\text{min}}} :=$
$H_{XY} {\sqrt{ \frac{ u^2 - 2 \, u \,v \, \text{Cos}(\phi) + v^2 + (u \, v)^2 (1 - (\text{Cos}(\phi))^2) (1 - (T_{AB} / H_{XY})^2) }{(1 - u^2) ((1 - u \, v \, \text{Cos}(\phi))^2 - (1 - u^2) (1 - v^2))} }} -$
$T_{AB} \frac{v \, (u \, \text{Cos}(\phi) - v)}{(1 - u \, v \, \text{Cos}(\phi))^2 - (1 - u^2) (1 - v^2)}$

and

$T_{A{\text{min}}} :=$
$\frac{2}{\sqrt{1 - u^2}} {\sqrt{ H_{XY}^2 + T_{AB}^2 \frac{(u \, v)^2 (1 - (\text{Cos}(\phi))^2)}{(1 - u \, v \, \text{Cos}(\phi))^2 - (1 - u^2) (1 - v^2)} }}$.

Similarly, starting from the ping duration $T_{Bj} \approx \beta_{Bj} \frac{\tau_B}{\sqrt{1 - v^2}}$, follows

$T_{B{\text{min}}} :=$
$\frac{2}{\sqrt{1 - v^2}} {\sqrt{ H_{XY}^2 + T_{AB}^2 \frac{(u \, v)^2 (1 - (\text{Cos}(\phi))^2)}{(1 - u \, v \, \text{Cos}(\phi))^2 - (1 - u^2) (1 - v^2)} }}$,

and consequently

$r := \tau_B / \tau_A = \frac{T_{B{\text{min}}} \sqrt{1 - v^2}}{\beta_{\text{min}}} / \frac{T_{A{\text{min}}} \sqrt{1 - u^2}}{\alpha_{\text{min}}} = \alpha_{\text{min}} / \beta_{\text{min}}$.

Still to do:

(1) Definition of how to assign the used coordinates, based on the geometric relations (in particular: flatness) stated in the narrative above.

(2) Definition of the parameters $H_{XY}$ and $T_{AB}$ in terms of the parameters (such as the integer $p_{BA}$) given as setup.

(3) Investigate further; for instance whether the number $\phi$ may be obtained, too, from given numbers $p, \alpha_{Ak}$, $\alpha_{Bk}$, $\beta_{Bj}$ and $\beta_{Aj}$?.