Lighthouse Simulation Program - The Formulas

Given:

Imagine two lighthouses, A and B, shining beams of light as they turn. The lights always turn in opposite directions (A turning anti-clockwise, B turning clockwise). The relative speeds of rotation can vary.

Suppose the two lights start facing each other. Sketch the path traced by the point of intersection of the beams if light A turns twice as quickly as light B. Investigate the path if the speed of rotation differ.

The two lighthouses are points on a cartesian plane, where lighthouse A is at (0, 0) and lighthouse B is at (d, 0).
A' is vertically above B and is on the ray of lighthouse A.
B' is vertically above A and is on the ray of lighthouse B.
k is a constant which is the multiplier or ratio between the speeds of lighthouses A and B.

From this we gather that:

the formula for the line AA' is
the formula for the line BB' is

And by simple substitution we get:

the formula for the x value of the point of intersection is
the formula for the y value of the point of intersection is

When we plot this as a parametric graph on a Casio Graphics Calculator (CFX-9850GB+) using the values:

k = 2
d = 1
View Window with X min = -1 max = 2, Y min = -1 max = 1, Range for T is 0 to 360, pitch 3.6

we obtain the following graph:

You may notice that the graph looks half as expected and half not. The fatal flaw with this formula, is the program is interpreting the lines as lines, and not rays as should be used. This means, even if the rays are pointing away from each other, with definitely no intersection point, the calculator will extrapolate the rays behind the point of origin and find a point of intersection.

A method to overcome this limitation has not yet been found. The range for T can be varied, but still we have no known way to be able to correct the range every time K changes. For values of K > 2, there are multiple ranges where the points intersect, so it is not possible on the graphic calculator.

The Lighthouse program uses the same formula as above in it's algorithm, but also checks for whether the lines will intersect by using the following method:

Assume a is angle of lighthouse A and b is angle of lighthouse B, relative to positive size of horizontal x-axis.
If b > 180 then let x = a - b else let x = b - a
If the x is positive, then there is an intersection point of the rays otherwise there is none.
Note that if the result is zero, then the rays are parallel.

How this works is not quite known, but it appears to hold true for all cases.

Using this method, the program can draw the same graph as the calculator but without the imaginary intersection points.

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