SIGNAL PATH 

Here I would like to show you the shapes, which communication signals that go towards a target form together, in (c+v)(c-v) mathematics. I named these shapes Signal Path. It is not a good naming, but I couldn’t find a corresponding name and couldn’t think of another name.

HOW DO SIGNALS TRAVEL IN (c+v)(c-v) MATHEMATICS?
It is quite explanatory and it provides convenience to show signals as sine waves in the animations. Therefore, I showed the communication signals as sine waves here as well. As can be seen in the animation, a signal tower sends signals to a plane.

Let’s first keep in mind that signals travel inside the plane’s field. Each point of the sine wave that sets out moves in a straight direction towards its arrival target completely independently of the other parts of the sine wave. The entirety of these points that are broadcast one after another and form the sine wave forms the Signal Path and gives it a peculiar shape. The fact that the Signal Path is comprised of parts which are in motion and independent of each other and that signals travel inside fields make the Signal Path sensitive to the movement of its arrival target and cause the Signal Path to have a dynamic structure that changes from moment to moment.

The red points on the signal show the points corresponding to 0 and 180 degrees while the sine wave is being generated. In the animation, only the lines that belong to these points are shown. However, although not shown in the animation, each point on the signal has its own line and each point follows its own line and reaches its target.

The red line on the signal tower shows that the signal tower broadcasts by following the plane. Its task is to ensure the sine wave is broadcast without deformation by changing its slope according to the position of the plane.

Animation controls:
In the animation, you can drag the Signal Tower, the Plane, and the Pink Point wherever you like and analyze how the signals move in different situations.
The Pink Point determines the route than the plane will follow. 
Numeric Stepper sets the speed of the plane. You can set the speed of the plane between 0 and 1 however you like. The value 1 is the speed of light.
The zoom function: You can see the details of the sine wave that is broadcast from the Signal Tower by using the slider bar.

As soon as you click on the Play button, the Signal Tower will start sending its signals.

THE SIGNAL PATHS THAT FORM WHILE REFERENCE SYSTEMS ARE APPROACHING EACH OTHER
It is the animation of the photo in the book.

If you thought “Why do signals move this way, forming arches?”, here (above) I have given you the answer. 

THE SIGNAL PATHS THAT FROM WHILE REFERENCE SYSTEMS ARE MOVING AWAY FROM EACH OTHER 
It is the animation of the photo in the book.

I would like to attract your attention to the fact that both sine waves are symmetrical to each other. This symmetry formed between Signal Paths is independent of the direction and the speeds of the planes. Even if one of the planes was motionless, this symmetry would form.

The Galilean Relativity Principle is preserved here as well. Whatever an observer on a plane measures and finds on the signal that comes to itself, the observer in the other plane will obtain the same result as long as it does the measurement in the same way. We can easily say this thanks to the symmetry.

SIGNAL PATH 360 DEGREES
It is the animation of the photo in the book.

I wanted to see the signals that come from different directions altogether. As a result, this animation came out. I included this situation in the book because it is an interesting case. 

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