SIGNAL PATH

I would like to show you the shapes created collectively by communication signals moving toward the target in (c+v)(c-v) mathematics. I named these shapes Signal Path. It may not be the best name, but I couldn’t find a corresponding word or think of another name.

HOW DO SIGNALS TRAVEL IN (c+v)(c-v) MATHEMATICS?
Representing signals in the form of a sine wave in animations is convenient and explanatory. Therefore, I have also represented communication signals here as sine waves. As seen in the animation, a signal tower is sending signals to an airplane.

Let’s first remember that the signals travel within the domain of the airplane. Each point of the sine wave that sets out travels in a straight line toward its target independently of the other parts of the sine wave. The entire set of these sequentially emitted points forming the sine wave constitutes the Signal Path and gives it its unique shape. The fact that the Signal Path consists of independent moving parts and that the signals travel within the domain makes the Signal Path sensitive to the motion of the target and turns it into a dynamic structure that changes moment by moment.

The red dots on the signal represent the points corresponding to 0 and 180 degrees when generating the sine wave. In the animation, only the lines corresponding to these points are shown. However, even if it is not shown in the animation, every point on the signal has its own line, and every point reaches its target by following its own line.

The red line on the Signal Tower shows that the tower broadcasts by following the airplane. Its function is to adjust its tilt according to the position of the airplane and ensure that the sine wave is emitted without deformation.

Animation controls:

• You can drag the Signal Tower, the Airplane, and the Pink Point to any location in the animation and analyze how the signals travel for different situations.
• The Pink Point determines the route the airplane will follow.
• The Numeric Stepper sets the speed of the airplane. You can adjust the speed of the airplane as you wish between 0 and 1. A value of 1 is the speed of light.
• Zoom function; You can use the scrollbar to see the details of the sine wave emitted by the Signal Tower.

Once you press the Start button, the Signal Tower will begin sending its signals.

SIGNAL PATHS FOR REFERENCE FRAMES MOVING TOWARD EACH OTHER
This is the animation of the image from the book.
"Why do signals travel like this, curving?" If you wondered about this, here (above), I have provided you with the answer.

SIGNAL PATHS FOR REFERENCE FRAMES MOVING AWAY FROM EACH OTHER
This is the animation of the image from the book.

Note that both sine waves are symmetric to each other. This symmetry between the Signal Paths is independent of the directions and speeds of the airplanes. Even if one of the airplanes were stationary, this symmetry would still occur.

The Galilean Principle of Relativity is also preserved here. Whatever one observer measures on the signal reaching them, the other observer will obtain the same result if they perform the same measurement on the signal reaching them. This symmetry makes it easy to state this.

SIGNAL PATH 360 DEGREES
This is the animation of the image from the book.
I wanted to see the signals coming from different directions collectively. As a result, this animation emerged. Since it is an interesting visual, I included it in the book.