N dog chase

Well, considering that there won't be a few people looking at this ghost thing I sent, I 'd like to try it out alone in the middle of the night.

Said to me during the summer vacationN dog chase

This is probably the description.

Sitting on a circle with N dogs, they are evenly distributed to the circumference. Sort the numbers (1 to n) in a counter-clockwise manner ). At a certain time point, they simultaneously move (1 to n) toward the dog at any time point, and all the dogs at the same rate at any time point. Ask the dog's action track.

Go back and think for yourself. Since all the dogs behave the same way, at any time, they must be on a circle with the same center, and the radius of the circle gradually decreases with time. In this way, the brain shows a circle that rotates and gradually decreases. Initially, the curve was "imagined". It has a self-similar property, and the enlarged part is the same as the overall (fragment ).

"They move to the dog at the same time." That is to say, the speed of the dog always points to the next dog at any time, in other words, the tangent slope of each point on the trajectory curve is always the same as that of the two dogs at that time. This geometric feature is much simpler to analyze using polar coordinates (n = 3 is used as an example)

Therefore, the split speed (Pointing to the center) and (Tangent to the circle) maintain the following relationship

Therefore, the split speed (Pointing to the center) and (Tangent to the circle) maintain the following relationship

(The addition of the negative number is because θ increases P but decreases)

Now, when the differential equations come out, the trajectory is very good. Finally, it is a logarithm screw.

The shape is like this.

Insert a segment here. The logarithm screw line is also called the growth screw line and the natural screw line. Because the shell of the parrot screw is like an angular screw,

 

Hawks and insects also approach their prey or light sources in the form of angular nuts, and there are

 

Low Pressure chart

 

Vortex galaxy

Beautiful mathematics, isn't it?

It is indeed not difficult as mentioned above, but it is necessary to select a quasi-coordinate system to grasp the geometric characteristics of the dog's motion conditions.

Thank you for drawing beetle.