MHB Crazy Circle Illusion: Amaze Your Friends!

[kaliprasad]

Crazy Circle Illusion! | Showyou

[youtube]pNe6fsaCVtI[/youtube]

 

[mathbalarka]

Surely this must be tricks of the eye? Anyone up for calculating the real path traversed by the vertices of an octagon when in such a motion?

 

[I like Serena]

Surely this must be tricks of the eye? Anyone up for calculating the real path traversed by the vertices of an octagon when in such a motion?

You can draw a circle with a cosine horizontally and a sine vertically.
This is how a cosine and a sine are defined on the unit circle.
It makes sense that if you create a whole bunch of sines on straight lines with the proper phase differences, that you'd get a circle.

 

[MarkFL]

Mesmerizing! :D

I have embedded the video so people can just watch it here.

 

[Deveno]

I think the "trick" to this is that the "inner circle" (polygon) has exactly half the radius of the outer circle.

Imagine we trace the path of a point on a circle of radius $r$ as it travels inside a circle of radius $2r$. Since it doesn't really matter "when" we start tracking it (the path is periodic), assume that both circles are touching at the point $(0,2r)$ at $t = 0$, and that the outer circle is centered at the origin.

As the inner circle "rolls" counter-clockwise, the point on the inner circle we are tracking moves CLOCKWISE around a shifting center.

This center is at: $((2r-r)\cos t,(2r-r)\sin t) = (r\cos t,r\sin t)$. Since the outer circle's circumference (which is directly proportional to radius) is twice that of the inner circle, as the center has moved through an angle of $t$, the point we are tracking makes an angle of $2t$ with the point of tangency. Half of this angle is $t$, the other half is the angle our tracked point makes to a horizontal line passing through the center of the inner circle.

It follows our tracked point has coordinates:

$(r\cos(-t),r\sin(-t)) + (r\cos t,r\sin t) = (2r\cos t,0)$.

As $t$ varies, the image $\{(x(t),y(t)): t \in \Bbb R_0^+\}$ is the interval $[-2r,2r]\times \{0\}$, which is a "straight-line" (segment).

 

[Nono713]

This was the topic of a recent code golf thread, which spawned many versions and variants of the concept (in different programming languages, but this is not relevant to this thread):

popularity contest - Make a circle illusion animation - Programming Puzzles & Code Golf Stack Exchange

By varying the phases you can create some really interesting animations. I wonder if you could get coherent aperiodic animations by also varying the frequency of the different balls...