       Re: rotating rings illusion

• To: mathgroup at smc.vnet.net
• Subject: [mg80801] Re: [mg80787] rotating rings illusion
• Date: Sun, 2 Sep 2007 02:53:56 -0400 (EDT)
• References: <200709010436.AAA26729@smc.twtelecom.net>

```Dear Imran,

I have a solution that works in Mathematica 6; first of all, create a torus

R = 1;
r = 0.1;

torus = ParametricPlot3D[{(R + r Cos[2 \[Pi] u])
Cos[2 \[Pi] t], (R + r Cos[2 \[Pi] u]) Sin[2 \[Pi] t],
r Sin[2 \[Pi] u]}, {u, 0, 1}, {t, 0, 1}, Mesh -> None]

then rotate the torus by an angle Theta in the horizontal plane to create
ring1; rotate ring 1 180 degrees around the vertical axis and translate it
by the adequate amount in the vertical direction; create a graphic adding
them (rings):

\[Theta] = 10 Degree;

ring1 = Graphics3D[Rotate[torus[], \[Theta], {1, 0, 0}]];
ring2 = Graphics3D[
Translate[
Rotate[ring1[], 180 Degree, {0, 0, 1}], {0, 0,
2 (R Sin[\[Theta]] + \[Rho])}]];
rings = Show[{ring1, ring2}, Boxed -> False, SphericalRegion -> True]

Finally, using manipulate, rotate the graphic "rings"along the vertical
axis:

Manipulate[
Graphics3D[Rotate[rings[], 2 \[Pi] \[Theta], {0, 0, 1}],
Boxed -> False], {\[Theta], 0, 1, 0.02}]

With the above parametrization, if you change  the variables R, r and Theta,
the rings will always touch.

On 9/1/07, Imran Akbar <imran at sulug.stanford.edu> wrote:
>
> Hi,
>    i'm just starting to use mathematica...
>
> i'm trying to mathematically model the two rotating rings illusion as
> shown
> in this video:
>
> 1)  how do you create a ring in mathematica?  i found the circle command,
> but that's only 2d.
> 2)  how can I orient each ring so that they're perpendicular to each
> other?
> 3)  how do I vary the # of rotations per minute in an animation?
>
> much obliged,
> imran
>
>
>

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