       Re: Re: rotating rings illusion

• To: mathgroup at smc.vnet.net
• Subject: [mg80820] Re: [mg80801] Re: [mg80787] rotating rings illusion
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Mon, 3 Sep 2007 06:11:33 -0400 (EDT)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <200709010436.AAA26729@smc.twtelecom.net> <200709020653.CAA25909@smc.twtelecom.net>

```\[Rho] below should be r.  Evidently a change of notation along the way!

> Dear Imran,
>
> I have a solution that works in Mathematica 6; first of all, create a torus
> with Radii R and r:
>
> 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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>

--
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

```

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