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I saw a chair that looked like this at the dentists office yesterday, and the pattern caught my interest, as the light patterns emulated thin film interference fringes formed by an object that has a radius of curvature (the spacing between the successive fringes approaches zero (not really here...) as we increase the order.

Upon looking closer, what is really happening, is that, depending on the angle of incidence, various intensities of light are allowed through the mesh holes (of course, each surface of the chair has a radius of curvature to make certain that this effect is attained).

If I were to form an equation to calculate the distance between the successive fringes (which would also require an equation for the radii of the rings ), from the center of the chair (or central fringe, depending on the angle of incidence), here are the variables I think that I would need to take into consideration:

  • equation to describe the radii of curvature for each of the surfaces
  • consideration for the angle of incidence
  • radius of the mesh holes
  • some sort of consideration for light intensity

Is there a way to do this (even if only in theory)?

I am thinking of just meticulously computing data, and doing a least squares fit, but I would rather derive the physically correct equation.

Exhibit A

Exhibit B

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  • $\begingroup$ It is a beating pattern between two fabrics with similar mesh width. It can be calculated fairly easily using vectors and a bit of geometry. Every reasonably useful CGI/raytraycing platform should be able to render this kind of system. $\endgroup$
    – CuriousOne
    Commented Oct 21, 2014 at 9:23
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    $\begingroup$ en.wikipedia.org/wiki/Moir%C3%A9_pattern $\endgroup$
    – Johannes
    Commented Oct 21, 2014 at 12:44
  • $\begingroup$ Thank you so much, I wouldn't have been able to find that. How did you first encounter the "Moir Pattern?" $\endgroup$
    – Gödel
    Commented Oct 21, 2014 at 17:32
  • $\begingroup$ @Johannes Now I can read this: amazon.com/exec/obidos/ASIN/079235950X/ref=nosim/weisstein-20 . I had no idea this was such an interesting phenomenon. $\endgroup$
    – Gödel
    Commented Oct 21, 2014 at 17:41

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This is not thin-film interference, which involves waves of light interfering. You're looking for a Moiré pattern.

This sort of phenomenon is difficult to model analytically, since it involves step functions with varying offsets.

Perhaps surprisingly, your eyes play an important role in this. Your receptors are arranged in a rough-Poisson-disk distribution, which turns out to be optimal for signal reconstruction from point samples. However, you still get aliasing due to undersampling. This is a fundamental problem.

The signal-processing-upshot of this is that, even if you can find some nasty analytic value for a summation over some area (like, for example, exists for a checkerboard, which is comparatively well-behaved), you'd actually still need to deal with a convolution to describe how it actually looks.

Therefore, I suspect a closed form does not exist.

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