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Selected highlights from the moiré demonstration kit

Each item in this moiré demonstration kit is provided in two different file formats: as a PostScript file, and as a PDF file. Both formats should give precisely the same printed results, but each of them has its own advantages: While PDF files are universal and easy to print from practically any computer, PostScript files are easily editable and hence more flexible, but they are less easy to print from Windows-based PCs.

Click on the highlighted "PS" or "PDF" indicator next to each of the items below to download the corresponding PostScript or PDF file. You can prepare your moiré demonstration kit by printing these PostScript or PDF files on transparencies (a printing resolution of at least 600 dpi is required). Using the PostScript format offers you the possibility to edit the downloaded files in order to modify the various adjustable parameters within them (angles, periods, etc.); this allows you to generate and print your own variants of the original transparencies for further experimentations. This editing possibility is not available, however, when using the PDF format. Further information on printing PostScript files is given here; further information on printing PDF files is given here.

For best visual results, lay the superposed transparencies on a light table, or hold them against a distant white background (for example, you may hold the superposed transparencies about 20 cm above a white sheet of paper). Alternatively, you may print one of the layers on a white paper and superpose on top of it the other layer, printed on a transparency. (Note that even the slightest undulations in the superposed transparencies may cause significant deformations in the resulting moiré shapes. These deformations can be largely reduced by printing the superposed layers on the same sheet of paper; but in this case the dynamic effect is obviously lost.)

A smiling face   smily   indicates moiré samples that are particularly cool.

This moiré demonstration kit accompanies the book The Theory of the Moiré Phenomenon by I. Amidror published by Springer in two volumes, as well as its original edition published by Kluwer Academic Publishers. All the references below are made to these books.



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Superposition of two line gratings:

Rotate grating2 on top of grating1 and observe the moiré bands. Their angle and their period are determined by formulas (2.9) on p. 20; see also Sec. 2.6 in the book.
Now, rotate grating1 on top of a copy of itself. Observe what happens to the moiré as the angle difference alpha between the two identical gratings tends to zero; note in particular that when alpha is small, any slight angle variation largely influences the moiré period. This is, indeed, a typical property of the moiré effect when it approaches its singular state (see Sec. 2.9 and Figs. 2.8(d)-(f) in the book; for a more comprehensive discussion see also Sec. 6.6).
You may also try to shift one of the gratings on top of the other while keeping the angle difference between them fixed. Note how even the slightest layer shift may cause a considerably larger shift in the moiré bands, especially when the angle difference alpha is small. This is explained in the book in Sec. 7.6 and illustrated in Fig. 7.3.
Thanks to its extreme sensitivity to the slightest variations in the original gratings, this simple moiré has found many different applications, including in high precision measurements of small angles, periods and displacements. These and other applications are described in Problems 2-19 to 2-30 and 7-1 to 7-17 in the book.


Superposition of two line grids:

Rotate grid2 on top of grid1 and observe the moiré patterns. Their angle and their period are determined by formulas (2.9) on p. 20; see also Sec. 2.11 and Fig. 2.10 in the book.
Now, rotate grid1 on top of a copy of itself. At what angles do you observe a moiré effect? Can you see the weak, second-order moiré around the angle difference of 36.87 degrees? See Fig. 2.10(c) in the book.
You may also try to slide one of the grids on top of the other while keeping the angle difference between them fixed. How do the moiré patterns react? This is explained in the book in Sec. 7.6 and in Fig. 7.6.


smily  Surprising moiré profile forms in dot-screen superpositions:

Superpose the pinhole screen scrnhole on top of scrncirc, scrntri or scrnone. What is the form of the resulting moiré intensity profile in each case? What happens when you slightly rotate the pinhole screen scrnhole clockwise or counterclockwise on top of scrnone? And finally, what happens if you use instead of the pinhole screen its "inverse video" scrndot? These phenomena are explained in the book in Sec. 4.4; see also Figs. 4.1 and 4.4. This is, in fact, a nice demonstration of the magnification property of the moiré effect.
Remark: A printer resolution of at least 600 dpi is required for these PostScript files.


smily  Superposition of a line grid and a circular grating - a flower-shaped moiré:

Superpose grid1 on top of each of the three circular gratings, circ1 circ2 or circ3, and observe the different moiré patterns. Now, rotate grid1 on top of circ3 and observe the moiré patterns; it is also interesting to watch the behaviour of the moiré patterns as you slide grid1 horizontally, vertically or diagonally on top of circ3. These flower-shaped moiré patterns are composed, in fact, of two perpendicular moiré patterns: The moiré pattern that is generated between the circular grating and one line grating, and the perpendicular moiré pattern that is generated between the same circular grating and a second line grating which is perpendicular to the first one.
Can you generate a similar flower-shaped moiré with more petals?


smily  Superposition of two different circular gratings - a lovely heart-shaped moiré:

Superpose circ2 on top of circ1 and observe the moiré patterns when both gratings share a common center. Now, slowly shift circ2 upwards on top of circ1, and observe how the moiré shapes evolve. The lovely heart-like moiré pattern that you have obtained consists of curves known as Cartesian Ovals (see Problem 11-8 and Fig. 11.4(c) in the book).


smily  Superposition of a line grid and a circular zone grating:

Superpose grid1 on top of the zone grating zone1 and observe the moiré patterns as you rotate and shift the line grid. Note that in addition to the simple higher order moirés generated by each of the orthogonal line gratings that make up grid1 (see the previous example), there also appear here mixed higher order moirés that are only generated when both gratings are present together.
Remark: A printer resolution of at least 600 dpi is required for printing the zone grating.


smily  Superposition of two curved dot-screens:

Curved dot-screen crvdscr1 is a non-linear geomrtric transformation of an original periodic dot-screen having 1-shaped dots, whereas crvdscr2 is a similar geomrtric transformation of an original periodic dot-screen consisting of tiny pinholes. Both screens have been geometrically transformed using the same coordinate transformation:
x' = 2xy,
y' = y*y - x*x.
Superpose crvdscr2 on top of crvdscr1 and observe the moiré obtained:
(1) When you shift crvdscr2 on top of crvdscr2 in various directions, without rotation;
(2) When you rotate crvdscr2 on top of crvdscr1;
(3) When you apply both rotations and shifts.
This case is explained in the book in Example 10.23 (p. 342) and Fig. 10.36.
Remark: A printer resolution of at least 600 dpi is required for printing these PostScript files.


smily  A surprise:

Print two copies of the argsinh-shaped grating argsinh1. Turn over one of the two transparencies and superpose it face down on top of the other; this gives you a superposition of an argsinh-shaped grating with a mirror-image copy of itself. Now, slowly rotate the inversed transparency clockwise, and observe the moiré pattern that starts to build up in the center as the angle difference approaches 90 degrees. Continue the clockwise rotation further on, and watch how the moiré shape in the center swells up, and finally gives birth to two (or even four) twins!



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Last modified: 2010/05/27