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 _{} 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.
Proceed to the full moiré demonstration kit
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- Superposition of two line gratings:
- Download grating1 (period: 0.03 inch) PS
PDF
- Download grating2 (period: 0.031 inch) PS
PDF
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:
- Download grid1 (period: 0.03 inch) PS
PDF
- Download grid2 (period: 0.031 inch) PS
PDF
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.
- _{}
Surprising moiré profile forms in dot-screen superpositions:
- Download scrncirc (a screen with circular dots; ~1.5 megabytes)
PS
PDF
- Download scrntri (a screen with triangular dots; ~1.5 megabytes)
PS
PDF
- Download scrnone (a screen with "1"-shaped dots; ~1.5 megabytes)
PS
PDF
- Download scrnhole (a pinhole screen; ~1.5 megabytes)
PS
PDF
- Download scrndot (a screen with small black dots; ~1.5 megabytes)
PS
PDF
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.
- _{}
Superposition of a line grid and a circular grating - a flower-shaped
moiré:
- Download grid1 (period: 0.03 inch) PS
PDF
- Download circ1 (radial period: 0.03 inch) PS
PDF
- Download circ2 (radial period: 0.032 inch) PS
PDF
- Download circ3 (radial period: 0.028 inch) PS
PDF
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?
- _{}
Superposition of two different circular gratings - a lovely heart-shaped
moiré:
- Download circ1 (radial period: 0.03 inch) PS
PDF
- Download circ2 (radial period: 0.032 inch) PS
PDF
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).
- _{}
Superposition of a line grid and a circular zone grating:
- Download grid1 PS
PDF
- Download zone1 (~1.5 megabytes) PS
PDF
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.
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Superposition of two curved dot-screens:
- Download crvdscr1 (~1.5 megabytes) PS
PDF
- Download crvdscr2 (~1.5 megabytes) PS
PDF
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.
- _{}
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