The moiré demonstration kit:
A guided tour
through the fascinating world of moiré effects
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.
Contents of the moiré demonstration kit:
- A. Superpositions of periodic layers
B. Superpositions of repetitive, curvilinear layers
C. Polychromatic moirés
D. Superpositions of random layers
Selected highlights from the moiré demonstration kit
- if you are in a hurry
Back to the moiré book homepage
- 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 three line gratings:
Superpose three copies of grating1 on top of each other, and
observe the different moirés that occur as you mutually rotate
them. Observe what happens when you superpose the three identical
gratings with equal angle differences of exactly 120 degrees between
each other. This is another example of a singular moiré. See Sec. 2.9
and Figs. 2.8(g)-(i) in the book.
- Superposition of a line grating and a line grid:
- Download grid1 (period: 0.03 inch) PS
PDF
- Download grating1 (period: 0.03 inch) PS
PDF
- Download grating2 (period: 0.031 inch) PS
PDF
- Download grating3 (period: 0.029 inch) PS
PDF
Rotate grating1 on top of grid1. At what angles do you see a moiré
effect? Observe what happens when you use grating2 or grating3 instead of grating1.
- 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.
- Superposition of two dot-screens:
- Download screen1 (period: 0.03 inch) PS
PDF
- Download screen2 (period: 0.031 inch) PS
PDF
Rotate screen2 on top of screen1 and observe the moiré patterns.
Their angle and their period are determined by formulas (2.9) on p. 20; see
also Sec. 2.12 and Figs. 2.10 and 3.3 in the book.
Now, rotate screen1 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.
Observe what happens as the angle difference between the two identical screens
tends to zero, and what happens when the angle difference tends to 36.87 degrees.
Both cases are, indeed, singular moirés; for a more formal and comprehensive
discussion on this subject see also Sec. 6.7 in the book.
- _{}
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 straight grating and a parabolic grating:
- Download grating1 (period: 0.03 inch) PS
PDF
- Download grating2 (period: 0.031 inch) PS
PDF
- Download grating3 (period: 0.029 inch) PS
PDF
- Download parabol1 (vertical period: 0.03 inch) PS
PDF
Rotate grating1 on top of parabol1 and observe the moiré patterns. Now, rotate
grating2 on top of parabol1 and observe the moiré patterns; note the different
behaviour of the two cases at the angle difference of zero. What do you think will
happen if the period of grating2 is reduced to 0.029 inch? You may give it a try
using grating3. See Sec. 10.7.3 and Figs. 10.9, 10.12 in the book.
- Superposition of two parabolic gratings:
- Download parabol1 (wider opening) PS
PDF
- Download parabol2 (medium opening) PS
PDF
- Download parabol3 (narrower opening) PS
PDF
Rotate parabol3 on top of a copy of itself and observe the moiré patterns.
What happens when the angle difference is 90 degrees? Now, superpose parabol3
on top of its own copy and slide it horizontally, without any rotations. Can
you see the perfectly periodic moiré bands that are generated in this
case? Repeat the same experiments with the other parabolic gratings
and with different pair combinations thereof. See Sec. 10.7.4 and Figs. 10.13,
10.14 and 10.33 in the book.
- Superposition of a line grating and a circular grating:
- Download grating1 (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 grating1 on top of each of the three circular gratings, circ1 circ2 or
circ3, and observe the difference between the moiré patterns in the three
cases (see Sec. 10.7.5 and Figs. 10.17 - 10.19 in the book). It is also interesting
to watch the behaviour of the moiré patterns as you rotate and slide grating1
on top of the circular grating.
- _{}
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 (see the previous example), 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 identical circular gratings:
Superpose circ1 on top of a copy of itself, center on center. Now, slowly shift
one of the circular gratings on top of the other, and observe how the moiré
shapes evolve. See Sec. 10.7.6 and Figs. 10.23, 10.24 and 10.26 in the book.
- _{}
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 an elliptic grating:
- Download grid2 (period: 0.031 inch) PS
PDF
- Download ellipse (small radial period: 0.03 inch) PS
PDF
Superpose grid2 on top of the elliptic grating ellipse and observe the moiré
patterns. Now, rotate grid2 on top of ellipse and observe how the moiré
patterns evolve. How do you explain the varying moiré shapes that you observe?
- Superposition of two elliptic gratings:
Superpose the elliptic grating ellipse on top of a copy of itself, center on center,
and observe how the moiré patterns vary as you slowly rotate one of the layers
on top of the other. Now, slowly shift one of the elliptic gratings on top of the
other, and observe how the moiré shapes evolve. Try also to superpose
the elliptic grating on top of a circular grating, say, circ1 (PS
PDF),
and observe the moiré patterns as you shift and rotate the elliptic grating.
You may also generate a new elliptic grating with a different eccentricity by modifying
the adjustable parameters in the PostScript file, and observe the intricate moiré
shapes that are generated between two different elliptic gratings.
- Superposition of a line grating and a circular zone grating:
- Download grating1 PS
PDF
- Download zone1 (~1.5 megabytes) PS
PDF
A zone grating (or zone plate) is a concentric circular grating in which the radius of
the n-th circle from the center is proportional to the square root of n
(see p. 438 in the book). Superpose grating1 on top of the zone grating zone1 and
observe the moiré patterns as you rotate and shift the line grating. Note in
particular the existence of higher order moirés, in contrast to the absence
of higher order moirés in the superposition of a line grating and a circular
grating. See Sec. 10.7.7 and Fig. 10.28 in the book.
Remark: A printer resolution of at least 600 dpi is required
for printing the zone grating.
- _{}
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
(cross harmonics) that are only generated due to the presence of both gratings together.
Remark: A printer resolution of at least 600 dpi is required
for printing the zone grating.
- Superposition of two identical circular zone gratings:
- Download zone1 (~1.5 megabytes) PS
PDF
Superpose the zone grating zone1 on top of a copy of itself, center on center. Now,
slowly shift one of the circular gratings on top of the other, and observe the
moiré shapes. Note that when the shift is small the predominant moiré
consists of a periodic straight line grating, but when the shift is larger the
predominant moirés have the form of zone gratings. This is explained in the
book in Sec. 10.7.8 and illustrated in Figs. 10.31 and 10.32.
Remark: A printer resolution of at least 600 dpi is required
for printing the zone grating.
- Superposition of two different circular zone gratings:
- Download zone1 (~1.5 megabytes) PS
PDF
- Download zone2 (~1.5 megabytes) PS
PDF
Superpose zone2 on top of zone1 and observe the moiré pattern when both zone
gratings share a common center. Now, slowly shift zone2 and observe how the
moiré pattern moves ahead of the shifted layer, in the same direction. Note
that when the shift is large and the main moiré runs beyond the figure borders,
new higher order zone grating moirés become visible in the superposition.
See Sec. 10.7.8 and Fig. 10.30 in the book.
Remark: A printer resolution of at least 600 dpi is required
for printing the zone gratings.
- Superposition of a straight grating and a bell-shaped grating:
- Download grating1 PS
PDF
- Download bell (~1.5 megabytes) PS
PDF
Superpose grating1 on top of the bell-shaped grating; at this point no moiré
patterns are visible. Now, rotate grating1 on top of the bell-shaped grating, and
observe the moiré patterns which start to appear. When the angle difference
reaches exactly 90 degrees, the moiré pattern draws the contour plot of the
distortion function g(x,y) that generates the given bell-shaped
grating from an initial straight line grating. This is explained in the book in
Problem 11-5 and Fig. 11.2.
Remark: A printer resolution of at least 600 dpi is required
for printing the bell-shaped grating.
- Superposition of two bell-shaped gratings:
- Download bell (~1.5 megabytes) PS
PDF
Superpose the bell-shaped grating on top of a copy of itself, and observe the
moiré patterns:
(1) When you rotate one grating on top of the other;
(2) When you shift one grating on top of the other without rotation;
(3) When you apply both rotations and shifts.
This case is explained in the book in Example 10.21 (p. 335) and Fig. 10.34.
Remark: A printer resolution of at least 600 dpi is required
for printing the bell-shaped grating.
- Superposition of two argsinh-shaped gratings:
The argsinh-shaped grating argsinh1 is generated by the bending function
g(x,y) = argsinh(x), while the grating
argsinh2 is generated by the bending function g(x,y) =
argsinh(x) + x/8. Superpose these two gratings on top of each other,
and observe the moiré patterns:
(1) When you rotate one grating on top of the other;
(2) When you shift one grating on top of the other without rotation;
(3) When you apply both rotations and shifts.
This case is explained in the book in Example 10.22 (p. 335) and Fig. 10.35.
- Superposition of a straight grating and a cosinusoidal grating:
- Download grating1 (period: 0.03 inch) PS
PDF
- Download grating2 (period: 0.031 inch) PS
PDF
- Download grating3 (period: 0.029 inch) PS
PDF
- Download cos1 (small amplitude; vertical period: 0.03 inch)
PS
PDF
- Download cos2 (large amplitude; vertical period: 0.03 inch)
PS
PDF
This case provides a nice demonstration of the magnification property of the
moiré effect. Superpose grating2 on top of cos1 and observe the moiré
patterns. The moiré curves that you see are, in fact, a largely magnified version
of the original cosinusoidal grating (see Problem 11-6 and Figure 11.3 in the book).
What happens now if you use instead of grating2, whose period is 0.031 inch, the line grating
grating1, whose period is 0.03 inch? What do you think will happen if you use a grating
whose period is 0.029 inch? You may give it a try using grating3.
You may also repeat the same experiments with cos2 instead of cos1.
- Superposition of two cosinusoidal gratings:
- Download cos1 (small amplitude) PS
PDF
- Download cos2 (large amplitude) PS
PDF
Rotate cos1 on top of a copy of itself and observe the moiré patterns. Repeat
the same experiment using two copies of cos2, and then with one copy of cos1 and
one copy of cos2.
- _{}
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!
The polychromatic samples of the moiré demonstration kit have been
assembled together into this separate section. These samples should be printed
on transparencies using a colour PostScript printer; the minimum printing
resolution required for most samples is 600 dpi.
- Superposition of colour line gratings:
- Download grating1 (black grating) PS
PDF
- Download gratr (red grating) PS
PDF
- Download gratg (green grating) PS
PDF
- Download gratb (blue grating) PS
PDF
- Download gratc (cyan grating) PS
PDF
- Download gratm (magenta grating) PS
PDF
- Download graty (yellow grating) PS
PDF
- Download gratrg (grating whose period consists of
R,G stripes) PS
PDF
- Download gratrgb (grating whose period consists of
R,G,B stripes) PS
PDF
- Download gratcmy (grating whose period consists of
C,M,Y stripes) PS
PDF
All of the above line gratings have identical periods (0.03 inch). The moiré
bands they generate have the same geometric properties (period, angle, etc.) as in the
black&white case, and only their colours vary from case to case.
Superpose the black grating grating1 on top of each of the colour gratings and observe
the moiré bands and their colours. You may also try to superpose different pairs
of colour gratings and observe the colours of the resulting moiré bands. This
simple polychromatic moiré is explained in the book in Sec. 9.5 and illustrated in
Colour Plate 2.
- _{}
Superposition of a line grid and a polychromatic circular grating:
- Download grid1 (period: 0.03 inch) PS
PDF
- Download grid2 (period: 0.031 inch) PS
PDF
- Download circcmy (radial period: 0.03 inch) PS
PDF
The polychromatic circular grating circcmy consists of
alternating cyan, magenta and yellow rings; in other words, its radial period is
composed of 3 equal parts: C, M and Y. Superpose grid1 or grid2 on top of this
polychromatic circular grating, and observe the different moiré patterns
in each case. It is also interesting to watch the behaviour of the moiré
patterns when you rotate grid1 or when you slide it horizontally, vertically or
diagonally on top of circcmy.
Can you generate a similar polychromatic flower-shaped moiré with
more petals?
- _{}
Superposition of two different circular gratings - a colourful heart-shaped
moiré:
- Download circcmy (radial period: 0.03 inch) PS
PDF
- Download circ2 (radial period: 0.032 inch) PS
PDF
Superpose circ2 on top of the polychromatic circular grating circcmy (see
previous example), and observe the moiré patterns when both gratings share
a common center. Now, slowly shift circ2 upwards
on top of circcmy, and observe how the moiré shapes evolve. The
colourful 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).
Moiré patterns that are generated in the superposition of correlated
random layers are known as Glass patterns. Note that non-correlated layers do not
generate Glass patterns in their superposition. Therefore, in order to guarantee
the generation of Glass patterns, the random layers in each of the examples
below are generated using the same seed in their random number generator, namely,
using the same sequence of random numbers. This is, indeed, a necessary requirement
for the generation of a Glass pattern in the superposition.
- Superposition of two random dot-screens:
- Download random dot-screen randscr1 PS
PDF
- Download random dot-screen randscr2 (scaled by 1.0333)
PS
PDF
- Download random dot-screen randscr3 (horizontally
scaled by 0.9677, and vertically scaled by 1.0333)
PS
PDF
Superpose randscr1 on top of a copy of itself, and observe the circular Glass pattern
which appears when you slightly rotate one of the layers. What happens to the Glass
pattern when you slightly shift one of the two layers horizontally or vertically?
Repeat the same experiment with randscr1 and its slightly scaled copy, randscr2; note
the spiral shape of the Glass pattern in this case. Now, repeat the same experiment
with randscr1 and its slightly scaled copy randscr3; note that randscr3 was slightly
scaled up vertically and slightly scaled down horizontally. Observe the hyperbolic
shape of the Glass pattern when the rotation angle is close to 0. When the angle is
such that the diagonal of randscr1 coincides with the diagonal of randscr3 the
hyperbolic Glass pattern turns into a diagonal linear Glass pattern; and when the
rotation angle continues to grow, the Glass pattern becomes elliptic, and then circular,
until it finally disappears.
Repeat now the same experiments when one of the two transparencies is turned face
down on top of the other. In this case there is no correlation between the two superposed
layers, and therefore no Glass patterns are generated in the superposition. This is
also what happens when the two superposed random screens are generated with different
seeds in their random number generator, i.e. with different sequences of random numbers.
To compare these results with those obtained between periodic layers, repeat the same
experiments using the corresponding periodic dot-screens instead of their random
counterparts:
- Download periodic dot-screen screen1 PS
PDF
- Download periodic dot-screen screen2 (scaled by 1.0333)
PS
PDF
- Download periodic dot-screen screen3 (horizontally
scaled by 0.9677, and vertically scaled by 1.0333)
PS
PDF
- _{}
Glass patterns with surprising profile forms in random dot-screen superpositions:
- Download randone (a random screen with "1"-shaped dots)
PS
PDF
- Download randtwo (a random screen with "2"-shaped dots)
PS
PDF
- Download randhole (a random pinhole screen)
PS
PDF
Superpose the random pinhole screen randhole on top of randone or randtwo.
What is the form of the resulting moiré intensity profile in each case?
What happens when you slightly rotate the pinhole screen randhole clockwise or
counterclockwise on top of randnone? Note that the random screens used here were obtained
by the addition of random noise to the dot locations of an underlying periodic screen.
But the same phenomenon occurs also in the superposition of screens with purely random
dot locations (provided that they use the same seed in their random number generator).
These phenomena are explained in a forthcoming paper by I. Amidror.
To compare these results with those obtained between periodic layers, repeat the same
experiments using the corresponding periodic dot-screens instead of their random
counterparts:
- Download perone (a periodic screen with "1"-shaped dots)
PS
PDF
- Download pertwo (a periodic screen with "2"-shaped dots)
PS
PDF
- Download perhole (a periodic pinhole screen)
PS
PDF
Remark: A printer resolution of at least 600 dpi is required
for these PostScript files.
- Superposition of a random dot-screen and a random parabolic dot-screen:
- Download random dot-screen randscr1 PS
PDF
- Download random dot-screen randscr2 (scaled by 1.0333)
PS
PDF
- Download random parabolic dot-screen randparh
PS
PDF
Superpose randscr1 on top of the random parabolic dot-screen randparh, center on center
and without rotation, and observe the linear Glass patterns as you slightly shift one
of the layers to the right or to the left. What happens when you apply a slight vertical
shift, too? And when you slightly rotate one of the layers on top of the other?
Repeat now the same experiments with the slightly scaled random dot-screen, randscr2.
Observe the two generated Glass patterns, one spiral-shaped and the other hyperbolic.
To compare these results with those obtained between periodic or repetitive layers,
repeat the same experiments using the corresponding periodic or repetitive dot-screens
instead of their random counterparts:
- Download periodic dot-screen screen1 (period: 0.03 inch)
PS
PDF
- Download periodic dot-screen screen2 (scaled by 1.0333)
PS
PDF
- Download parabolic dot-screen scrnpar (original
period before applying the parabolic transformation: 0.03 inch)
PS
PDF
- _{}
Superposition of two orthogonal random parabolic dot-screens:
- Download random horizontal parabolic dot-screen randparh
PS
PDF
- Download random vertical parabolic dot-screen randparv
PS
PDF
Superpose randparh and randparv on top of each other, and shift them with respect to
each other horizontally and vertically. Observe the four Glass patterns which are
simultaneously generated in the superposition: two circular (or elliptic) Glass patterns
and two hyperbolic ones. What happens when you also apply rotation?
Repeat the same experiment with two copies of randparh, rotating one of them by
90 degrees to replace randparv. In this case, no Glass patterns will appear in the
superposition, since the 90 degrees rotation completely destroys the correlation between
the two layers.
To compare these results with those obtained between repetitive layers, repeat the
same experiments using two copies of the corresponding repetitive parabolic dot-screen:
- Download parabolic dot-screen scrnpar (original
period before applying the parabolic transformation: 0.03 inch)
PS
PDF
Why can we use here two identical copies of the parabolic dot-screen and rotate one
of them by 90 degrees?
- Superposition of two random line gratings:
- Download random line grating randgrt1 PS
PDF
- Download random line grating randgrt2 (scaled by 1.0333)
PS
PDF
Superpose randgrt1 on top of a copy of itself, and observe the linear Glass pattern
which appears when you slightly rotate one of the layers. What happens to the Glass
pattern when you slightly shift one of the two layers horizontally or vertically?
Repeat the same experiment with randgrt1 and its slightly scaled copy, randgrt2.
Repeat now the same experiments when one of the two transparencies is rotated by 180
degrees on top of the other. In this case there is no correlation between the two
superposed layers, and therefore no Glass patterns are generated in the superposition.
This is also what happens when the two superposed random gratings are generated with
different seeds in their random number generator, i.e. with different sequences of
random numbers.
To compare these results with those obtained between periodic layers, repeat the same
experiments using the corresponding periodic line gratings instead of their random
counterparts:
- Download periodic line grating grating1 PS
PDF
- Download periodic line grating grating2 (scaled by 1.0333)
PS
PDF
- Superposition of two random parabolic line gratings:
- Download random parabolic line grating randpar1 PS
PDF
Superpose randpar1 on top of a copy of itself, and observe the linear Glass pattern
which appears when you slightly rotate one of the layers. What happens to the Glass
pattern when you slightly shift one of the two layers horizontally or vertically?
To compare these results with those obtained between repetitive layers, repeat the same
experiments using the corresponding repetitive parabolic line gratings instead of their
random counterparts:
- Download parabolic line grating parabol1 PS
PDF
Back to the moiré book homepage
Last modified: 2010/05/27