In physics, a moiré arrangement (play /mwɑrˈeɪ/; French: mwaʁe) is an arrest arrangement created, for example, if two grids are overlaid at an angle, or if they accept hardly altered cobweb sizes.
Tuesday, February 7, 2012
Etymology
The appellation originates from moire (or moiré in its French form), a blazon of textile, commonly of cottony but now aswell of affection or constructed fiber, with a bouncing or 'watered' appearance.
The history of the chat moiré is complicated. The ancient agreed agent is the Arabic mukhayyar (مُخَيَّر in Arabic, which agency chosen), a bolt fabricated from the absolute of the Angora goat, from khayyara (خيّر in Arabic), 'he chose' (hence 'a choice, or excellent, cloth'). It has aswell been appropriate that the Arabic chat was formed from the Latin marmoreus, acceptation 'like marble'. By 1570 the chat had begin its way into English as mohair. This was again adopted into French as mouaire, and by 1660 (in the writings of Samuel Pepys) it had been adopted aback into English as moire or moyre. Meanwhile the French mouaire had mutated into a verb, moirer, acceptation 'to aftermath a watered bolt by aberrant or pressing', which by 1823 had spawned the adjective moiré. Moire (pronounced "mwar") and moiré (pronounced "mwar-ay") are now acclimated somewhat interchangeably in English, admitting moire is added generally acclimated for the bolt and moiré for the pattern.
"Watered textile" refers to laying allotment of the bolt on top of addition part, and acute the two layers if wet. The affinity of the agreement of alone accoutrement (warp and woof), which is, however, not absolute spacing, creates appropriate patterns if the layers are apprenticed together; if dry, the patterns remain.
The history of the chat moiré is complicated. The ancient agreed agent is the Arabic mukhayyar (مُخَيَّر in Arabic, which agency chosen), a bolt fabricated from the absolute of the Angora goat, from khayyara (خيّر in Arabic), 'he chose' (hence 'a choice, or excellent, cloth'). It has aswell been appropriate that the Arabic chat was formed from the Latin marmoreus, acceptation 'like marble'. By 1570 the chat had begin its way into English as mohair. This was again adopted into French as mouaire, and by 1660 (in the writings of Samuel Pepys) it had been adopted aback into English as moire or moyre. Meanwhile the French mouaire had mutated into a verb, moirer, acceptation 'to aftermath a watered bolt by aberrant or pressing', which by 1823 had spawned the adjective moiré. Moire (pronounced "mwar") and moiré (pronounced "mwar-ay") are now acclimated somewhat interchangeably in English, admitting moire is added generally acclimated for the bolt and moiré for the pattern.
"Watered textile" refers to laying allotment of the bolt on top of addition part, and acute the two layers if wet. The affinity of the agreement of alone accoutrement (warp and woof), which is, however, not absolute spacing, creates appropriate patterns if the layers are apprenticed together; if dry, the patterns remain.
Pattern formation
Moiré patterns are generally an causeless antiquity of images produced by assorted agenda imaging and computer cartoon techniques, for archetype if scanning a halftone account or ray archetype a checky even (the closing getting a appropriate case of aliasing, due to undersampling a accomplished approved pattern).1 This can be affected in arrangement mapping through the use of mipmapping and anisotropic filtering
The cartoon on the high appropriate shows a moiré pattern. The curve could represent fibers in moiré silk, or curve fatigued on cardboard or on a computer screen. The nonlinear alternation of the optical patterns of curve creates a absolute and arresting arrangement of almost alongside aphotic and ablaze bands, the moiré pattern, superimposed on the lines.2
More circuitous bandage moiré patterns are created if the curve are arced or not absolutely parallel. Moiré patterns absolute circuitous shapes, or sequences of symbols anchored in one of the layers (in anatomy of periodically again aeroembolism shapes) are created with appearance moiré, contrarily alleged bandage moiré patterns. One of the a lot of important backdrop of appearance moiré is its adeptness to aggrandize tiny shapes forth either one or both axes, that is, stretching. A accepted 2D archetype of moiré deepening occurs if examination a chain-link fence through a additional chain-link fence of identical design. The accomplished anatomy of the architecture is arresting even at abundant distances.
The cartoon on the high appropriate shows a moiré pattern. The curve could represent fibers in moiré silk, or curve fatigued on cardboard or on a computer screen. The nonlinear alternation of the optical patterns of curve creates a absolute and arresting arrangement of almost alongside aphotic and ablaze bands, the moiré pattern, superimposed on the lines.2
More circuitous bandage moiré patterns are created if the curve are arced or not absolutely parallel. Moiré patterns absolute circuitous shapes, or sequences of symbols anchored in one of the layers (in anatomy of periodically again aeroembolism shapes) are created with appearance moiré, contrarily alleged bandage moiré patterns. One of the a lot of important backdrop of appearance moiré is its adeptness to aggrandize tiny shapes forth either one or both axes, that is, stretching. A accepted 2D archetype of moiré deepening occurs if examination a chain-link fence through a additional chain-link fence of identical design. The accomplished anatomy of the architecture is arresting even at abundant distances.
Calculations
Moiré of alongside patterns
edit Geometrical approach
the patterns are superimposed in the mid-width of the figure
Moiré acquired by the superimposition of two agnate patterns rotated by an bend α
Let us accede two patterns fabricated of alongside and centermost lines, e.g., vertical lines. The footfall of the aboriginal arrangement is p, the footfall of the additional is p+δp, with 0<δ<1.
If the curve of the patterns are superimposed at the larboard of the figure, the about-face amid the curve access if traveling to the right. After a accustomed amount of lines, the patterns are opposed: the curve of the additional arrangement are amid the curve of the aboriginal pattern. If we attending from a far distance, we accept the activity of anemic zones if the curve are superimposed, (there is white amid the lines), and of aphotic zones if the curve are "opposed".
The average of the aboriginal aphotic area is if the about-face is according to p/2. The nth band of the additional arrangement is confused by n·δp compared to the nth band of the aboriginal network. The average of the aboriginal aphotic area appropriately corresponds to
n·δp = p/2
that is
n = \frac{p}{2 \delta p}.
The ambit d amid the average of a anemic area and a aphotic area is
d = n \cdot p = \frac{p^2}{2 \delta p}
the ambit amid the average of two aphotic zones, which is aswell the ambit amid two anemic zones, is
2d = \frac{p^2}{\delta p}
From this formula, we can see that :
the bigger the step, the bigger the ambit amid the anemic and aphotic zones;
the bigger the alterity δp, the afterpiece the aphotic and anemic zones; a abundant agreement amid aphotic and anemic zones beggarly that the patterns accept actual abutting steps.
Of course, if δp = p/2, we accept a analogously blah figure, with no contrast.
The assumption of the moiré is agnate to the Vernier scale.
edit Geometrical approach
the patterns are superimposed in the mid-width of the figure
Moiré acquired by the superimposition of two agnate patterns rotated by an bend α
Let us accede two patterns fabricated of alongside and centermost lines, e.g., vertical lines. The footfall of the aboriginal arrangement is p, the footfall of the additional is p+δp, with 0<δ<1.
If the curve of the patterns are superimposed at the larboard of the figure, the about-face amid the curve access if traveling to the right. After a accustomed amount of lines, the patterns are opposed: the curve of the additional arrangement are amid the curve of the aboriginal pattern. If we attending from a far distance, we accept the activity of anemic zones if the curve are superimposed, (there is white amid the lines), and of aphotic zones if the curve are "opposed".
The average of the aboriginal aphotic area is if the about-face is according to p/2. The nth band of the additional arrangement is confused by n·δp compared to the nth band of the aboriginal network. The average of the aboriginal aphotic area appropriately corresponds to
n·δp = p/2
that is
n = \frac{p}{2 \delta p}.
The ambit d amid the average of a anemic area and a aphotic area is
d = n \cdot p = \frac{p^2}{2 \delta p}
the ambit amid the average of two aphotic zones, which is aswell the ambit amid two anemic zones, is
2d = \frac{p^2}{\delta p}
From this formula, we can see that :
the bigger the step, the bigger the ambit amid the anemic and aphotic zones;
the bigger the alterity δp, the afterpiece the aphotic and anemic zones; a abundant agreement amid aphotic and anemic zones beggarly that the patterns accept actual abutting steps.
Of course, if δp = p/2, we accept a analogously blah figure, with no contrast.
The assumption of the moiré is agnate to the Vernier scale.
Interferometric approach
Let us accede now two cellophane patterns with a adverse I that varies with a sinusoidal law:
I_1(x) = I_0 \cdot \sin (2\pi \cdot k_1 \cdot x)
I_2(x) = I_0 \cdot \sin (2\pi \cdot k_2 \cdot x)
(the accomplish are appropriately p1 = 1/k1 and p2 = 1/k2), if the patterns are superimposed, the consistent acuteness (interference) is
I(x) = I_0 \cdot ( \sin (2\pi \cdot k_1 \cdot x) + \sin (2\pi \cdot k_2 \cdot x) )
with the Euler's formula:
I(x) = I_0 \cdot 2 \cos \left ( 2\pi \frac{(k_1-k_2)}{2} \cdot x \right ) \cdot \sin \left ( 2\pi \frac{(k_1+k_2)}{2} \cdot x \right )
We can see that the consistent acuteness is fabricated of a atrium law with a top "spatial frequency" (wave number) which is the boilerplate of the spatial frequencies of the two patterns, and of a atrium law with a low spatial abundance which is the bisected of the aberration amid the spatial frequencies of the two patterns. This additional basic is an "envelope" for the aboriginal atrium law. The amicableness λ of this basic is the changed of the spatial frequency
\frac{1}{\lambda} = \frac{k_1 - k_2}{2} = \frac{1}{2} \cdot \left ( \frac{1}{p_1} - \frac{1}{p_2} \right )
if we accede that's p1 = p and p2 = p+δp:
\lambda = 2\frac{p_1 p_2}{p_2 - p_1} \approx 2\frac{p^2}{\delta p} .
The ambit amid the zeros of this envelope is λ/2, and the maxima of amplitude are aswell spaced by λ/2; we appropriately access the aforementioned after-effects as the geometrical approach, with a alterity of p/2 which is the ambiguity affiliated to the advertence that is considered: arrangement 1 or arrangement 2. This alterity is negligible if δp << p.
This abnormality is agnate to the stroboscopy.
edit Rotated patterns
Let us accede two patterns with the aforementioned footfall p, but the additional arrangement is angry by an bend α. Seen from far, we can aswell see aphotic and anemic lines: the anemic curve accord to the curve of nodes, that is, curve casual through the intersections of the two patterns.
If we accede a corpuscle of the "net", we can see that the corpuscle is a rhombus: it is a parallelogram with the four abandon according to d = p/sin α; (we accept a appropriate triangle which hypothenuse is d and the ancillary against to the α bend is p).
Unit corpuscle of the "net"; "ligne claire" agency "pale line"
Effect of alteration angle.
The anemic curve accord to the baby askew of the rhombus. As the diagonals are the bisectors of the neighbouring sides, we can see that the anemic band makes an bend according to α/2 with the erect of the curve of anniversary pattern.
Additionally, the agreement amid two anemic curve is D, the bisected of the big diagonal. The big askew 2D is the hypothenuse of a appropriate triangle and the abandon of the appropriate bend are d(1+cos α) and p. The Pythagorean assumption gives:
(2D)2 : d2(1+cos α)2 + p2
id est
(2D)^2 = \frac{p^2}{\sin^2 \alpha}(1+ \cos \alpha)^2 + p^2 = p^2 \cdot \left ( \frac{(1 + \cos \alpha)^2}{\sin^2 \alpha} + 1\right )
thus
(2D)^2 = 2 p^2 \cdot \frac{1+\cos \alpha}{\sin^2 \alpha}
Effect on arced lines.
When α is actual baby (α < π/6), the afterward approximations can be done:
sin α ≈ α
cos α ≈ 1
thus
D ≈ p / α.
We can see that the abate the α, the extreme the anemic lines; if the both patterns are alongside (α = 0), the agreement amid the anemic curve is "infinite" (there is no anemic line).
There are appropriately two means to actuate α: by the acclimatization of the anemic curve and by their spacing
α ≈ p / D
If we accept to admeasurement the angle, the final absurdity is proportional to the altitude error. If we accept to admeasurement the spacing, the final absurdity is proportional to the changed of the spacing. Thus, for the baby angles, it is best to admeasurement the spacing.
I_1(x) = I_0 \cdot \sin (2\pi \cdot k_1 \cdot x)
I_2(x) = I_0 \cdot \sin (2\pi \cdot k_2 \cdot x)
(the accomplish are appropriately p1 = 1/k1 and p2 = 1/k2), if the patterns are superimposed, the consistent acuteness (interference) is
I(x) = I_0 \cdot ( \sin (2\pi \cdot k_1 \cdot x) + \sin (2\pi \cdot k_2 \cdot x) )
with the Euler's formula:
I(x) = I_0 \cdot 2 \cos \left ( 2\pi \frac{(k_1-k_2)}{2} \cdot x \right ) \cdot \sin \left ( 2\pi \frac{(k_1+k_2)}{2} \cdot x \right )
We can see that the consistent acuteness is fabricated of a atrium law with a top "spatial frequency" (wave number) which is the boilerplate of the spatial frequencies of the two patterns, and of a atrium law with a low spatial abundance which is the bisected of the aberration amid the spatial frequencies of the two patterns. This additional basic is an "envelope" for the aboriginal atrium law. The amicableness λ of this basic is the changed of the spatial frequency
\frac{1}{\lambda} = \frac{k_1 - k_2}{2} = \frac{1}{2} \cdot \left ( \frac{1}{p_1} - \frac{1}{p_2} \right )
if we accede that's p1 = p and p2 = p+δp:
\lambda = 2\frac{p_1 p_2}{p_2 - p_1} \approx 2\frac{p^2}{\delta p} .
The ambit amid the zeros of this envelope is λ/2, and the maxima of amplitude are aswell spaced by λ/2; we appropriately access the aforementioned after-effects as the geometrical approach, with a alterity of p/2 which is the ambiguity affiliated to the advertence that is considered: arrangement 1 or arrangement 2. This alterity is negligible if δp << p.
This abnormality is agnate to the stroboscopy.
edit Rotated patterns
Let us accede two patterns with the aforementioned footfall p, but the additional arrangement is angry by an bend α. Seen from far, we can aswell see aphotic and anemic lines: the anemic curve accord to the curve of nodes, that is, curve casual through the intersections of the two patterns.
If we accede a corpuscle of the "net", we can see that the corpuscle is a rhombus: it is a parallelogram with the four abandon according to d = p/sin α; (we accept a appropriate triangle which hypothenuse is d and the ancillary against to the α bend is p).
Unit corpuscle of the "net"; "ligne claire" agency "pale line"
Effect of alteration angle.
The anemic curve accord to the baby askew of the rhombus. As the diagonals are the bisectors of the neighbouring sides, we can see that the anemic band makes an bend according to α/2 with the erect of the curve of anniversary pattern.
Additionally, the agreement amid two anemic curve is D, the bisected of the big diagonal. The big askew 2D is the hypothenuse of a appropriate triangle and the abandon of the appropriate bend are d(1+cos α) and p. The Pythagorean assumption gives:
(2D)2 : d2(1+cos α)2 + p2
id est
(2D)^2 = \frac{p^2}{\sin^2 \alpha}(1+ \cos \alpha)^2 + p^2 = p^2 \cdot \left ( \frac{(1 + \cos \alpha)^2}{\sin^2 \alpha} + 1\right )
thus
(2D)^2 = 2 p^2 \cdot \frac{1+\cos \alpha}{\sin^2 \alpha}
Effect on arced lines.
When α is actual baby (α < π/6), the afterward approximations can be done:
sin α ≈ α
cos α ≈ 1
thus
D ≈ p / α.
We can see that the abate the α, the extreme the anemic lines; if the both patterns are alongside (α = 0), the agreement amid the anemic curve is "infinite" (there is no anemic line).
There are appropriately two means to actuate α: by the acclimatization of the anemic curve and by their spacing
α ≈ p / D
If we accept to admeasurement the angle, the final absurdity is proportional to the altitude error. If we accept to admeasurement the spacing, the final absurdity is proportional to the changed of the spacing. Thus, for the baby angles, it is best to admeasurement the spacing.
Implications & applications
Printing full-color images
BeatTrackMoirePattern.ogg
The artefact of two "beat tracks" of hardly altered speeds overlaid, bearing an aural moiré pattern; if the beats of one clue accord to area in amplitude a atramentous dot or band exists and the beats of the added clue accord to the credibility in amplitude area a camera is sampling light, because the frequencies are not absolutely the aforementioned and accumbent altogether together, beats (or samples) will adjust carefully at some moments in time and far afar at added times. The afterpiece calm beats are, the darker it is at that spot; the further apart, the lighter. The aftereffect is alternate in the aforementioned way as a clear moiré pattern.
In clear arts and prepress, the accepted technology for press full-color images involves the superimposition of halftone screens. These are approved ellipsoidal dot patterns—often four of them, printed in cyan, yellow, magenta, and black. Some affectionate of moiré arrangement is inevitable, but in favorable affairs the arrangement is "tight;" that is, the spatial abundance of the moiré is so top that it is not noticeable. In the clear arts, the appellation moiré agency an badly arresting moiré pattern. Part of the prepress art consists of selecting awning angles and halftone frequencies which abbreviate moiré. The afterimage of moiré is not absolutely predictable. The aforementioned set of screens may aftermath acceptable after-effects with some images, but arresting moiré with others.
In accomplishment industries, these patterns are acclimated for belief diminutive ache in materials: by deforming a filigree with account to a advertence filigree and barometer the moiré pattern, the accent levels and patterns can be deduced. This address is adorable because the calibration of the moiré arrangement is abundant beyond than the angle that causes it, authoritative altitude easier.
BeatTrackMoirePattern.ogg
The artefact of two "beat tracks" of hardly altered speeds overlaid, bearing an aural moiré pattern; if the beats of one clue accord to area in amplitude a atramentous dot or band exists and the beats of the added clue accord to the credibility in amplitude area a camera is sampling light, because the frequencies are not absolutely the aforementioned and accumbent altogether together, beats (or samples) will adjust carefully at some moments in time and far afar at added times. The afterpiece calm beats are, the darker it is at that spot; the further apart, the lighter. The aftereffect is alternate in the aforementioned way as a clear moiré pattern.
In clear arts and prepress, the accepted technology for press full-color images involves the superimposition of halftone screens. These are approved ellipsoidal dot patterns—often four of them, printed in cyan, yellow, magenta, and black. Some affectionate of moiré arrangement is inevitable, but in favorable affairs the arrangement is "tight;" that is, the spatial abundance of the moiré is so top that it is not noticeable. In the clear arts, the appellation moiré agency an badly arresting moiré pattern. Part of the prepress art consists of selecting awning angles and halftone frequencies which abbreviate moiré. The afterimage of moiré is not absolutely predictable. The aforementioned set of screens may aftermath acceptable after-effects with some images, but arresting moiré with others.
In accomplishment industries, these patterns are acclimated for belief diminutive ache in materials: by deforming a filigree with account to a advertence filigree and barometer the moiré pattern, the accent levels and patterns can be deduced. This address is adorable because the calibration of the moiré arrangement is abundant beyond than the angle that causes it, authoritative altitude easier.
Television screens and photographs
Moiré patterns are frequently apparent on television screens anon if a being is cutting a shirt or anorak of a accurate braid or pattern, such as a houndstooth jacket, due to sampling problems in the television camera. As the being moves about, the Moiré arrangement is absolutely noticeable. Because of this, newscasters and added professionals who arise on TV consistently are instructed to abstain accouterment which could could cause the effect, added frequently accepted as 'strobing'.
Photographs of a TV awning taken with a agenda camera generally display moiré patterns.
edit Marine navigation
The Moiré aftereffect is acclimated in shoreside beacons to mark underwater hazards (usually pipelines or cables).3 The Moiré aftereffect creates arrows that 'point' arise an abstract band appearance the hazard. As you canyon over the hazard, the arrows on the alarm arise to become vertical bands afore 'changing' aback to arrows pointing in the about-face direction. An archetype can be begin in the UK on the East bank of Southampton water, adverse Fawley oil refinery (50°51′21.63″N 1°19′44.77″W). Similar Moiré aftereffect beacons can be acclimated to adviser mariners to the centre point of an advancing arch (here you would position the barge so that you could see vertical curve - cogent the 'centreline').
Photographs of a TV awning taken with a agenda camera generally display moiré patterns.
edit Marine navigation
The Moiré aftereffect is acclimated in shoreside beacons to mark underwater hazards (usually pipelines or cables).3 The Moiré aftereffect creates arrows that 'point' arise an abstract band appearance the hazard. As you canyon over the hazard, the arrows on the alarm arise to become vertical bands afore 'changing' aback to arrows pointing in the about-face direction. An archetype can be begin in the UK on the East bank of Southampton water, adverse Fawley oil refinery (50°51′21.63″N 1°19′44.77″W). Similar Moiré aftereffect beacons can be acclimated to adviser mariners to the centre point of an advancing arch (here you would position the barge so that you could see vertical curve - cogent the 'centreline').
Strain measurement
The moiré aftereffect can be acclimated in ache measurement: the abettor just has to draw a arrangement on the object, and blanket the advertence arrangement to the askew arrangement on the askew object.
A agnate aftereffect can be acquired by the superposition of an holographic angel of the article to the article itself: the hologram is the advertence step, and the aberration with the article are the deformations, which arise as anemic and aphotic lines.
See also: approach of elasticity, ache tensor and holographic interferometry.
A agnate aftereffect can be acquired by the superposition of an holographic angel of the article to the article itself: the hologram is the advertence step, and the aberration with the article are the deformations, which arise as anemic and aphotic lines.
See also: approach of elasticity, ache tensor and holographic interferometry.
Image processing
Some angel scanner disciplinarian programs accommodate an alternative filter, alleged a "descreen" filter, to abolish Moiré-pattern artifacts which would contrarily be produced if scanning printed halftone images to aftermath agenda images.4
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