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2858:. The effect occurs when there is mismatch between the lattice parameter or angle of the 2D layer and that of the underlying substrate, or another 2D layer, such as in 2D material heterostructures. The phenomenon is exploited as a means of engineering the electronic structure or optical properties of materials, which some call moiré materials. The often significant changes in electronic properties when twisting two atomic layers and the prospect of electronic applications has led to the name
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This phase moiré effect and the classical moiré effect from opaque lines are two ends of a continuous spectrum in optics, which is called the universal moiré effect. The phase moiré effect is the basis for a type of broadband interferometer in x-ray and particle wave applications. It also provides a way to reveal hidden patterns in invisible layers.
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the frequencies are not exactly the same and aligned perfectly together, beats (or samples) will align closely at some moments in time and far apart at other times. The closer together beats are, the darker it is at that spot; the farther apart, the lighter. The result is periodic in the same way as a graphic moiré pattern. See:
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effect creates arrows that point towards an imaginary line marking the hazard or line of safe passage; as navigators pass over the line, the arrows on the beacon appear to become vertical bands before changing back to arrows pointing in the reverse direction. An example can be found in the UK on the eastern shore of
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The moiré effect is used in shoreside beacons called "Inogon leading marks" or "Inogon lights", manufactured by Inogon Licens AB, Sweden, to designate the safest path of travel for ships heading to locks, marinas, ports, etc., or to indicate underwater hazards (such as pipelines or cables). The moiré
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The essence of the moiré effect is the (mainly visual) perception of a distinctly different third pattern which is caused by inexact superimposition of two similar patterns. The mathematical representation of these patterns is not trivially obtained and can seem somewhat arbitrary. In this section we
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Line moiré is one type of moiré pattern; a pattern that appears when superposing two transparent layers containing correlated opaque patterns. Line moiré is the case when the superposed patterns comprise straight or curved lines. When moving the layer patterns, the moiré patterns transform or move at
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The moiré effect also occurs between overlapping transparent objects. For example, an invisible phase mask is made of a transparent polymer with a wavy thickness profile. As light shines through two overlaid masks of similar phase patterns, a broad moiré pattern occurs on a screen some distance away.
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The visibility of these patterns is dependent on the medium or substrate in which they appear, and these may be opaque (as for example on paper) or transparent (as for example in plastic film). For purposes of discussion we shall assume the two primary patterns are each printed in greyscale ink on a
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If the lines of the patterns are superimposed at the left of the figure, the shift between the lines increases when going to the right. After a given number of lines, the patterns are opposed: the lines of the second pattern are between the lines of the first pattern. If we look from a far distance,
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often exhibit moiré patterns. Since both the TV screen and the digital camera use a scanning technique to produce or to capture pictures with horizontal scan lines, the conflicting sets of lines cause the moiré patterns. To avoid the effect, the digital camera can be aimed at an angle of 30 degrees
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Moiré patterns appear in many situations. In printing, the printed pattern of dots can interfere with the image. In television and digital photography, a pattern on an object being photographed can interfere with the shape of the light sensors to generate unwanted artifacts. They are also sometimes
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of MX-type (M = Ti, Nb; X = C, N) overlapping with austenitic matrix. Both phases, MX and the matrix, have face-centered cubic crystal structure and cube-on-cube orientation relationship. However, they have significant lattice misfit of about 20 to 24% (based on the chemical composition of alloy),
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screens. These are regular rectangular dot patterns—often four of them, printed in cyan, yellow, magenta, and black. Some kind of moiré pattern is inevitable, but in favorable circumstances the pattern is "tight"; that is, the spatial frequency of the moiré is so high that it is not noticeable. In
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The product of two "beat tracks" of slightly different speeds overlaid, producing an audible moiré pattern; if the beats of one track correspond to where in space a black dot or line exists and the beats of the other track correspond to the points in space where a camera is sampling light, because
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imaging by the interference of diffracting crystal lattice planes that are overlapping, and which might have different spacing and/or orientation. Most of the moiré contrast observations reported in the literature are obtained using high-resolution phase contrast imaging in TEM. However, if probe
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We shall also choose to represent the opacity of the pattern resulting from printing one pattern atop the other at a given point on the paper as the average (i.e. the arithmetic mean) of each pattern's opacity at that position, which is half their sum, and, as calculated, does not exceed 1. (This
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We now consider the "printing" superimposition of two almost similar, sinusoidally varying, grey-scale patterns to show how they produce a moiré effect in first printing one pattern on the paper, and then printing the other pattern over the first, keeping their coordinate axes in register. We
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in materials: by deforming a grid with respect to a reference grid and measuring the moiré pattern, the stress levels and patterns can be deduced. This technique is attractive because the scale of the moiré pattern is much larger than the deflection that causes it, making measurement easier.
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choice is not unique. Any other method to combine the functions that satisfies keeping the resultant function value within the bounds will also serve; arithmetic averaging has the virtue of simplicity—with hopefully minimal damage to one's concepts of the printmaking process.)
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moiré pattern. Part of the prepress art consists of selecting screen angles and halftone frequencies which minimize moiré. The visibility of moiré is not entirely predictable. The same set of screens may produce good results with some images, but visible moiré with others.
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If we choose to measure the angle, the final error is proportional to the measurement error. If we choose to measure the spacing, the final error is proportional to the inverse of the spacing. Thus, for the small angles, it is best to measure the spacing.
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282:, with a rippled or "watered" appearance. Moire, or "watered textile", is made by pressing two layers of the textile when wet. The similar but imperfect spacing of the threads creates a characteristic pattern which remains after the fabric dries.
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frequency tone which is heard when two pure notes of almost identical pitch are sounded simultaneously. This is an acoustic version of the moiré effect in the one dimension of time: the original two notes are still present—but the listener's
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with transparent gaps is overlaid on another similar pattern. For the moiré interference pattern to appear, the two patterns must not be completely identical, but rather displaced, rotated, or have slightly different pitch.
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patterns. One of the most important properties of shape moiré is its ability to magnify tiny shapes along either one or both axes, that is, stretching. A common 2D example of moiré magnification occurs when viewing a
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represents the periodic variation (i.e., spatial frequency) of the pattern's grey intensity, measured as the number of intensity cycles per unit distance. Since the sine function is cyclic over argument changes of
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shall give a mathematical example of two parallel patterns whose superimposition forms a moiré pattern, and show one way (of many possible ways) these patterns and the moiré effect can be rendered mathematically.
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The fine lines that make up the sky in this image create moiré patterns when shown at some resolutions for the same reason that photographs of televisions exhibit moiré patterns: the lines are not absolutely
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When graphene is grown on the (111) surface of iridium, its long-wavelength height modulation can be thought of as a moiré pattern arising from the superposition of the two mismatched hexagonal lattices.
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2103:{\displaystyle {\begin{aligned}(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)\end{aligned}}}
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Moiré patterns revealing complex shapes, or sequences of symbols embedded in one of the layers (in form of periodically repeated compressed shapes) are created with shape moiré, otherwise called
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is of two pitches that are the average of and half the difference of the frequencies of the two notes. Aliasing in sampling of time-varying signals also belongs to this moiré paradigm.
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1436:{\displaystyle {\begin{aligned}f_{3}&={\frac {f_{1}+f_{2}}{2}}\\&={\frac {1}{2}}+{\frac {\sin(k_{1}x)+\sin(k_{2}x)}{4}}\\&={\frac {1+\sin(Ax)\cos(Bx)}{2}}\end{aligned}}}
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exploit the tendency of digital scanners to produce moiré patterns by including fine circular or wavy designs that are likely to exhibit a moiré pattern when scanned and printed.
2256:{\displaystyle {\begin{aligned}(2D)^{2}&=2p^{2}\cdot {\frac {1+\cos \alpha }{\sin ^{2}\alpha }}\\D&={\frac {\frac {p}{2}}{\sin {\frac {\alpha }{2}}}}.\end{aligned}}}
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image of the object to the object itself: the hologram is the reference step, and the difference with the object are the deformations, which appear as pale and dark lines.
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Use of the moiré effect in strain measurement: case of uniaxial traction (top) and of pure shear (bottom); the lines of the patterns are initially horizontal in both cases
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Tang, Shujie; Wang, Haomin; Zhang, Yu; Li, Ang; Xie, Hong; Liu, Xiaoyu; Liu, Lianqing; Li, Tianxin; Huang, Fuqiang; Xie, Xiaoming; Jiang, Mianheng (16 September 2013).
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represent the grey intensity in each pattern by a positive opacity function of distance along a fixed direction (say, the x-coordinate) in the paper plane, in the form
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interaction of the optical patterns of lines creates a real and visible pattern of roughly parallel dark and light bands, the moiré pattern, superimposed on the lines.
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Shape moiré is one type of moiré pattern demonstrating the phenomenon of moiré magnification. 1D shape moiré is the particular simplified case of 2D shape moiré.
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The drawing on the upper right shows a moiré pattern. The lines could represent fibers in moiré silk, or lines drawn on paper or on a computer screen. The
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Yin, Xi; Liu, Xinhong; Pan, Yung-Tin; Walsh, Kathleen A.; Yang, Hong (10 December 2014). "Hanoi Tower-like
Multilayered Ultrathin Palladium Nanosheets".
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we have the feeling of pale zones when the lines are superimposed (there is white between the lines), and of dark zones when the lines are "opposed".
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3776:"Atomic resolution characterization of strengthening nanoparticles in a new high-temperature-capable 43Fe-25Ni-22.5Cr austenitic stainless steel"
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Liu, Yuan; Weiss, Nathan O.; Duan, Xidong; Cheng, Hung-Chieh; Huang, Yu; Duan, Xiangfeng (2016). "Van der Waals heterostructures and devices".
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white sheet, where the opacity (e.g., shade of grey) of the "printed" part is given by a value between 0 (white) and 1 (black) inclusive, with
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Moiré patterns are commonly seen on television screens when a person is wearing a shirt or jacket of a particular weave or pattern, such as a
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Kobayashi, Katsuyoshi (1996-01-01). "Moiré pattern in scanning tunneling microscopy: Mechanism in observation of subsurface nanostructures".
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K. Schouteden, N. Galvanetto (2015). "Scanning probe microscopy study of chemical vapor deposition grown graphene transferred to Au(111)".
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The average of these two functions, representing the superimposed printed image, evaluates as follows (see reverse identities here :
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Le Ster, Maxime; Maerkl, Tobias; Kowalczyk, Pawel J.; Brown, Simon A. (2019). "Moiré patterns in van der Waals heterostructures".
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Heczko, M.; Esser, B.D.; Smith, T.M.; Beran, P.; Mazánová, V.; McComb, D.W.; Kruml, T.; Polák, J.; Mills, M.J. (14 March 2018).
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where the presence of 1 keeps the function positive definite, and the division by 2 prevents function values greater than 1.
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through a second chain-link fence of identical design. The fine structure of the design is visible even at great distances.
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1189:{\displaystyle {\begin{aligned}f_{1}&={\frac {1+\sin(k_{1}x)}{2}}\\f_{2}&={\frac {1+\sin(k_{2}x)}{2}}\end{aligned}}}
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representing neutral grey. Any value less than 0 or greater than 1 using this grey scale is essentially "unprintable".
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Consider two patterns made of parallel and equidistant lines, e.g., vertical lines. The step of the first pattern is
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measurement: the operator just has to draw a pattern on the object, and superimpose the reference pattern to the
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Miao, Houxun; Panna, Alireza; Gomella, Andrew A.; Bennett, Eric E.; Znati, Sami; Chen, Lei; Wen, Han (2016).
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of the crystal, or when a thin layer of a second crystal is on the surface, e.g. single-layer, double-layer
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2393:{\displaystyle {\begin{aligned}\sin \alpha &\approx \alpha \\\cos \alpha &\approx 1\end{aligned}}}
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A moiré pattern formed by two units of parallel lines, one unit rotated 5° clockwise relative to the other
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the distance between the middle of two dark zones, which is also the distance between two pale zones, is
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3536:"Precisely aligned graphene grown on hexagonal boron nitride by catalyst free chemical vapor deposition"
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Consider now two such patterns, where one has a slightly different periodic variation from the other:
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N’Diaye, Alpha T. (2006-01-01). "Two-Dimensional Ir
Cluster Lattice on a Graphene Moiré on Ir(111)".
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3001:"Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy"
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1651:. Seen from afar, we can also see darker and paler lines: the pale lines correspond to the lines of
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Strong moiré visible in this photo of a parrot's feathers (more pronounced in the full-size image)
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jacket. This is due to interlaced scanning in televisions and non-film cameras, referred to as
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3606:"Silane-catalysed fast growth of large single-crystalline graphene on hexagonal boron nitride"
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lines on top of a layer containing a complex shape which is periodically repeating along the
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299:, "to produce a watered textile by weaving or pressing", by the 18th century. The adjective
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Difference in distance of the front and rear picket fence on a bridge create moiré patterns.
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A live demonstration of the moiré effect that stems from interferences between circles
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a fine regular pattern). This can be overcome in texture mapping through the use of
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2804:(TEM), translational moiré fringes can be seen as parallel contrast lines formed in
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th line of the first network. The middle of the first dark zone thus corresponds to
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Moiré pattern arising from the superposition of two graphene lattices twisted by 4°
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A series of oil paintings based on moiré principles by
British artist, Pip Dickens
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Moiré obtained by the superimposition of two similar patterns rotated by an angle
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is in use from the 17th century, for "watered silk". It was a loan of the
English
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of the neighbouring sides, we can see that the pale line makes an angle equal to
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808:{\displaystyle d=n\cdot (p+\delta p)={\frac {p^{2}}{2\delta p}}+{\frac {p}{2}}}
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Optical
Inspection of Arrays and Periodic Structures Using Moire Magnification
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the bigger the step, the bigger the distance between the pale and dark zones;
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In condensed matter physics, the moiré phenomenon is commonly discussed for
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is the hypotenuse of a right triangle and the sides of the right angle are
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3103:"A universal moiré effect and application in X-ray phase-contrast imaging"
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Moiré pattern (bottom) created by superimposing two grids (top and middle)
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A universal moiré effect and application in X-ray phase-contrast imaging
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Transmission electron microscopy : a textbook for materials science
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is, the farther apart the pale lines; when both patterns are parallel (
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Counterfeit deterrent features for the next-generation currency design
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Energy
Minimization Methods in Computer Vision and Pattern Recognition
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exhibits superconductivity and other important electronic properties.
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patterns are created if the lines are curved or not exactly parallel.
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If we consider a cell of the lattice formed, we can see that it is a
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Moiré pattern created by overlapping two sets of concentric circles
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295:(attested 1610). In French usage, the noun gave rise to the verb
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3164:. IEE Colloquium, Microengineering in Optics and Optoelectronics.
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Figueiredo, Mário; Zerubia, Josiane (2001). Jain, Anil K. (ed.).
2773:, moiré fringes appear if surface atomic layers have a different
3376:. Washington DC: National Materials Advisory Board. p. 61.
3051:"Scanning Images in Books/Magazines/Newspapers (Moire patterns)"
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The middle of the first dark zone is when the shift is equal to
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Moiré pattern appearing on scaled camera captures of LCD screen
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Kamal, Hala; Völkel, Reinhard; Alda, Javier (November 1998).
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industries, these patterns are used for studying microscopic
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The patterns are superimposed in the mid-width of the figure.
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a faster speed. This effect is called optical moiré speedup.
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APS Physics - Trend: Bilayer
Graphene's Wicked, Twisted Road
2456:: by the orientation of the pale lines and by their spacing
3326:(3 ed.). Bingley, England: Emerald. pp. 204–205.
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A similar effect can be obtained by the superposition of a
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Line moiré with slow movement of the revealing layer upward
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of this field. A prominent example is in twisted bi-layer
1888:{\displaystyle (2D)^{2}=d^{2}(1+\cos \alpha )^{2}+p^{2}}
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Other one-dimensional moiré effects include the classic
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Moiré pattern seen over a cage in the San
Francisco Zoo
30:"Moiré" and "Moire" redirect here. For other uses, see
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than the bulk crystal. This can for example be due to
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a checkered plane (the latter being a special case of
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between the middle of a pale zone and a dark zone is
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Additionally, the spacing between two pale lines is
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formed from this verb is in use from at least 1823.
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3730:Williams, David B.; Carter, C. Barry (2009-01-01).
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3604:Tang, Shujie; Wang, Haomin; Wang, Huishan (2015).
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3926:An interactive example of various moiré patterns
1647:, but the second pattern is rotated by an angle
3944:"The Moiré Effect Lights That Guide Ships Home"
3348:"Scanning Images in Magazines/Books/Newspapers"
2877:, known examples exhibiting moiré contrast are
1779:, half of the long diagonal. The long diagonal
1772:with the perpendicular of each pattern's line.
3935:Use arrow keys and mouse to manipulate layers.
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2820:Materials science and condensed matter physics
2429:{\displaystyle D\approx {\frac {p}{\alpha }}.}
862:{\displaystyle 2d={\frac {p^{2}}{\delta p}}+p}
672:{\displaystyle n\cdot \delta p={\frac {p}{2}}}
3302:The Moiré Effect Lights That Guide Ships Home
2482:{\displaystyle \alpha \approx {\frac {p}{D}}}
884:The principle of the moiré is similar to the
216:that can be produced when a partially opaque
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627:th line of the second pattern is shifted by
3954:about the Moiré Inogon light in Southampton
1557:{\displaystyle B={\frac {k_{1}-k_{2}}{2}}.}
496:. Unsourced material may be challenged and
97:Moiré pattern of double layered shading net
3158:Hutley, M.C.; Stevens, R.F. (1999-11-16).
1495:{\displaystyle A={\frac {k_{1}+k_{2}}{2}}}
430:patterns may appear when superimposing an
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716:{\displaystyle n={\frac {p}{2\delta p}}.}
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516:Learn how and when to remove this message
2789:heterostructure of graphene and hBN, or
3283:International Hydrographic Organization
2956:
2906:Barrier grid animation and stereography
1744:The pale lines correspond to the small
3322:Kazda, Antonín; Caves, Robert (2015).
3769:
3767:
3725:
3723:
3669:
3667:
3599:
3597:
3441:
3439:
3395:
3393:
2452:There are thus two ways to determine
195:
7:
3780:Materials Science and Engineering: A
869:From this formula, we can see that:
494:adding citations to reliable sources
1697:and the side opposite to the angle
2764:structured illumination microscopy
2734:images to produce digital images.
2552:Television screens and photographs
101:In mathematics, physics, and art,
25:
3372:Sincerbox, Glenn T., ed. (1991).
434:layer containing tiny horizontal
232:In physics, its manifestation is
3177:"Properties of Moiré Magnifiers"
3020:10.1046/j.1365-2818.2000.00710.x
2969:Longman Pronunciation Dictionary
2840:
2828:
2802:transmission electron microscopy
2704:pattern on the deformed object.
2696:The moiré effect can be used in
2572:
2560:
2287:
2278:
2269:
1730:
1712:
554:
542:
466:
156:
113:
53:
41:
2886:which produces a moiré effect.
2131:
2121:
2055:
2036:
1983:
1964:
1916:
1906:
1863:
1844:
1822:
1812:
1445:where it is easily shown that
1420:
1411:
1402:
1393:
1359:
1343:
1331:
1315:
1173:
1157:
1112:
1096:
972:
963:
892:Mathematical function approach
761:
746:
1:
3478:10.1103/PhysRevLett.97.215501
2999:Gustafsson, M. G. L. (2000).
2811:high-angle annular dark field
2771:scanning tunneling microscopy
2762:, using a technique known as
2494:Implications and applications
1662:with the four sides equal to
351:techniques, for example when
3521:10.1016/j.carbon.2015.08.033
3324:Airport design and operation
2438:We can see that the smaller
575:, the step of the second is
335:Moiré patterns are often an
2756:super-resolution microscopy
2539:the graphic arts, the term
876:the bigger the discrepancy
4005:
3837:10.1038/natrevmats.2016.42
3793:10.1016/j.msea.2018.02.004
3696:10.1103/PhysRevB.99.075422
2672:Holographic interferometry
2661:
2499:Printing full-color images
2331:small-angle approximations
529:Moiré of parallel patterns
419:
397:
29:
3422:10.1103/PhysRevB.53.11091
2945:Multidimensional sampling
2856:two-dimensional materials
1719:Unit cell of the "net"; "
1005:, the distance increment
256:The term originates from
225:created deliberately; in
3817:Nature Reviews Materials
3271:"inogen light [
2971:(3rd ed.). Pearson
2642:50.8560083°N 1.3291028°W
1737:Effect of changing angle
270:, traditionally made of
3448:Physical Review Letters
3279:Hydrographic Dictionary
1566:This function average,
3281:(5 ed.). Monaco:
3250:"Prohibited anchorage"
2779:surface reconstruction
2681:
2647:50.8560083; -1.3291028
2523:
2483:
2430:
2394:
2297:Effect on curved lines
2257:
2104:
1889:
1558:
1496:
1437:
1190:
985:
901:
863:
809:
717:
673:
332:
324:
316:
238:double-slit experiment
236:like that seen in the
98:
81:
73:
32:Moire (disambiguation)
3738:. Springer. pp.
3610:Nature Communications
3224:"US Patent 4,629,325"
3006:Journal of Microscopy
2901:Angle-sensitive pixel
2809:aberration-corrected
2679:
2509:
2484:
2431:
2395:
2258:
2105:
1890:
1559:
1497:
1438:
1191:
986:
899:
864:
810:
718:
674:
377:anisotropic filtering
330:
322:
314:
274:but now also made of
214:interference patterns
96:
79:
71:
3285:. 2003. p. 113.
2940:Moiré Phase Tracking
2726:provide an optional
2664:Theory of elasticity
2601:screen taken with a
2511:Warning: audiogenic
2460:
2404:
2337:
2114:
1899:
1809:
1723:" means "pale line".
1693:whose hypotenuse is
1510:
1451:
1227:
1057:
939:
819:
731:
683:
641:
534:Geometrical approach
490:improve this section
343:produced by various
285:In French, the noun
27:Interference pattern
3883:2014NanoL..14.7188Y
3829:2016NatRM...116042L
3688:2019PhRvB..99g5422L
3632:2015NatCo...6.6499T
3562:2013NatSR...3E2666T
3470:2006PhRvL..97u5501N
3414:1996PhRvB..5311091K
3408:(16): 11091–11099.
3196:1998OptEn..37.3007K
3184:Optical Engineering
3119:2016NatPh..12..830M
2935:Lenticular printing
2638: /
2621:Fawley oil refinery
2545:excessively visible
1803:Pythagorean theorem
1689:; (we have a right
3958:"The Moiré Museum"
3931:2011-07-24 at the
3640:10.1038/ncomms7499
3540:Scientific Reports
3248:Alexander Trabas.
2806:phase-contrast TEM
2682:
2658:Strain measurement
2606:to the TV screen.
2524:
2479:
2426:
2390:
2388:
2253:
2251:
2100:
2098:
1885:
1554:
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1433:
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1186:
1184:
981:
902:
859:
805:
713:
669:
333:
325:
317:
99:
82:
74:
3891:10.1021/nl503879a
3877:(12): 7188–7194.
3676:Physical Review B
3570:10.1038/srep02666
3402:Physical Review B
3252:. Light of Lights
3190:(11): 3007–3014.
3127:10.1038/nphys3734
2982:978-1-4058-8118-0
2875:materials science
2775:crystal structure
2760:diffraction limit
2724:computer programs
2617:Southampton Water
2610:Marine navigation
2597:Photographs of a
2592:interline twitter
2513:epileptic seizure
2507:
2477:
2421:
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2241:
2225:
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2083:
1962:
1549:
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1281:
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1119:
979:
851:
803:
790:
708:
667:
526:
525:
518:
349:computer graphics
307:Pattern formation
234:wave interference
94:
16:(Redirected from
3996:
3946:, an article on
3903:
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3855:
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3461:cond-mat/0609286
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3309:
3299:(5 March 2018),
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3204:10.1117/1.601889
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3098:
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3071:
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3062:
3061:
3055:www.scantips.com
3047:
3041:
3040:
3022:
2996:
2987:
2986:
2967:(3 April 2008).
2961:
2917:Euclid's orchard
2912:Beat (acoustics)
2844:
2832:
2797:nanostructures.
2715:Image processing
2653:
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2396:
2391:
2389:
2329:) the following
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2008:
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1646:
1639:Rotated patterns
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1286:
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1243:
1242:
1219:Prosthaphaeresis
1213:
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1187:
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670:
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660:
636:
633:compared to the
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595:
584:
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521:
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452:chain-link fence
212:are large-scale
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3933:Wayback Machine
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2307:is very small (
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428:One-dimensional
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345:digital imaging
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280:synthetic fiber
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242:beat phenomenon
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3910:External links
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3228:Google Patents
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3113:(9): 830–834.
3107:Nature Physics
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1257:
1250:
1247:
1245:
1241:
1237:
1233:
1232:
1210:
1203:
1179:
1175:
1172:
1167:
1163:
1159:
1156:
1153:
1150:
1147:
1141:
1138:
1136:
1132:
1128:
1124:
1123:
1118:
1114:
1111:
1106:
1102:
1098:
1095:
1092:
1089:
1086:
1080:
1077:
1075:
1071:
1067:
1063:
1062:
978:
974:
971:
968:
965:
962:
959:
956:
953:
947:
944:
893:
890:
882:
881:
874:
858:
855:
849:
846:
840:
836:
830:
827:
824:
802:
799:
794:
788:
785:
782:
776:
772:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
736:
712:
706:
703:
700:
696:
691:
688:
666:
663:
658:
655:
652:
649:
646:
560:
553:
552:
548:
541:
540:
539:
538:
537:
535:
532:
530:
527:
524:
523:
474:
472:
465:
459:
456:
420:Main article:
417:
414:
398:Main article:
395:
392:
308:
305:
253:
250:
103:moiré patterns
59:
52:
51:
47:
40:
39:
38:
37:
36:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4001:
3990:
3987:
3985:
3982:
3980:
3977:
3975:
3972:
3971:
3969:
3959:
3956:
3953:
3949:
3945:
3942:
3940:
3937:
3934:
3930:
3927:
3924:
3922:
3919:
3917:
3914:
3913:
3909:
3900:
3896:
3892:
3888:
3884:
3880:
3876:
3872:
3865:
3862:
3859:
3854:
3851:
3846:
3842:
3838:
3834:
3830:
3826:
3822:
3818:
3811:
3808:
3803:
3799:
3794:
3789:
3785:
3781:
3777:
3770:
3768:
3764:
3759:
3755:
3751:
3749:9780387765013
3745:
3741:
3736:
3735:
3726:
3724:
3720:
3715:
3711:
3706:
3701:
3697:
3693:
3689:
3685:
3682:(7): 075422.
3681:
3677:
3670:
3668:
3664:
3659:
3655:
3650:
3645:
3641:
3637:
3633:
3629:
3624:
3619:
3615:
3611:
3607:
3600:
3598:
3594:
3589:
3585:
3580:
3575:
3571:
3567:
3563:
3559:
3554:
3549:
3545:
3541:
3537:
3530:
3527:
3522:
3518:
3514:
3510:
3503:
3500:
3495:
3491:
3487:
3483:
3479:
3475:
3471:
3467:
3462:
3457:
3453:
3449:
3442:
3440:
3436:
3431:
3427:
3423:
3419:
3415:
3411:
3407:
3403:
3396:
3394:
3390:
3385:
3383:9780309050289
3379:
3375:
3368:
3365:
3353:
3349:
3343:
3340:
3335:
3333:9781784418700
3329:
3325:
3318:
3315:
3304:
3303:
3298:
3292:
3289:
3284:
3280:
3276:
3275:
3267:
3264:
3251:
3244:
3241:
3229:
3225:
3219:
3216:
3205:
3201:
3197:
3193:
3189:
3185:
3178:
3171:
3168:
3163:
3162:
3154:
3151:
3146:
3142:
3137:
3132:
3128:
3124:
3120:
3116:
3112:
3108:
3104:
3097:
3094:
3089:
3087:9783540425236
3083:
3079:
3078:
3070:
3067:
3056:
3052:
3046:
3043:
3038:
3034:
3030:
3026:
3021:
3016:
3012:
3008:
3007:
3002:
2995:
2993:
2989:
2984:
2978:
2974:
2970:
2966:
2960:
2957:
2950:
2946:
2943:
2941:
2938:
2936:
2933:
2931:
2928:
2926:
2924:
2920:
2918:
2915:
2913:
2910:
2907:
2904:
2902:
2899:
2897:
2894:
2893:
2889:
2887:
2884:
2883:nanoparticles
2880:
2876:
2871:
2869:
2865:
2861:
2857:
2843:
2831:
2819:
2817:
2815:
2812:
2807:
2803:
2798:
2796:
2792:
2788:
2787:Van der Waals
2784:
2780:
2776:
2772:
2767:
2765:
2761:
2757:
2749:
2747:
2745:
2737:
2735:
2733:
2729:
2725:
2722:
2721:image scanner
2714:
2712:
2710:
2705:
2703:
2699:
2694:
2691:
2687:
2686:manufacturing
2678:
2673:
2669:
2668:Strain tensor
2665:
2657:
2655:
2651:
2630:50°51′21.63″N
2622:
2618:
2609:
2607:
2604:
2600:
2595:
2593:
2589:
2575:
2563:
2551:
2549:
2546:
2542:
2537:
2533:
2529:
2521:
2516:
2514:
2498:
2493:
2491:
2474:
2471:
2466:
2463:
2450:
2446:
2436:
2423:
2418:
2415:
2410:
2407:
2383:
2380:
2378:
2373:
2370:
2367:
2360:
2357:
2355:
2350:
2347:
2344:
2333:can be made:
2332:
2311:
2290:
2281:
2272:
2263:
2246:
2238:
2235:
2230:
2227:
2222:
2219:
2213:
2211:
2206:
2196:
2193:
2188:
2184:
2178:
2175:
2172:
2169:
2166:
2160:
2155:
2151:
2147:
2144:
2142:
2135:
2127:
2124:
2092:
2088:
2085:
2079:
2076:
2071:
2067:
2059:
2051:
2048:
2045:
2042:
2039:
2029:
2025:
2020:
2016:
2012:
2010:
2000:
1996:
1992:
1987:
1979:
1976:
1973:
1970:
1967:
1958:
1955:
1950:
1946:
1939:
1935:
1929:
1927:
1920:
1912:
1909:
1880:
1876:
1872:
1867:
1859:
1856:
1853:
1850:
1847:
1839:
1835:
1831:
1826:
1818:
1815:
1804:
1794:
1790:
1784:
1773:
1760:
1751:
1747:
1733:
1722:
1715:
1706:
1692:
1683:
1674:
1666:
1661:
1656:
1654:
1638:
1636:
1634:
1629:
1624:
1618:
1609:
1602:
1592:
1583:
1570:
1564:
1551:
1546:
1540:
1536:
1532:
1527:
1523:
1516:
1513:
1505:
1502:
1487:
1481:
1477:
1473:
1468:
1464:
1457:
1454:
1446:
1443:
1424:
1417:
1414:
1408:
1405:
1399:
1396:
1390:
1387:
1384:
1381:
1375:
1373:
1363:
1356:
1351:
1347:
1340:
1337:
1334:
1328:
1323:
1319:
1312:
1309:
1303:
1298:
1295:
1290:
1288:
1278:
1272:
1268:
1264:
1259:
1255:
1248:
1246:
1239:
1235:
1222:
1220:
1215:
1209:
1202:
1196:
1177:
1170:
1165:
1161:
1154:
1151:
1148:
1145:
1139:
1137:
1130:
1126:
1116:
1109:
1104:
1100:
1093:
1090:
1087:
1084:
1078:
1076:
1069:
1065:
1052:
1049:
1042:
1028:
1020:
1016:
1010:
996:The quantity
994:
991:
976:
969:
966:
960:
957:
954:
951:
945:
942:
934:
930:
926:
906:
898:
891:
889:
887:
886:Vernier scale
875:
872:
871:
870:
856:
853:
847:
844:
838:
834:
828:
825:
822:
800:
797:
792:
786:
783:
780:
774:
770:
764:
758:
755:
752:
749:
743:
740:
737:
734:
723:The distance
710:
704:
701:
698:
694:
689:
686:
664:
661:
656:
653:
650:
647:
644:
631:
611:
601:
597:
594:
590:
583:
579:
557:
545:
533:
528:
520:
517:
509:
499:
495:
491:
485:
484:
480:
475:This section
473:
469:
464:
463:
457:
455:
453:
448:
443:
441:
440:vertical axis
437:
433:
429:
423:
415:
413:
411:
408:More complex
406:
401:
393:
391:
387:
385:
380:
378:
374:
370:
369:undersampling
366:
362:
358:
354:
350:
346:
342:
338:
329:
321:
313:
306:
304:
302:
298:
294:
293:
288:
283:
281:
277:
273:
269:
265:
261:
260:
251:
249:
247:
243:
239:
235:
230:
228:
222:
219:
218:ruled pattern
215:
211:
210:moiré fringes
204:
198:
190:
189:
180:
153:
147:
146:
137:
110:
104:
78:
70:
56:
44:
33:
19:
18:Moire pattern
3979:Interference
3874:
3871:Nano Letters
3870:
3864:
3853:
3823:(9): 16042.
3820:
3816:
3810:
3783:
3779:
3733:
3679:
3675:
3613:
3609:
3543:
3539:
3529:
3512:
3508:
3502:
3451:
3447:
3405:
3401:
3373:
3367:
3355:. Retrieved
3352:ScanTips.com
3351:
3342:
3323:
3317:
3306:, retrieved
3301:
3291:
3278:
3272:
3266:
3254:. Retrieved
3243:
3231:. Retrieved
3227:
3218:
3207:. Retrieved
3187:
3183:
3170:
3160:
3153:
3110:
3106:
3096:
3080:. Springer.
3076:
3069:
3058:. Retrieved
3054:
3045:
3013:(2): 82–87.
3010:
3004:
2968:
2959:
2922:
2872:
2867:
2853:
2799:
2768:
2753:
2741:
2718:
2706:
2695:
2683:
2633:1°19′44.77″W
2613:
2596:
2585:
2544:
2540:
2528:graphic arts
2525:
2510:
2451:
2444:
2437:
2309:
2302:
1792:
1788:
1782:
1774:
1758:
1743:
1721:ligne claire
1720:
1681:
1672:
1664:
1657:
1642:
1632:
1625:
1616:
1607:
1600:
1590:
1581:
1568:
1565:
1506:
1503:
1447:
1444:
1223:
1216:
1207:
1200:
1197:
1053:
1050:
1040:
1026:
1018:
1014:
1008:
995:
992:
935:
931:
927:
907:
903:
883:
629:
609:
602:
598:
592:
588:
581:
577:
570:
512:
503:
488:Please help
476:
458:Calculations
446:
444:
425:
407:
403:
388:
381:
334:
300:
296:
290:
286:
284:
263:
257:
255:
231:
223:
209:
102:
100:
3705:10092/16978
3546:(1): 2666.
3515:: 318–322.
2965:Wells, John
2930:Kell factor
2925:(sculpture)
2868:magic angle
2860:twistronics
2709:holographic
2645: /
2619:, opposite
2588:houndstooth
436:transparent
422:Shape moiré
416:Shape moiré
361:ray tracing
359:picture or
323:Shape moiré
227:micrometers
3968:Categories
3623:1503.02806
3297:Scott, Tom
3256:9 December
3209:2021-03-30
3060:2020-02-27
2951:References
2908:(kinegram)
2879:thin films
2750:Microscopy
2662:See also:
1633:perception
1198:such that
506:April 2018
447:band moiré
410:line moiré
400:Line moiré
394:Line moiré
373:mipmapping
3952:Tom Scott
3845:2058-8437
3802:0921-5093
3786:: 49–60.
3758:876600051
3714:128326155
3553:1309.0172
2744:banknotes
2738:Banknotes
2543:means an
2467:≈
2464:α
2419:α
2411:≈
2381:≈
2374:α
2371:
2361:α
2358:≈
2351:α
2348:
2236:α
2231:
2197:α
2194:
2179:α
2176:
2161:⋅
2080:α
2077:
2052:α
2049:
2026:⋅
1980:α
1977:
1959:α
1956:
1895:that is:
1860:α
1857:
1791:(1 + cos
1750:bisectors
1533:−
1409:
1391:
1341:
1313:
1155:
1094:
961:
845:δ
784:δ
756:δ
744:⋅
702:δ
651:δ
648:⋅
477:does not
384:nonlinear
367:, due to
252:Etymology
246:acoustics
3989:Printing
3984:Patterns
3974:Geometry
3929:Archived
3899:25369350
3658:25757864
3616:: 6499.
3588:24036628
3494:19791195
3486:17155746
3357:22 April
3277:]".
3145:27746823
3029:10810003
2923:Guardian
2896:Aliasing
2890:See also
2864:graphene
2795:antimony
2783:graphene
2732:halftone
2702:deformed
2536:halftone
2532:prepress
1746:diagonal
1691:triangle
679:that is
365:aliasing
357:halftone
353:scanning
337:artifact
240:and the
3948:YouTube
3879:Bibcode
3825:Bibcode
3684:Bibcode
3649:4382696
3628:Bibcode
3579:3773621
3558:Bibcode
3466:Bibcode
3430:9982681
3410:Bibcode
3308:5 March
3233:5 March
3192:Bibcode
3136:5063246
3115:Bibcode
3037:9257781
2973:Longman
2791:bismuth
2520:phasing
2515:hazard.
2326:
2314:
1805:gives:
1769:
1755:
1686:
1669:
1660:rhombus
1045:
1031:
923:
911:
620:
606:
587:0 <
585:, with
498:removed
483:sources
268:textile
193:French:
3897:
3843:
3800:
3756:
3746:
3742:–397.
3712:
3656:
3646:
3586:
3576:
3509:Carbon
3492:
3484:
3428:
3380:
3330:
3143:
3133:
3084:
3035:
3027:
2979:
2728:filter
2698:strain
2690:strain
2670:, and
1801:. The
623:. The
432:opaque
341:images
297:moirer
292:mohair
276:cotton
61:level.
3710:S2CID
3618:arXiv
3548:arXiv
3490:S2CID
3456:arXiv
3180:(PDF)
3033:S2CID
2785:, or
2742:Many
2719:Some
2541:moiré
2400:thus
2312:<
2303:When
2110:thus
1653:nodes
1023:, or
591:<
301:moiré
287:moire
264:moiré
259:moire
208:) or
185:mwah-
3895:PMID
3841:ISSN
3798:ISSN
3754:OCLC
3744:ISBN
3654:PMID
3584:PMID
3482:PMID
3426:PMID
3378:ISBN
3359:2010
3328:ISBN
3310:2018
3258:2016
3235:2018
3141:PMID
3082:ISBN
3025:PMID
2977:ISBN
2793:and
2530:and
1797:and
1680:sin
1628:beat
1614:and
1599:cos(
1588:and
1504:and
1021:= 2π
630:n δp
481:any
479:cite
375:and
347:and
272:silk
145:-ray
143:MWAH
3950:by
3887:doi
3833:doi
3788:doi
3784:719
3740:393
3700:hdl
3692:doi
3644:PMC
3636:doi
3574:PMC
3566:doi
3517:doi
3474:doi
3418:doi
3274:sic
3200:doi
3131:PMC
3123:doi
3015:doi
3011:198
2881:or
2873:In
2800:In
2769:In
2754:In
2684:In
2526:In
2447:= 0
2368:cos
2345:sin
2228:sin
2185:sin
2173:cos
2068:sin
2046:cos
1974:cos
1947:sin
1854:cos
1705:).
1701:is
1406:cos
1388:sin
1338:sin
1310:sin
1221:):
1152:sin
1091:sin
958:sin
492:by
339:of
278:or
244:in
187:RAY
3970::
3893:.
3885:.
3875:14
3873:.
3839:.
3831:.
3819:.
3796:.
3782:.
3778:.
3766:^
3752:.
3722:^
3708:.
3698:.
3690:.
3680:99
3678:.
3666:^
3652:.
3642:.
3634:.
3626:.
3612:.
3608:.
3596:^
3582:.
3572:.
3564:.
3556:.
3542:.
3538:.
3513:95
3511:.
3488:.
3480:.
3472:.
3464:.
3452:97
3450:.
3438:^
3424:.
3416:.
3406:53
3404:.
3392:^
3350:.
3226:.
3198:.
3188:37
3186:.
3182:.
3139:.
3129:.
3121:.
3111:12
3109:.
3105:.
3053:.
3031:.
3023:.
3009:.
3003:.
2991:^
2975:.
2766:.
2666:,
2599:TV
1667:=
1601:Bx
1214:.
1206:≈
1048:.
1034:2π
1029:=
1003:2π
888:.
878:δp
596:.
589:δp
582:δp
580:+
442:.
379:.
355:a
248:.
191:,
176:eɪ
167:ɑː
154::
152:US
148:,
133:eɪ
127:ɑː
111::
109:UK
3901:.
3889::
3881::
3847:.
3835::
3827::
3821:1
3804:.
3790::
3760:.
3716:.
3702::
3694::
3686::
3660:.
3638::
3630::
3620::
3614:6
3590:.
3568::
3560::
3550::
3544:3
3523:.
3519::
3496:.
3476::
3468::
3458::
3432:.
3420::
3412::
3386:.
3361:.
3336:.
3260:.
3237:.
3212:.
3202::
3194::
3147:.
3125::
3117::
3090:.
3063:.
3039:.
3017::
2985:.
2623:(
2522:.
2475:D
2472:p
2454:α
2445:α
2440:α
2424:.
2416:p
2408:D
2384:1
2323:6
2320:/
2317:π
2310:α
2305:α
2247:.
2239:2
2223:2
2220:p
2214:=
2207:D
2189:2
2170:+
2167:1
2156:2
2152:p
2148:2
2145:=
2136:2
2132:)
2128:D
2125:2
2122:(
2093:)
2089:1
2086:+
2072:2
2060:2
2056:)
2043:+
2040:1
2037:(
2030:(
2021:2
2017:p
2013:=
2001:2
1997:p
1993:+
1988:2
1984:)
1971:+
1968:1
1965:(
1951:2
1940:2
1936:p
1930:=
1921:2
1917:)
1913:D
1910:2
1907:(
1881:2
1877:p
1873:+
1868:2
1864:)
1851:+
1848:1
1845:(
1840:2
1836:d
1832:=
1827:2
1823:)
1819:D
1816:2
1813:(
1799:p
1795:)
1793:α
1789:d
1783:D
1781:2
1777:D
1766:2
1763:/
1759:α
1703:p
1699:α
1695:d
1682:α
1677:/
1673:p
1665:d
1649:α
1645:p
1620:2
1617:k
1611:1
1608:k
1603:)
1594:2
1591:k
1585:1
1582:k
1577:A
1572:3
1569:f
1552:.
1547:2
1541:2
1537:k
1528:1
1524:k
1517:=
1514:B
1488:2
1482:2
1478:k
1474:+
1469:1
1465:k
1458:=
1455:A
1425:2
1421:)
1418:x
1415:B
1412:(
1403:)
1400:x
1397:A
1394:(
1385:+
1382:1
1376:=
1364:4
1360:)
1357:x
1352:2
1348:k
1344:(
1335:+
1332:)
1329:x
1324:1
1320:k
1316:(
1304:+
1299:2
1296:1
1291:=
1279:2
1273:2
1269:f
1265:+
1260:1
1256:f
1249:=
1240:3
1236:f
1211:2
1208:k
1204:1
1201:k
1178:2
1174:)
1171:x
1166:2
1162:k
1158:(
1149:+
1146:1
1140:=
1131:2
1127:f
1117:2
1113:)
1110:x
1105:1
1101:k
1097:(
1088:+
1085:1
1079:=
1070:1
1066:f
1041:k
1037:/
1027:x
1025:Δ
1019:x
1017:Δ
1015:k
1009:x
1007:Δ
998:k
977:2
973:)
970:x
967:k
964:(
955:+
952:1
946:=
943:f
920:2
917:/
914:1
857:p
854:+
848:p
839:2
835:p
829:=
826:d
823:2
801:2
798:p
793:+
787:p
781:2
775:2
771:p
765:=
762:)
759:p
753:+
750:p
747:(
741:n
738:=
735:d
725:d
711:.
705:p
699:2
695:p
690:=
687:n
665:2
662:p
657:=
654:p
645:n
635:n
625:n
617:2
614:/
610:p
593:p
578:p
573:p
563:α
519:)
513:(
508:)
504:(
500:.
486:.
262:(
179:/
173:r
170:ˈ
164:w
161:m
158:/
136:/
130:r
124:w
121:m
118:ˈ
115:/
105:(
34:.
20:)
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