Editing Moiré pattern (section)

=== Moiré of parallel patterns ===

==== Geometrical approach ====
{{multiple image
| total_width = 320
| align = right
| image1 = Moire parallel.svg
| alt1 = 
| caption1 = The patterns are superimposed in the mid-width of the figure.
| image2 = Moire ecart angulaire.png
| alt2 = 
| caption2 = Moiré obtained by the superimposition of two similar patterns rotated by an angle {{mvar|α}}
| footer = 
}}

Consider two patterns made of parallel and equidistant lines, e.g., vertical lines. The step of the first pattern is {{mvar|p}}, the step of the second is {{math|''p'' + ''δp''}}, with {{math|0 < ''δp'' < ''p''}}.

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, 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".

The middle of the first dark zone is when the shift is equal to {{math|{{sfrac|''p''|2}}}}. The {{mvar|n}}th line of the second pattern is shifted by {{math|''n δp''}} compared to the {{mvar|n}}th line of the first network. The middle of the first dark zone thus corresponds to
<math display="block">n \cdot \delta p = \frac{p}{2}</math>
that is
<math display="block">n = \frac{p}{2 \delta p}.</math>
The distance {{mvar|d}} between the middle of a pale zone and a dark zone is
<math display="block">d = n \cdot (p+\delta p) = \frac{p^2}{2 \delta p}+\frac{p}{2}</math>
the distance between the middle of two dark zones, which is also the distance between two pale zones, is
<math display="block">2d = \frac{p^2}{\delta p}+p</math>
From this formula, we can see that:
* the bigger the step, the bigger the distance between the pale and dark zones;
* the bigger the discrepancy {{mvar|δp}}, the closer the dark and pale zones; a great spacing between dark and pale zones mean that the patterns have very close steps.

The principle of the moiré is similar to the [[Vernier scale]].

==== Mathematical function approach ====
[[File:Moiré grid.svg|thumb|upright|Moiré pattern (bottom) created by superimposing two grids (top and middle)]]

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 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.

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 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 {{sfrac|1|2}} representing neutral grey. Any value less than 0 or greater than 1 using this grey scale is essentially "unprintable".

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 choice is not unique. Any other method to combine the functions that satisfies keeping the resultant function value within the bounds [0,1] will also serve; arithmetic averaging has the virtue of simplicity—with hopefully minimal damage to one's concepts of the printmaking process.)

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 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

<math display="block">f = \frac{1 + \sin(k x)}{2}</math>

where the presence of 1 keeps the function positive definite, and the division by 2 prevents function values greater than 1.

The quantity {{mvar|k}} 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 {{math|2π}}, the distance increment {{math|Δ''x''}} per intensity cycle (the wavelength) obtains when {{math|''k'' Δ''x'' {{=}} 2π}}, or {{math|Δ''x'' {{=}} {{sfrac|2π|''k''}}}}.

Consider now two such patterns, where one has a slightly different periodic variation from the other:

<math display="block">\begin{align}
 f_1 &= \frac{1 + \sin(k_1 x)}{2} \\[4pt]
 f_2 &= \frac{1 + \sin(k_2 x)}{2}
\end{align}</math>

such that {{math|''k''<sub>1</sub> ≈ ''k''<sub>2</sub>}}.

The average of these two functions, representing the superimposed printed image, evaluates as follows (see reverse identities here :[[Prosthaphaeresis]] ):

<math display="block">\begin{align} f_3 &= \frac{f_1 + f_2}{2} \\[5pt] &= \frac12 + \frac{\sin(k_1 x) + \sin(k_2 x)}{4} \\[5pt] &= \frac{1 + \sin(A x) \cos(B x)}{2} \end{align}</math>

where it is easily shown that

<math display="block">A = \frac{k_1 + k_2}{2}</math>

and

<math display="block">B = \frac{k_1 - k_2}{2}.</math>

This function average, {{math|''f''<sub>3</sub>}}, clearly lies in the range [0,1]. Since the periodic variation {{mvar|A}} is the average of and therefore close to {{math|''k''<sub>1</sub>}} and {{math|''k''<sub>2</sub>}}, the moiré effect is distinctively demonstrated by the sinusoidal envelope "beat" function {{math|cos(''Bx'')}}, whose periodic variation is half the difference of the periodic variations {{math|''k''<sub>1</sub>}} and {{math|''k''<sub>2</sub>}} (and evidently much lower in frequency).

Other one-dimensional moiré effects include the classic [[beat (acoustics)|beat]] 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 ''perception'' 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.