Editing Moiré pattern (section)

=== Rotated patterns ===
Consider two patterns with the same step {{mvar|p}}, but the second pattern is rotated by an angle {{mvar|α}}. Seen from afar, we can also see darker and paler lines: the pale lines correspond to the lines of [[Node (physics)|nodes]], that is, lines passing through the intersections of the two patterns.

If we consider a cell of the lattice formed, we can see that it is a [[rhombus]] with the four sides equal to {{math|''d'' {{=}} {{sfrac|''p''|sin ''α''}}}}; (we have a right [[triangle]] whose hypotenuse is {{mvar|d}} and the side opposite to the angle {{mvar|α}} is {{mvar|p}}).

{{multiple image
| direction = vertical
| total_width = 160
| align = right
| image1 = Moire calcul angle.png
| alt1 = 
| caption1 = Unit cell of the "net"; "''ligne claire''" means "pale line".
| image2 = Moire02.gif
| alt2 = 
| caption2 = Effect of changing angle
| footer = 
}}

The pale lines correspond to the small [[diagonal]] of the rhombus. As the diagonals are the [[Bisection|bisectors]] of the neighbouring sides, we can see that the pale line makes an angle equal to {{math|{{sfrac|''α''|2}}}} with the perpendicular of each pattern's line.

Additionally, the spacing between two pale lines is {{mvar|D}}, half of the long diagonal. The long diagonal {{math|2''D''}} is the hypotenuse of a right triangle and the sides of the right angle are {{math|''d''(1 + cos ''α'')}} and {{mvar|p}}. The [[Pythagorean theorem]] gives:
<math display="block">(2 D)^2 = d^2 (1 + \cos \alpha)^2 + p^2</math>
that is:
<math display="block">\begin{align} (2D)^2 &= \frac{p^2}{\sin^2 \alpha}(1+ \cos \alpha)^2 + p^2 \\[5pt]
&= p^2 \cdot \left ( \frac{(1 + \cos \alpha)^2}{\sin^2 \alpha} + 1\right) \end{align}</math>
thus
<math display="block">\begin{align} (2D)^2 &= 2 p^2 \cdot \frac{1+\cos \alpha}{\sin^2 \alpha} \\[5pt] D &= \frac{\frac{p}{2}}{\sin\frac{\alpha}{2}}. \end{align}</math>

{{multiple image
| total_width = 320
| align = right
| image1 = Moire Circles.svg
| alt1 = 
| caption1 = 
| image2 = Moire Lines.svg
| alt2 = 
| caption2 = 
| image3 = Moire Lines and Circles.svg
| alt3 = 
| caption3 = 
| footer = Effect on curved lines
}}

When {{mvar|α}} is very small ({{math|''α'' < {{sfrac|π|6}}}}) the following [[small-angle approximation]]s can be made:
<math display="block">\begin{align} \sin \alpha &\approx \alpha \\ \cos \alpha &\approx 1 \end{align}</math>
thus
<math display="block">D \approx \frac{p}{\alpha}.</math>

We can see that the smaller {{mvar|α}} is, the farther apart the pale lines; when both patterns are parallel ({{math|''α'' {{=}} 0}}), the spacing between the pale lines is infinite (there is no pale line).

There are thus two ways to determine {{mvar|α}}: by the orientation of the pale lines and by their spacing
<math display="block">\alpha \approx \frac{p}{D}</math>
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.