# 莫列波紋

## 計算

### 平行圖樣中的摩列

#### 幾何手法

the patterns are superimposed in the mid-width of the figure

The middle of the first dark zone is when the shift is equal to ${\displaystyle {\frac {p}{2}}}$. 第二個圖樣第${\displaystyle n}$th 線與第一圖樣第${\displaystyle n}$th 線相比，移動了 ${\displaystyle n\cdot \delta p}$。於是，第一暗區的中央為：${\displaystyle n\cdot \delta p={\frac {p}{2}}}$

${\displaystyle n={\frac {p}{2\delta p}}.}$

${\displaystyle d=n\cdot p={\frac {p^{2}}{2\delta p}}}$

${\displaystyle 2d={\frac {p^{2}}{\delta p}}}$

#### 數學方程手法

${\displaystyle f={\frac {1+\sin(kx)}{2}}}$

1的存在使得方程恆正，而除以2避免方程結果大於1。

${\displaystyle k}$為強度週期/單位距離， 表示圖樣灰階強度的週期變動。因正弦方程對 ${\displaystyle 2\pi }$有循環，當 ${\displaystyle k\Delta x=2\pi }$，或 ${\displaystyle \Delta x={\frac {2\pi }{k}}}$時，可得每強度週期（波長）之距離增長。

${\displaystyle f_{1}={\frac {1+\sin(k_{1}x)}{2}}}$
${\displaystyle f_{2}={\frac {1+\sin(k_{2}x)}{2}}}$

${\displaystyle f_{3}={\frac {f_{1}+f_{2}}{2}}}$
${\displaystyle ={\frac {1}{2}}+{\frac {\sin(k_{1}x)+\sin(k_{2}x)}{4}}}$
${\displaystyle ={\frac {1+\sin(Ax)\cos(Bx)}{2}}}$

${\displaystyle A={\frac {k_{1}+k_{2}}{2}}}$

${\displaystyle B={\frac {k_{1}-k_{2}}{2}}.}$

### 旋轉圖樣

"網"單位格; "ligne claire" ， "亮紋"

${\displaystyle (2D)^{2}=d^{2}(1+\cos \alpha )^{2}+p^{2}}$

id est

${\displaystyle (2D)^{2}={\frac {p^{2}}{\sin ^{2}\alpha }}(1+\cos \alpha )^{2}+p^{2}=p^{2}\cdot \left({\frac {(1+\cos \alpha )^{2}}{\sin ^{2}\alpha }}+1\right)}$

${\displaystyle (2D)^{2}=2p^{2}\cdot {\frac {1+\cos \alpha }{\sin ^{2}\alpha }}}$ or ${\displaystyle D={\frac {p}{2}}/\sin {\frac {\alpha }{2}}.}$

${\displaystyle \alpha }$ 極小 (${\displaystyle \alpha <{\frac {\pi }{6}}}$)， 可做以下近似：

${\displaystyle \sin \alpha \approx \alpha }$
${\displaystyle \cos \alpha \approx 1}$

${\displaystyle D\approx {\frac {p}{\alpha }}.}$

${\displaystyle \alpha \approx {\frac {p}{D}}}$

## 意義與應用

### 列印全彩圖案

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 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: phase (music).

## 參考

1. ^ Alexander Trabas. Beacons. Online-list-of-lights.info. [2012-10-30]. （原始内容存档于2012年3月9日）.