Moiré patterns (Spatial Beats)

double guardrails Rungs

phantom rungsspatial beats

The animation shows a perspective sketch of two rows of rungs moving left to right. There is the illusion of much thicker rungs, widely spaced. You can see between them, but you cannot see through them. You should see the illusion of several stationary objects, although all of the objects are in motion.

Open the Java applet. It allows you to change the perspective of the observer, and the size and spacing of the rungs. [Applet by Paul Kunkel]

Analysis

Let the near-rung frequency be f (spokes/° visual angle). Eg., if the observer distance is 10 times the inter-rail distance, then the far-rung frequency will be 1.1 f due to the perspective. Therefore, visual beats will occur at Δf = 0.1f — phantom rungs will appear at every 10th near-rung.

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Geometrical Moiré patterns

Specimen grating rotation angle = 2°

Moiré patterns form when two images with a similar spatial repetition are overlayed. Slide the mouse on the window to move the left pattern onto and off from the right pattern. The apparent images that flash, move vertically, or at angles to the motion are moiré patterns.

These represent nearly the simplest moiré patterns. You can see similar patterns by holding two combs, or pieces of screen in front of one another and sliding one back and forth.

Analysis
 
Let the master grating pitch be p, specimen grating rotation θ, and moiré pitch δ. Considering one diagonal segment of specimen grating, length '/' = p/sin(θ) = δ/sin(f-θ). The moiré fringes bisect the obtuse angle between the master and specimen gratings; thus the moiré fringe angle f = 90° + θ/2.

\ δ = p sin(f-θ)/sin(θ) = p cos(θ/2)/sin(θ). (= 28.65p if θ = 2°)

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