Line moiré
Simple moiré patterns can be observed when superposing two transparent layers comprising periodically repeating opaque parallel lines as shown in Figure 1.
The superposition image of Figure 1 outlines periodically repeating dark parallel bands, called moiré lines.
Light bands of the superposition image correspond to the zones where the lines of both layers overlap.
Figure 3 shows a detailed diagram of the superposition image between two adjacent zones with overlapping lines of the revealing and base layers (i.e., between two light bands).
According to the formula for computing pm, the closer the periods of the two layers, the stronger the magnification factor is.
The GIF animation shown in Figure 4 corresponds to a slow movement of the revealing layer.
The negative value of the ratio computed according to this formula signifies a movement in the reverse direction.
The difference between the vertical periods pb, pr, and the distances Tb, Tr is shown in the diagram of Figure 8.
The inclination degree of layer lines may change along the horizontal axis forming curves.
Layer periods pb and pr represent the distances between the curves along the vertical axis.
Correspondingly, the period pm (along the vertical axis) computed with the basic formula also remains the same.
The intersections of the lines of the base and the revealing layers (marked in the figure by two arrows) lie on a central axis of a light moiré band.
From Figure 8 we deduce the following two equations: From these equations we deduce the equation for computing the inclination of moiré lines as a function of the inclinations of the base layer and the revealing layer lines: The true pattern periods Tb, Tr, and Tm (along the axes perpendicular to pattern lines) are computed as follows (see Figure 8): From here, using the formula for computing tan(αm) with periods p, we deduce a well known formula for computing the moiré angle αm with periods T:[4][5][6] From formula for computing pm we deduce another well known formula for computing the period Tm of moiré pattern (along the axis perpendicular to moiré bands): In the particular case when Tb=Tr=T, the formula for the period Tm is reduced into well known formula: And the formula for computing αm is reduced to: Here is the equation for computing the revealing layer line inclination αr for a given base layer line inclination αb, and a desired moiré line inclination αm: For any given base layer line inclination, this equation permits us to obtain a desired moiré line inclination by properly choosing the revealing layer inclination.
The revealing pattern of Figure 9 is computed by interpolating the curves into connected straight lines, where for each position along the horizontal axis, the revealing line’s inclination angle αr is computed as a function of αb and αm according to the equation above.
Figure 11 shows an animation where we obtain a superposition image with a constant inclination pattern of moiré lines (+30, –30, +30, –30, +30) for continuously modifying pairs of base and revealing layers.
