Superposition of layers with periodically repeating parallel lines
Simple moiré patterns can be observed when superposing two transparent layers comprising periodically repeating opaque parallel lines as shown in Figure 1. The lines of one layer are parallel to the lines of the second layer. The superposition image does not change if transparent layers with their opaque patterns are inverted. When considering printed samples, one of the layers is denoted as theSpeedup of movements with moiré
The moiré bands of Figure 1 will move if we displace the revealing layer. When the revealing layer moves perpendicularly to layer lines, the moiré bands move along the same axis, but several times faster than the movement of the revealing layer. TheSuperposition of layers with inclined lines
Here we present patterns with inclined lines. When we are interested in optical speedup we can represent the case of inclined patterns such that the formulas for computing moiré periods and optical speedups remain valid in their current simplest form. For this purpose, the values of periods ''p''r, ''p''b, and ''p''m correspond to the distances between the lines along the axis of movements (the vertical axis in the animated example of Figure 4). When the layer lines are perpendicular to the movement axis, the periods (''p'') are equal to the distances (denoted as ''T'') between the lines (as in Figure 4). If the lines are inclined, the periods (''p'') along the axis of the movement are not equal to the distances (''T'') between the lines.Computing moiré lines’ inclination as function of the inclination of layers’ lines
The superposition of two layers with identically inclined lines forms moiré lines inclined at the same angle. Figure 5 is obtained from Figure 1 with a vertical shearing. In Figure 5 the layer lines and the moiré lines are inclined by 10 degrees. Since the inclination is not a rotation, during the inclination the distance (''p'') between the layer lines along the vertical axis is conserved, but the true distance (''T'') between the lines (along an axis perpendicular to these lines) is changed. The difference between the vertical periods ''p''b, ''p''r, and the distances ''T''b, ''T''r is shown in the diagram of Figure 8. The inclination degree of layer lines may change along the horizontal axis forming curves. The superposition of two layers with identical inclination pattern forms moiré curves with the same inclination pattern. In Figure 6 the inclination degree of layer lines gradually changes according to the following sequence of degrees (+30, –30, +30, –30, +30). Layer periods ''p''b and ''p''r represent the distances between the curves along the vertical axis. The presented formulas for computing the period ''p''m (the vertical distance between the moiré curves) and the optical speedup (along the vertical axis) are valid for Figure 6. More interesting is the case when the inclination degrees of layer lines are not the same for the base and revealing layers. Figure 7 shows an animation of a superposition images where the inclination degree of base layer lines is constant (10 degrees), but the inclination of the revealing layer lines oscillates between 5 and 15 degrees. The periods of layers along the vertical axis ''p''b and ''p''r are the same all the time. Correspondingly, the period ''p''m (along the vertical axis) computed with the basic formula also remains the same. Figure 8 helps to compute the inclination degree of moiré optical lines as a function of the inclination of the revealing and the base layer lines. We draw the layer lines schematically without showing their true thicknesses. The bold lines of the diagram inclined by ''α''b degrees are the base layer lines. The bold lines inclined by ''α''r degrees are the revealing layer lines. The base layer lines are vertically spaced by a distance equal to ''p''b, and the revealing layer lines are vertically spaced by a distance equal to ''p''r. The distances ''T''b and ''T''r represent the true space between the base layer and revealing layer lines, correspondingly. 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. The dashed line of Figure 8 corresponds to the axis of the light moiré band. The inclination degree of moiré lines is therefore the inclination ''α''m of the dashed line. 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: :Deducing other known formulas
The true pattern periods ''T''b, ''T''r, and ''T''m (along the axes perpendicular to pattern lines) are computed as follows (see Figure 8): : From here, using the formula for computing tan(''α''m) with periods ''p'', we deduce a well known formula for computing the moiré angle ''α''m with periods ''T'': : From formula for computing ''p''m we deduce another well known formula for computing the period ''T''m ofThe revealing lines inclination as a function of the superposition image’s lines inclination
Here is the equation for computing the revealing layer line inclination ''α''r for a given base layer line inclination ''α''b, and a desired moiré line inclination ''α''m: : 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. In Figure 6 we showed an example where the curves of layers follow an identical inclination pattern forming a superposition image with the same inclination pattern. The inclination degrees of the layers’ and moiré lines change along the horizontal axis according to the following sequence of alternating degree values (+30, –30, +30, –30, +30). In Figure 9 we obtain the same superposition pattern as in Figure 6, but with a base layer comprising straight lines inclined by –10 degrees. 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 ''α''r is computed as a function of ''α''b and ''α''m according to the equation above. Figure 9 demonstrates that the difference between the inclination angles of revealing and base layer lines has to be several times smaller than the difference between inclination angles of moiré and base layer lines. Another example forming the same superposition patterns as in Figure 6 and Figure 9 is shown in Figure 10. In Figure 10 the desired inclination pattern (+30, –30, +30, –30, +30) is obtained using a base layer with an inverted inclination pattern (–30, +30, –30, +30, –30). 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. The base layer inclination pattern gradually changes and the revealing layer inclination pattern correspondingly adapts such that the superposition image’s inclination pattern remains the same.References
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