Computing moiré lines’ inclination as function of the inclination of layers’ lines

Figure 5. Identical inclination of layer 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:

${\displaystyle {\begin{cases}\tan \alpha _{m}={\frac {p_{b}+l\cdot \tan \alpha _{b}}{l}}\\\tan \alpha _{r}={\frac {p_{b}-p_{r}+l\cdot \tan \alpha _{b}}{l}}\end{cases}}}$

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:

${\displaystyle \tan \alpha _{m}={\frac {p_{b}\cdot \tan \alpha _{r}-p_{r}\cdot \tan \alpha _{b}}{p_{b}-p_{r}}}}$

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):

${\displaystyle T_{b}=p_{b}\cdot \cos \alpha _{b},\ T_{r}=p_{r}\cdot \cos \alpha _{r},\ T_{m}=p_{m}\cdot \cos \alpha _{m}}$

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:^[4][5][6]

${\displaystyle \alpha _{m}=\arctan \left({\frac {T_{b}\cdot \sin \alpha _{r}-T_{r}\cdot \sin \alpha _{b}}{T_{b}\cdot \cos \alpha _{r}-T_{r}\cdot \cos \alpha _{b}}}\right)}$

From formula for computing p_m we deduce another well known formula for computing the period T_m of moiré pattern (along the axis perpendicular to moiré bands):

${\displaystyle T_{m}={\frac {T_{b}\cdot T_{r}}{\sqrt {T_{b}^{2}+T_{r}^{2}-2\cdot T_{b}\cdot T_{r}\cdot \cos(\alpha _{r}-\alpha _{b})}}}}$

In the particular case when T_b=T_r=T, the formula for the period T_m is reduced into well known formula:

${\displaystyle T_{m}={\frac {T}{2\cdot \sin \left({\frac {\alpha _{r}-\alpha _{b}}{2}}\right)}}}$

And the formula for computing α_m is reduced to:

${\displaystyle \alpha _{m}=90^{\circ }+{\frac {\alpha _{r}+\alpha _{b}}{2}}}$

The 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:

${\displaystyle \tan \alpha _{r}={\frac {p_{r}}{p_{b}}}\cdot \tan \alpha _{b}+\left(1-{\frac {p_{r}}{p_{b}}}\right)\cdot \tan \alpha _{m}.}$

Figure 9. Moiré curves with straight base layer lines

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.

Figure 10. Inversed base layer and moiré 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. The same moiré curves with modifying layer patterns

Effect on circular lines.

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.