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When one family of curves is superposed on another family of curves, a new family called the moiré pattern appears.
To produce moiré patterns, the lines of the overlapping figures must cross at an angle of less than about 45°. The moiré lines are then the locus of points of intersection. The illustration shows the case of two identical figures of simple gratings of alternate black and white bars of equal spacing. When the figures are crossed at 90°, a checkerboard pattern with no moiré effect is seen. At crossing angles of less than 45°, however, one sees a moiré pattern of equispaced lines, the moiré fringes. The spacing of the fringes increases with decreasing crossing angle. This provides one with a simple method for measuring extremely small angles (down to 1 second of arc). As the angle of crossing approaches zero, the moiré fringes approach 90° with respect to the original figures.
Even when the spacings of the original figures are far below the resolution of the eye, the moiré fringes will still be readily seen. This phenomenon provides a means of checking the fidelity of a replica of a diffraction grating. See Diffraction grating
Moiré techniques are widely used in the stress analysis of metals, in the examination of large optical surfaces, in investigating aberrations of lenses, and in determining a refractive index gradient (for example, that of sugar molecules diffusing into water).