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Moir´ Methods. Triangulation e Figure 3.2 is an illustration of two interfering plane waves. Let us look at the figure for what it really is, namely two gratings that lie in contact, with a small angle between the grating lines. As a result, we see a fringe pattern of much lower frequency than the individual gratings. This is an example of the moir´ effect and the resulting fringes are e called moir´ fringes or a moir´ pattern. Figures 3.4, 3.8 and 3.9 are examples of the same e e effect. The mathematical description of moir´ patterns resulting from the superposition e. | Optical Metrology. Kjell J. Gasvik Copyright 2002 John Wiley Sons Ltd. ISBN 0-470-84300-4 7 Moire Methods. Triangulation 7.1 INTRODUCTION Figure 3.2 is an illustration of two interfering plane waves. Let us look at the figure for what it really is namely two gratings that lie in contact with a small angle between the grating lines. As a result we see a fringe pattern of much lower frequency than the individual gratings. This is an example of the moire effect and the resulting fringes are called moire fringes or a moire pattern. Figures 3.4 3.8 and 3.9 are examples of the same effect. The mathematical description of moire patterns resulting from the superposition of sinusoidal gratings is the same as for interference patterns formed by electromagnetic waves. The moire effect is therefore often termed mechanical interference. The main difference lies in the difference in wavelength which constitutes a factor of about 102 and greater. The moire effect can be observed in our everyday surroundings. Examples are folded fine-meshed curtains moire means watered silk rails on each side of a bridge or staircase nettings etc. Moire as a measurement technique can be traced many years back. Today there is little left of the moire effect but techniques applying gratings and other type of fringes are widely used. In this chapter we go through the theory for superposition of gratings with special emphasis on the fringe projection technique. The chapter ends with a look at a triangulation probe. 7.2 SINUSOIDAL GRATINGS Often gratings applied in moire methods are transparencies with transmittances given by a square-wave function. Instead of square-wave functions we describe linear gratings by sinusoidal transmittances reflectances bearing in mind that all types of periodic gratings can be described as a sum of sinusoidal gratings. A sinusoidal grating of constant frequency is given by t1 x y a a cos x P J 7.1 where p is the grating period and where 0 a 2. The principle behind .