2046:
616:
2054:
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
72:
2083:
2095:
35:
26:; a pattern that appears when superposing two transparent layers containing correlated opaque patterns. Line moiré is the case when the superposed patterns comprise straight or curved lines. When moving the layer patterns, the moiré patterns transform or move at a faster speed. This effect is called optical moiré speedup.
1469:
946:
50:
and the other one as the revealing layer. It is assumed that the revealing layer is printed on a transparency and is superimposed on top of the base layer, which can be printed either on a transparency or on an opaque paper. The periods of the two layer patterns are close. We denote the period of the
759:
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
300:
For the case when the revealing layer period is longer than the base layer period, the distance between moiré bands is the absolute value computed by the formula. The superposition of two layers comprising parallel lines forms an optical image comprising parallel moiré lines with a magnified period.
2053:
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
79:
Light bands of the superposition image correspond to the zones where the lines of both layers overlap. The dark bands of the superposition image forming the moiré lines correspond to the zones where the lines of the two layers interleave, hiding the white background. The labels of Figure 2 show the
623:
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
690:
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
664:
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
2110:
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
568:
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
2040:
94:
of moiré lines is the distance from one point where the lines of both layers overlap (at the bottom of the figure) to the next such point (at the top). Let us count the layer lines, starting from the bottom point. At the count 0 the lines of both layers overlap. Since in our case
1657:
2090:
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).
716:
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
1267:
1080:
559:
In case the period of the revealing layer is longer than the period of the base layer, the optical image moves in the opposite direction. The negative value of the ratio computed according to this formula signifies a movement in the reverse direction.
326:
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.
1300:
1779:
776:
42:
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.
1864:
2078:
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.
554:
1901:
1498:
342:
shown in Figure 4 corresponds to a slow movement of the revealing layer. The GIF file repeatedly animates an upward movement of the revealing layer (perpendicular to layer lines) across a distance equal to
453:
295:
590:
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
68:
The superposition image of Figure 1 outlines periodically repeating dark parallel bands, called moiré lines. Spacing between the moiré lines is much larger than the periods of lines in the two layers.
1117:
360:) of its pattern, the superposition optical image must be the same as the initial one. It means that the moiré lines traverse a distance equal to the period of the superposition image
957:
109:, for the same number of counted lines, the base layer lines with a long period advance faster than the revealing layer lines with a short period. At the halfway of the distance
84:
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).
46:
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 the
2364:: The basics of line moiré patterns and optical speedup; equations for computing the contours and the velocities of moiré curves; circular patterns and rotational movements
951:
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:
1464:{\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)}
941:{\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}}}
1693:
2128:
350:. The animation demonstrates that the moiré lines of the superposition image move up at a speed, much faster than the movement speed of the revealing layer.
80:
passages from light zones with overlapping layer lines to dark zones with interleaving layer lines. The light and dark zones are periodically interchanging.
2277:
331:
2172:
2241:
1794:
2374:
311:
The thicknesses of layer lines affect the overall darkness of the superposition image and the thickness of the moiré bands, but the period
2035:{\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}.}
473:
1652:{\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})}}}}
2189:
387:
213:
2103:
686:(the vertical distance between the moiré curves) and the optical speedup (along the vertical axis) are valid for Figure 6.
123:/2) of the revealing layer lines, due to which the lines are interleaving, forming a dark moiré band. At the full distance
1262:{\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}}
632:) between the lines (along an axis perpendicular to these lines) is changed. The difference between the vertical periods
2094:
2045:
615:
339:
679:
represent the distances between the curves along the vertical axis. The presented formulas for computing the period
2147:
2300:
71:
1075:{\displaystyle \tan \alpha _{m}={\frac {p_{b}\cdot \tan \alpha _{r}-p_{r}\cdot \tan \alpha _{b}}{p_{b}-p_{r}}}}
2195:
2252:
2082:
2367:
2111:
pattern correspondingly adapts such that the superposition image’s inclination pattern remains the same.
731:
degrees are the revealing layer lines. The base layer lines are vertically spaced by a distance equal to
2398:
2378:
2361:
330:
381:=0), the following equation represents the ratio of the optical speed to the revealing layer’s speed:
34:
2333:
785:
2278:"Moiré patterns: their application to refractive index and refractive index gradient measurements"
2220:
1774:{\displaystyle T_{m}={\frac {T}{2\cdot \sin \left({\frac {\alpha _{r}-\alpha _{b}}{2}}\right)}}}
2408:
2403:
2393:
2185:
353:
When the revealing layer is shifted up perpendicularly to the layer lines by one full period (
2341:
2292:
2139:
1870:
The revealing lines inclination as a function of the superposition image’s lines inclination
1489:
23:
691:
lines oscillates between 5 and 15 degrees. The periods of layers along the vertical axis
2337:
1111:(along the axes perpendicular to pattern lines) are computed as follows (see Figure 8):
2219:
Emin
Gabrielyan (2007-03-08). "The basics of line moiré patterns and optical speedup".
628:) between the layer lines along the vertical axis is conserved, but the true distance (
2345:
308:, the closer the periods of the two layers, the stronger the magnification factor is.
2387:
760:
light moiré band. The inclination degree of moiré lines is therefore the inclination
137:, so the lines of the layers again overlap. The base layer lines gain the distance
611:
Computing moiré lines’ inclination as function of the inclination of layers’ lines
712:(along the vertical axis) computed with the basic formula also remains the same.
130:, the base layer lines are ahead of the revealing layer lines by a full period
116:, the base layer lines are ahead the revealing layer lines by a half a period (
599:) between the lines (as in Figure 4). If the lines are inclined, the periods (
47:
738:, and the revealing layer lines are vertically spaced by a distance equal to
1859:{\displaystyle \alpha _{m}=90^{\circ }+{\frac {\alpha _{r}+\alpha _{b}}{2}}}
2296:
2143:
2225:
1874:
Here is the equation for computing the revealing layer line inclination
2102:
665:
following sequence of degrees (+30, –30, +30, –30, +30). Layer periods
2181:
367:
while the revealing layer traverses the distance equal to its period
549:{\displaystyle {\frac {v_{m}}{v_{r}}}={\frac {p_{b}}{p_{b}-p_{r}}}.}
2101:
30:
Superposition of layers with periodically repeating parallel lines
603:) along the axis of the movement are not equal to the distances (
2240:
Stanley Morse; August J. Durelli; Cesar A. Sciammarella (1961).
1283:, we deduce a well known formula for computing the moiré angle
200:+ 1. From here we obtain the well known formula for the period
2098:
Figure 11. The same moiré curves with modifying layer patterns
1481:
we deduce another well known formula for computing the period
448:{\displaystyle {\frac {v_{m}}{v_{r}}}={\frac {p_{m}}{p_{r}}}.}
290:{\displaystyle p_{m}={\frac {p_{b}\cdot p_{r}}{p_{b}-p_{r}}}.}
724:
degrees are the base layer lines. The bold lines inclined by
934:
2249:
Transactions of the
American Society of Civil Engineers
2136:
Transactions of the
American Society of Civil Engineers
705:
are the same all the time. Correspondingly, the period
2049:
Figure 9. Moiré curves with straight base layer lines
1904:
1797:
1696:
1501:
1303:
1120:
960:
779:
770:
From Figure 8 we deduce the following two equations:
476:
390:
334:
Figure 4. Slow movement of the revealing layer upward
216:
2034:
1858:
1773:
1651:
1463:
1261:
1074:
940:
548:
447:
289:
2324:G. Oster; Y. Nishijima (1963). "Moiré patterns".
2129:"Moiré fringes as a means of analyzing strains"
1492:(along the axis perpendicular to moiré bands):
1272:From here, using the formula for computing tan(
318:does not depend on the layer lines’ thickness.
2242:"Geometry of moiré fringes in strain analysis"
2086:Figure 10. Inversed base layer and moiré lines
619:Figure 5. Identical inclination of layer lines
158:) as the number of the revealing layer lines (
8:
374:. Assuming that the base layer is immobile (
75:Figure 2. Overlapping and interleaving zones
564:Superposition of layers with inclined lines
2285:Journal of the Optical Society of America
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58:and the period of the revealing layer as
2127:C.A. Sciammarella; A.J. Durelli (1962).
2093:
2081:
2044:
1881:for a given base layer line inclination
614:
329:
70:
38:Figure 1. Two layers with parallel lines
33:
2119:
1888:, and a desired moiré line inclination
301:According to the formula for computing
2251:. 126, part I: 250–271. Archived from
660:is shown in the diagram of Figure 8.
7:
1687:is reduced into well known formula:
172:) for the same distance minus one:
2373:Mirrors of line moiré intro page:
2174:The Theory of the Moiré Phenomenon
14:
2346:10.1038/scientificamerican0563-54
2075:according to the equation above.
2276:Y. Nishijima; G. Oster (1964).
1784:And the formula for computing α
322:Speedup of movements with moiré
1643:
1617:
1:
2370:: Aperiodic random line moiré
2061:is computed as a function of
1680:, the formula for the period
1086:Deducing other known formulas
1662:In the particular case when
207:of the superposition image:
1474:From formula for computing
2425:
624:inclination the distance (
465:with its formula, we have
2106:Effect on circular lines.
1090:The true pattern periods
2138:. 127, part I: 582–587.
2297:10.1364/JOSA.54.000001
2171:Isaac Amidror (2000).
2144:10.1061/TACEAT.0008466
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607:) between the lines.
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16:Type of moiré pattern
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1118:
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767:of the dashed line.
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214:
144:with as many lines (
2362:Line moiré patterns
2338:1963SciAm.208e..54O
2326:Scientific American
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2368:Random line moiré
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2146:. Archived from
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2226:physics/0703098
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1279:) with periods
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258:
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2356:External links
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51:base layer as
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28:
15:
13:
10:
9:
6:
4:
3:
2:
2421:
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2320:
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2306:on 2007-10-13
2302:
2298:
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2279:
2272:
2269:
2258:on 2007-10-08
2254:
2250:
2243:
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2222:
2215:
2212:
2201:on 2007-10-13
2197:
2193:
2191:0-7923-5950-X
2187:
2183:
2176:
2175:
2167:
2164:
2153:on 2007-12-11
2149:
2145:
2141:
2137:
2130:
2123:
2120:
2114:
2112:
2104:
2096:
2092:
2084:
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2071:
2064:
2057:
2047:
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2024:
2020:
2016:
2013:
2010:
2006:
1998:
1994:
1988:
1984:
1978:
1975:
1971:
1967:
1962:
1958:
1954:
1951:
1948:
1941:
1937:
1931:
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1532:
1528:
1523:
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1490:moiré pattern
1484:
1477:
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1290:with periods
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1126:
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1016:
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977:
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968:
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961:
954:
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952:
926:
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903:
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895:
891:
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858:
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840:
836:
832:
829:
826:
823:
820:
815:
811:
804:
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791:
788:
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773:
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771:
768:
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748:
741:
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494:
490:
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467:
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458:By replacing
442:
435:
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384:
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377:
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340:GIF animation
332:
328:
321:
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105:
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90:
85:
81:
73:
69:
66:
61:
54:
49:
44:
36:
29:
27:
25:
24:moiré pattern
21:
2399:Interference
2329:
2325:
2319:
2308:. Retrieved
2301:the original
2288:
2284:
2271:
2260:. Retrieved
2253:the original
2248:
2235:
2214:
2203:. Retrieved
2196:the original
2173:
2166:
2155:. Retrieved
2148:the original
2135:
2122:
2109:
2089:
2077:
2069:
2062:
2055:
2052:
1889:
1882:
1875:
1873:
1783:
1681:
1677:
1670:
1663:
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1291:
1284:
1280:
1273:
1271:
1105:
1098:
1091:
1089:
950:
769:
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753:
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629:
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375:
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110:
103:
96:
88:
86:
82:
78:
67:
59:
52:
45:
41:
19:
18:
2379:Switzerland
87:The period
2388:Categories
2310:2007-03-19
2291:(1): 1–5.
2262:2007-03-19
2205:2007-03-19
2157:2007-03-19
2115:References
48:base layer
20:Line moiré
2021:α
2017:
2011:⋅
1979:−
1959:α
1955:
1949:⋅
1913:α
1909:
1842:α
1829:α
1817:∘
1800:α
1750:α
1746:−
1737:α
1726:
1720:⋅
1635:α
1631:−
1622:α
1615:
1609:⋅
1596:⋅
1583:⋅
1577:−
1529:⋅
1446:α
1442:
1436:⋅
1423:−
1414:α
1410:
1404:⋅
1383:α
1379:
1373:⋅
1360:−
1351:α
1347:
1341:⋅
1321:
1306:α
1251:α
1247:
1241:⋅
1203:α
1199:
1193:⋅
1155:α
1151:
1145:⋅
1057:−
1036:α
1032:
1026:⋅
1013:−
1004:α
1000:
994:⋅
969:α
965:
917:α
913:
907:⋅
888:−
863:α
859:
837:α
833:
827:⋅
796:α
792:
528:−
269:−
244:⋅
2409:Printing
2404:Patterns
2394:Geometry
2334:Bibcode
2188:
2182:Kluwer
1318:arctan
1215:
1167:
1104:, and
583:, and
2304:(PDF)
2281:(PDF)
2256:(PDF)
2245:(PDF)
2221:arXiv
2199:(PDF)
2178:(PDF)
2151:(PDF)
2132:(PDF)
2186:ISBN
2068:and
752:and
698:and
672:and
338:The
102:<
2375:USA
2342:doi
2330:208
2293:doi
2140:doi
2014:tan
1952:tan
1906:tan
1723:sin
1612:cos
1488:of
1439:cos
1407:cos
1376:sin
1344:sin
1244:cos
1196:cos
1148:cos
1029:tan
997:tan
962:tan
910:tan
856:tan
830:tan
789:tan
576:,
2390::
2377:,
2340:.
2328:.
2289:54
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2247:.
2184:.
2180:.
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2348:.
2344::
2336::
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2265:.
2229:.
2223::
2208:.
2160:.
2142::
2073:m
2070:α
2066:b
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2059:r
2056:α
2030:.
2025:m
2007:)
1999:b
1995:p
1989:r
1985:p
1976:1
1972:(
1968:+
1963:b
1942:b
1938:p
1932:r
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1922:=
1917:r
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1883:α
1879:r
1876:α
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1838:+
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1809:=
1804:m
1786:m
1765:)
1760:2
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1741:r
1730:(
1717:2
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1618:(
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1513:=
1508:m
1504:T
1486:m
1483:T
1479:m
1476:p
1458:)
1450:b
1431:r
1427:T
1418:r
1399:b
1395:T
1387:b
1368:r
1364:T
1355:r
1336:b
1332:T
1325:(
1315:=
1310:m
1292:T
1288:m
1285:α
1281:p
1277:m
1274:α
1255:m
1236:m
1232:p
1228:=
1223:m
1219:T
1212:,
1207:r
1188:r
1184:p
1180:=
1175:r
1171:T
1164:,
1159:b
1140:b
1136:p
1132:=
1127:b
1123:T
1109:m
1106:T
1102:r
1099:T
1095:b
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1065:r
1061:p
1052:b
1048:p
1040:b
1021:r
1017:p
1008:r
989:b
985:p
978:=
973:m
927:l
921:b
904:l
901:+
896:r
892:p
883:b
879:p
872:=
867:r
847:l
841:b
824:l
821:+
816:b
812:p
805:=
800:m
783:{
765:m
762:α
757:r
754:T
750:b
747:T
743:r
740:p
736:b
733:p
729:r
726:α
722:b
719:α
710:m
707:p
703:r
700:p
696:b
693:p
684:m
681:p
677:r
674:p
670:b
667:p
658:r
655:T
651:b
648:T
644:r
641:p
637:b
634:p
630:T
626:p
605:T
601:p
597:T
593:p
591:(
588:m
585:p
581:b
578:p
574:r
571:p
544:.
536:r
532:p
523:b
519:p
512:b
508:p
502:=
495:r
491:v
485:m
481:v
463:m
460:p
443:.
436:r
432:p
426:m
422:p
416:=
409:r
405:v
399:m
395:v
379:b
376:v
372:r
369:p
365:m
362:p
358:r
355:p
348:r
345:p
316:m
313:p
306:m
303:p
285:.
277:r
273:p
264:b
260:p
252:r
248:p
239:b
235:p
228:=
223:m
219:p
205:m
202:p
198:b
195:p
193:/
191:m
188:p
184:r
181:p
179:/
177:m
174:p
170:r
167:p
165:/
163:m
160:p
156:b
153:p
151:/
149:m
146:p
142:m
139:p
135:r
132:p
128:m
125:p
121:r
118:p
114:m
111:p
107:b
104:p
100:r
97:p
92:m
89:p
63:r
60:p
56:b
53:p
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