Abstract

Using geometrical optics, an oblique-shadow method is shown to generate moiré patterns, which under specific conditions can be interpreted as contour lines of the surface. The oblique-shadow method consists of illuminating a viewing grid with equal spaced parallel lines. The light can be either a point source or parallel light, the observation point can be either at finite distance or at infinity. Conditions for which moiré patterns become contour lines of the surface are given.

© 1975 Optical Society of America

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References

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  1. H. Takasaki, Appl. Opt. 9, 1467 (1970).
    [CrossRef] [PubMed]
  2. H. Takasaki, Appl. Opt. 12, 845 (1973).
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    [CrossRef] [PubMed]
  4. P. S. Theocaris, J. Sci. Instrum. 41, 133 (1964).
    [CrossRef]
  5. P. S. Theocarus, Exp. Mech. 5, 153 (1964).
    [CrossRef]
  6. C. Chiang, Brit. J. App. Phys. (J. Phys. D), Ser. 2, 2, 287 (1969).
    [CrossRef]
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    [CrossRef]

1973 (1)

1970 (2)

1969 (1)

C. Chiang, Brit. J. App. Phys. (J. Phys. D), Ser. 2, 2, 287 (1969).
[CrossRef]

1967 (1)

1964 (2)

P. S. Theocaris, J. Sci. Instrum. 41, 133 (1964).
[CrossRef]

P. S. Theocarus, Exp. Mech. 5, 153 (1964).
[CrossRef]

Appl. Opt. (3)

Brit. J. App. Phys. (J. Phys. D), Ser. 2 (1)

C. Chiang, Brit. J. App. Phys. (J. Phys. D), Ser. 2, 2, 287 (1969).
[CrossRef]

Exp. Mech. (1)

P. S. Theocarus, Exp. Mech. 5, 153 (1964).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Sci. Instrum. (1)

P. S. Theocaris, J. Sci. Instrum. 41, 133 (1964).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

A diagram showing the parallel lines on the viewing grid and their shadow on the surface which interfere to form the moiré patterns. S is the illumination point and O is the observation point.

Equations (27)

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x = m a + c 1 ,
x = n b + c 2 ,
x = x 0 d ( x 1 , y 1 ) z 0 + d ( x 1 , y 1 ) + ( n b + c 2 ) z 0 z 0 + d ( x 1 , y 1 ) ,
m n = k ,
x = d ( x 1 , y 1 ) x 0 z 0 + d ( x 1 , y 1 ) + [ ( x c 1 ) b / a k b + c 2 ] z 0 z 0 + d ( x 1 , y 1 ) ,
d ( x 1 , y 1 ) = ( a b ) x a c 2 + b c 1 + a b k a ( x 0 x ) z 0 .
x 0 x 1 x 0 x = y 1 y = z 0 + d ( x 1 , y 1 ) z 0 .
x 1 = ( b x + a c 2 b c 1 a b k ) / a ,
y 1 = y ( a x 0 b x a c 2 + b c 1 + a b k ) / a ( x 0 x ) .
d [ b x + a c 2 b c a b k a , ( a x 0 b x a c 2 + b c 1 + a b k ) y a ( x 0 x ) ] = ( a b ) x a c 2 + b c 1 + a b k a ( x 0 x ) z 0 .
x = n a + c 1 ,
x = ( n a + c 1 ) z 2 + d ( x 1 , y 1 ) z 2 .
b = z 2 + d ( x 1 , y 1 ) z 2 a ,
c 2 = z 2 + d ( x 1 , y 1 ) z 2 c 1 .
d { ( 1 + d / z 2 ) ( x a k ) , [ 1 + a k ( 1 + d / z 2 ) d x / z 2 x 0 x ] y } = a k z 0 z 2 x 0 z 2 a k z 0 + x ( z 0 z 2 ) .
d { ( 1 + d / z 2 ) ( x a k ) , [ 1 + a k ( 1 + d / z 2 ) d x / z 2 x 0 x ] y } = a k z 2 / ( x 0 a k ) .
d { ( 1 + d / z 2 ) ( x a k ) , [ 1 + a k ( 1 + d / z 2 ) x d / z 2 x 0 x ] y } = a k z 0 / x 0 .
x 1 = ( 1 + d / z 2 ) ( x a k ) ,
y 1 = [ 1 + a k ( 1 + d / z 2 ) d x / z 2 x 0 x ] y .
d ( x a k , y ) = a k z 0 / x 0 .
( x 0 x ) / z 0 = tan β .
d { ( 1 + d / z 2 ) ( x a k ) , [ 1 + a k ( 1 + d / z 2 ) d x / z 2 x 0 x ] y } = a k z 2 z 2 tan β + x 0 a k .
d ( x a k , y ) = a k z 2 z 2 tan β + x 0 .
d ( x a k + d tan α , x 0 x + a k d tan α x 0 x y ) = a k z 0 z 0 tan α + x 0 x .
d ( x a k + d tan α , y ) = a z 0 k z 0 tan α + x 0 .
d ( x a k + d tan α , x 0 x + a k d tan α x 0 x y ) = a k ( tan α + tan β ) .
d ( x , y ) = a k / ( tan α + tan β ) .

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