Abstract

For a number of years, observations have been made of contour-like patterns created by the moiré interference of a grid with its shadow cast onto a surface. The conditions under which these patterns correspond to planar contours are derived, and systems to generate these contours are suggested. It is shown that moiré contour patterns provide a useful and flexible means of contouring surfaces. Experimental results are presented.

© 1970 Optical Society of America

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References

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  1. K. A. Haines, B. P. Hildebrand, Appl. Opt. 8, 595 (1969).
  2. P. S. Theocarus, Exp. Mech. 5, 153 (1964).
    [CrossRef]

1969 (1)

K. A. Haines, B. P. Hildebrand, Appl. Opt. 8, 595 (1969).

1964 (1)

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

Haines, K. A.

K. A. Haines, B. P. Hildebrand, Appl. Opt. 8, 595 (1969).

Hildebrand, B. P.

K. A. Haines, B. P. Hildebrand, Appl. Opt. 8, 595 (1969).

Theocarus, P. S.

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

Appl. Opt. (1)

K. A. Haines, B. P. Hildebrand, Appl. Opt. 8, 595 (1969).

Exp. Mech. (1)

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

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Figures (6)

Fig. 1
Fig. 1

Moiré contouring system with the source and observer at infinity.

Fig. 2
Fig. 2

Moiré contouring system with the source and observer at finite distances from the grid. When h1 = h2, the moiré patterns formed correspond to true contours.

Fig. 3
Fig. 3

Laboratory setup used to generate the moiré contour photographs in Figs. 46. The apparatus consists of a 25-line/in. (~1-line/mm) grid, two quartz–iodine lamps, and a camera.

Fig. 4
Fig. 4

A 46-cm scale model of the C-5A contoured to a resolution of 5 mm using moiré patterns.

Fig. 5
Fig. 5

A model car is contoured with resolutions of 6.3 mm, 2.16 mm, 1.04 mm, and 0.63 mm, respectively, illustrating the flexibility of moiré contouring systems.

Fig. 6
Fig. 6

A conical object is contoured using moiré patterns with resolutions ranging from 1.02 mm at the top to 1.07 mm at the bottom.

Equations (34)

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T ( x ) = 1 2 + 1 2 sin ( 2 π x / p ) ,
I 0 ( x , y ) = ( B / 2 ) cos ϕ [ x , y , z ( x , y ) ] { sin ( 2 π / p ) [ x z ( x , y ) tan α ] + 1 } .
I 0 ( x , y ) = K { sin ( 2 π / p ) [ x z ( x , y ) tan α ] + 1 } ,
I 1 ( x , y ) = ( K / 2 ) [ sin ( 2 π / p ) ( x z tan α ) + 1 ] [ sin ( 2 π / p ) × ( x + z tan β ) + 1 ] = ( K / 2 ) [ 1 + sin ( 2 π / p ) × ( x z tan α ) + sin ( 2 π / p ) ( x + z tan β ) + sin ( 2 π / p ) ( x z tan α ) sin ( 2 π / p ) ( x + z tan β ) ] .
I 1 ( x , y ) = ( K / 2 ) [ 1 + sin ( 2 π / p ) ( x z tan α ) + ( 2 π / p ) ( x + z tan β ) 1 2 cos ( 2 π / p ) [ 2 x + z ( tan β tan α ) ] + 1 2 cos ( 2 π z / p ) ( tan α + tan β ) ] .
Δ z = N p / ( tan α + tan β ) .
p = { [ h 1 + z ( x , y ) ] / h 1 } p .
I 2 ( x , y ) = I r 2 [ x , y , z ( x , y ) ] ( 1 2 + 1 2 sin { 2 π h 1 x [ h 1 + z ( x , y ) ] p } ) × cos ϕ [ x , y , z ( x , y ) ] ,
I 3 ( x , y ) = I r 2 cos ϕ [ x , y , z ( x , y ) ] [ 1 2 + 1 2 sin 2 π h 1 x p ( h 1 + z ) ] × [ 1 2 + 1 2 sin 2 π p ( d z + h 2 x h 2 + z ) ] = I 4 r 2 cos ϕ ( x , y , z ) [ 1 + sin 2 π p ( h 1 x h 1 + z ) + sin 2 π p ( d z + h 2 x h 2 + z ) + sin 2 π p ( h 1 x h 1 + z ) sin 2 π p ( d z + h 2 x h 2 + z ) ] .
I 3 ( x , y ) = I 4 r 2 cos ϕ ( x , y , z ) [ 1 + sin 2 π p ( h 1 x h 1 + z ) + sin 2 π p ( d z + h 2 x h 2 + z ) + 1 2 cos 2 π p ( h 1 x h 1 + z d z + h 2 x h 2 + z ) 1 2 cos 2 π p ( h 1 x h 1 + z + d z + h 2 x h 2 + z ) ] .
[ I / 4 r 2 ( x , y , z ) ] cos ϕ ( x , y , z ) = C
I 3 ( x , y ) = C [ 1 + sin 2 π p ( h x h + z ) + sin 2 π p ( d z + h x h + z ) 1 2 cos 2 π p ( 2 h x + d z h + z ) + 1 2 cos 2 π p ( d z h + z ) ] .
( C / 2 ) cos [ 2 π d z / p ( h + z ) ] ,
2 π d z 1 / p ( h + z 1 ) = 2 π N 1
2 π d z 2 / p ( h + z 2 ) = 2 π N 2 .
Δ z = N 2 p h d p N 2 N 1 p h d p N 1 .
( C / 2 ) cos [ 2 π d z / p ( h + z ) ]
2 π z d p ( h + z ) = 2 π d p ( z h z 2 h 2 + z 3 h 3 + ) 2 π d z h p .
( C / 2 ) cos ( 2 π d z / p h ) .
Δ z = z 2 z 1 = N p h / d .
( C / 2 ) cos [ 2 π d z / p ( h + z ) ]
T ( x ) = 1 2 + 1 2 g ( 2 π x / p ) ,
I 4 ( x , y ) = C { 1 + g [ 2 π h x p ( h + z ) ] } { 1 + g [ 2 π ( d z + h x ) p ( h + z ) ] } = C [ 1 + g ( 2 π h x p ( h + z ) ) + g ( 2 π ( d z + h x ) p ( h + z ) ) + g ( 2 π h x p ( h + z ) ) g ( 2 π ( d z + h x ) p ( h + z ) ) ] .
g ( 2 π x p ) = 1 a n sin 2 π n x p + b n cos 2 π n x p ,
I 4 ( x , y ) = C { 1 + 1 a n sin 2 π x n h p ( h + z ) + b n cos 2 π x n h p ( h + z ) + 1 a n sin 2 π n ( d z + h x ) p ( h + z ) + b n cos 2 π n ( d z + h x ) p ( h + z ) + [ 1 a n sin 2 π n p ( d z + h x h + z ) + b n cos 2 π n p ( d z + h x h + z ) ] × [ 1 a n sin 2 π n h x p ( h + z ) + b n cos 2 π n h x p ( h + z ) ] } .
I 4 ( x , y ) = C { 1 + g [ 2 π h x p ( h + z ) ] + g [ 2 π ( d z + h x ) p ( h + z ) ] + 1 ( a n 2 + b n 2 2 ) cos [ 2 π n d x p ( h + z ) ] + a n b n sin 2 π n p ( d z + 2 h x h + z ) + b n 2 a n 2 2 cos 2 π n p ( d z + 2 h x h + z ) + 1 1 m n ( a m a n + b n b m 2 ) cos 2 π p [ ( m n ) h x n d z h + z ] + ( b m b n a m a n 2 ) cos 2 π p [ ( m + n ) h x + n d z h + z ] + a m b n sin 2 π p [ ( m + n ) h x + n d z h + z ] } .
f ( z ) = C [ 1 + 1 1 2 ( a n 2 + b n 2 ) cos 2 π p ( n d z h + z ) ] .
2 π d p ( z n z + h ) = 2 π n d p ( z h z 2 h 2 + z 3 h 3 + ) = 2 π n d p ( z h ) .
f ( z ) = C [ 1 + 1 1 2 ( a n 2 + b n 2 ) cos 2 π n d z p h ] .
Δ z = z 2 z 1 = N 2 p h d p N 2 N 1 p h d p N 1
Δ z = N p h / d ,
c n = 1 2 ( a n 2 + b n 2 ) .
T ( x ) = 1 2 + 2 π n = 1 n odd 1 n sin 2 π n x p .
f ( z ) = C [ 1 + 2 π 2 n = 1 n odd 1 n 2 cos 2 π p ( n d z h + z ) ] .

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