Abstract

We apply the screw triangle method (STM), which was conventionally used in the field of mechanism design, to design a prism to produce a specified image orientation change (IOC). Compared to existing methods [ Tsai and Lin, Appl. Opt. 45, 3951 (2006) ; Appl. Opt. 46, 3087 (2007) ], the proposed method is both simpler and more efficient, since its derivations are essentially vector-based calculations. The validity of the proposed approach is demonstrated via an illustrative example.

© 2008 Optical Society of America

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References

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  1. W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 100-121.
  2. C. Y. Tsai and P. D. Lin, Appl. Opt. 45, 3951 (2006).
    [CrossRef] [PubMed]
  3. C. Y. Tsai and P. D. Lin, Appl. Opt. 46, 3087 (2007).
    [CrossRef] [PubMed]
  4. B. Roth, J. Eng. Ind. 89, 102 (1967).
    [CrossRef]
  5. L. W. Tsai and B. Roth, Mech. Mach. Theory 7, 85 (1972).
    [CrossRef]
  6. J. S. Dai, Mech. Mach. Theory 41, 41 (2006).
    [CrossRef]
  7. H. Lipkin and J. Duffy, Proc. Inst. Mech. Eng., IMechE Conf. 216, 1 (2002).
    [CrossRef]
  8. R. P. Paul, Robot Manipulators--Mathematics, Programming and Control (MIT Press, 1982).
  9. M. R. Spiegel, Schaum's Outline of Vector Analysis (McGraw-Hill, 1989).

2007 (1)

2006 (2)

2002 (1)

H. Lipkin and J. Duffy, Proc. Inst. Mech. Eng., IMechE Conf. 216, 1 (2002).
[CrossRef]

2001 (1)

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 100-121.

1989 (1)

M. R. Spiegel, Schaum's Outline of Vector Analysis (McGraw-Hill, 1989).

1982 (1)

R. P. Paul, Robot Manipulators--Mathematics, Programming and Control (MIT Press, 1982).

1972 (1)

L. W. Tsai and B. Roth, Mech. Mach. Theory 7, 85 (1972).
[CrossRef]

1967 (1)

B. Roth, J. Eng. Ind. 89, 102 (1967).
[CrossRef]

Dai, J. S.

J. S. Dai, Mech. Mach. Theory 41, 41 (2006).
[CrossRef]

Duffy, J.

H. Lipkin and J. Duffy, Proc. Inst. Mech. Eng., IMechE Conf. 216, 1 (2002).
[CrossRef]

Lin, P. D.

Lipkin, H.

H. Lipkin and J. Duffy, Proc. Inst. Mech. Eng., IMechE Conf. 216, 1 (2002).
[CrossRef]

Paul, R. P.

R. P. Paul, Robot Manipulators--Mathematics, Programming and Control (MIT Press, 1982).

Roth, B.

L. W. Tsai and B. Roth, Mech. Mach. Theory 7, 85 (1972).
[CrossRef]

B. Roth, J. Eng. Ind. 89, 102 (1967).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 100-121.

Spiegel, M. R.

M. R. Spiegel, Schaum's Outline of Vector Analysis (McGraw-Hill, 1989).

Tsai, C. Y.

Tsai, L. W.

L. W. Tsai and B. Roth, Mech. Mach. Theory 7, 85 (1972).
[CrossRef]

Appl. Opt. (2)

J. Eng. Ind. (1)

B. Roth, J. Eng. Ind. 89, 102 (1967).
[CrossRef]

Mech. Mach. Theory (2)

L. W. Tsai and B. Roth, Mech. Mach. Theory 7, 85 (1972).
[CrossRef]

J. S. Dai, Mech. Mach. Theory 41, 41 (2006).
[CrossRef]

Proc. Inst. Mech. Eng., IMechE Conf. (1)

H. Lipkin and J. Duffy, Proc. Inst. Mech. Eng., IMechE Conf. 216, 1 (2002).
[CrossRef]

Other (3)

R. P. Paul, Robot Manipulators--Mathematics, Programming and Control (MIT Press, 1982).

M. R. Spiegel, Schaum's Outline of Vector Analysis (McGraw-Hill, 1989).

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 100-121.

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Figures (4)

Fig. 1
Fig. 1

Achieving the specified image orientation change, rot ( m 0 , Φ 0 ) , via two successive rotations, namely, rot ( m 2 , Φ 2 ) followed by rot ( m 3 , Φ 3 ) .

Fig. 2
Fig. 2

Screw triangle geometry [4, 5].

Fig. 3
Fig. 3

Implementation of Fig. 1 for a prism with two reflectors using the STM applied in the mechanical engineering field for mechanism design [4, 5].

Fig. 4
Fig. 4

Illustrative example of prism design.

Equations (14)

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Γ 0 = [ a 0 b 0 c 0 ] = [ a 0 x b 0 x c 0 x a 0 y b 0 y c 0 y a 0 z b 0 z c 0 z ] .
Γ 0 = rot ( m 0 , Φ 0 ) .
tan Φ 0 2 = m 3 ( m 0 × m 2 ) ( m 3 × m 0 ) ( m 0 × m 2 ) ,
tan Φ 2 2 = m 3 ( m 0 × m 2 ) ( m 0 × m 2 ) ( m 2 × m 3 ) ,
tan Φ 3 2 = m 3 ( m 0 × m 2 ) ( m 2 × m 3 ) ( m 3 × m 0 ) .
Γ 0 = [ a 0 b 0 c 0 ] = l 3 ( n 3 ) l 2 l 2 ( n 2 ) l 1 ,
Γ 0 = rot ( m 0 , Φ 0 ) .
Γ 0 = rot ( m 0 , Φ 0 ) = rot ( m 3 , Φ 3 ) rot ( m 2 , Φ 2 ) .
m 3 { ( m 0 × m 2 ) + ( tan Φ 0 2 ) [ m 0 × ( m 0 × m 2 ) ] } = 0 .
m 3 b 0 = 0 .
m 3 = { ( m 0 × m 2 ) + ( tan Φ 0 2 ) [ m 0 × ( m 0 × m 2 ) ] } × b 0 { ( m 0 × m 2 ) + ( tan Φ 0 2 ) [ m 0 × ( m 0 × m 2 ) ] } × b 0 ,
Γ 0 = l 4 ( n 4 ) l 3 l 3 ( n 3 ) l 2 l 2 ( n 2 ) l 1 .
Γ 0 = l 3 ( n 3 ) l 2 l 2 ( n 2 ) l 1 = [ l 4 ( n 4 ) l 3 ] 1 Γ 0
Γ 0 = [ a 0 b 0 c 0 ] = [ 0.229316 0.843802 0.485192 0.842691 0.421574 0.334883 0.487120 0.332073 0.807739 ] .

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