Abstract

The problem of controlling a single ray bundle with a single reflector is not generally solvable, but approximate solutions may often be found that are acceptable for applications. We introduce a new technique for finding such approximations and apply it to the design of a driver-side mirror for an automobile that has no blind spot and minimal distortion.

© 2008 Optical Society of America

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References

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  1. D. G. Burkhard and D. L. Shealy, Sol. Energy 17, 221 (1975).
    [CrossRef]
  2. R. Winston, Selected Papers on Nonimaging Optics (SPIE, 1995).
  3. R. Hicks, J. Opt. Soc. Am. A 22, 323 (2005).
    [CrossRef]
  4. H. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993).
  5. R. A. Hicks and R. Perline, in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision Pattern Recognition (IEEE, 2001), p. 584.
  6. R. Hicks and R. Perline, Appl. Opt. 44, 3893 (2005).
    [CrossRef] [PubMed]
  7. M. Halstead, B. Barsky, S. Klein, and R. Mandell, in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), p. 335.
  8. R. Klette and K. Schluns, Proc. SPIE 2908, 204 (1996).
    [CrossRef]

2005

1996

R. Klette and K. Schluns, Proc. SPIE 2908, 204 (1996).
[CrossRef]

1975

D. G. Burkhard and D. L. Shealy, Sol. Energy 17, 221 (1975).
[CrossRef]

Barsky, B.

M. Halstead, B. Barsky, S. Klein, and R. Mandell, in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), p. 335.

Buchdahl, H.

H. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993).

Burkhard, D. G.

D. G. Burkhard and D. L. Shealy, Sol. Energy 17, 221 (1975).
[CrossRef]

Halstead, M.

M. Halstead, B. Barsky, S. Klein, and R. Mandell, in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), p. 335.

Hicks, R.

Hicks, R. A.

R. A. Hicks and R. Perline, in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision Pattern Recognition (IEEE, 2001), p. 584.

Klein, S.

M. Halstead, B. Barsky, S. Klein, and R. Mandell, in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), p. 335.

Klette, R.

R. Klette and K. Schluns, Proc. SPIE 2908, 204 (1996).
[CrossRef]

Mandell, R.

M. Halstead, B. Barsky, S. Klein, and R. Mandell, in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), p. 335.

Perline, R.

R. Hicks and R. Perline, Appl. Opt. 44, 3893 (2005).
[CrossRef] [PubMed]

R. A. Hicks and R. Perline, in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision Pattern Recognition (IEEE, 2001), p. 584.

Schluns, K.

R. Klette and K. Schluns, Proc. SPIE 2908, 204 (1996).
[CrossRef]

Shealy, D. L.

D. G. Burkhard and D. L. Shealy, Sol. Energy 17, 221 (1975).
[CrossRef]

Winston, R.

R. Winston, Selected Papers on Nonimaging Optics (SPIE, 1995).

Appl. Opt.

J. Opt. Soc. Am. A

Proc. SPIE

R. Klette and K. Schluns, Proc. SPIE 2908, 204 (1996).
[CrossRef]

Sol. Energy

D. G. Burkhard and D. L. Shealy, Sol. Energy 17, 221 (1975).
[CrossRef]

Other

R. Winston, Selected Papers on Nonimaging Optics (SPIE, 1995).

H. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993).

R. A. Hicks and R. Perline, in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision Pattern Recognition (IEEE, 2001), p. 584.

M. Halstead, B. Barsky, S. Klein, and R. Mandell, in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), p. 335.

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Figures (5)

Fig. 1
Fig. 1

Given a correspondence, T, that assigns lines in a bundle to points on a target surface, one can define a vector field N that is hopefully normal to a mirror surface that realizes the correspondence.

Fig. 2
Fig. 2

Given a vector field N, one may construct a surface by integrating radially out from an initial condition.

Fig. 3
Fig. 3

Coordinates of the driver-side mirror problem.

Fig. 4
Fig. 4

Prototype driver-side mirror.

Fig. 5
Fig. 5

Upper image is from a flat driver-side mirror. The lower image is from a free-form mirror designed using Eq. (5), for the same scene.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

back = [ x , y , z ] [ x , y , z ] ,
forward = T ( x z , y z ) [ x , y , z ] T ( x z , y z ) [ x , y , z ] .
d z d r = f x ( r cos θ , r sin θ ) cos θ + f y ( r cos θ , r sin θ ) sin θ .
W ( x , y , z ) = [ W 1 , W 2 , 1 ] = [ N 1 N 3 , N 2 N 3 , 1 ] .
d z d r = W 1 ( r cos θ , r sin θ , z ( r ) ) cos θ W 2 ( r cos θ , r sin θ , z ( r ) ) sin θ .
q = [ a 1 , y a x , z a 1 ] .
T ( q ) = λ ( k + a ) y a x u + λ ( k + a ) z a x v + w .

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