Abstract

We address the problem of determining the reflection point on a specular surface where a light ray that travels from a source to a target is reflected. The specular surfaces considered are those expressed by a quadratic equation. So far, there is no closed form explicit equation for the general solution of this determination of the reflection point, and the usual approach is to use the Snell law or the Fermat principle whose equations are derived in multidimensional nonlinear minimizations. We prove in this Letter that one can impose a set of three restrictions to the reflection point that can impose a set of three restrictions that culminates in a very elegant formalism of searching the reflection point in a unidimensional curve in space. This curve is the intersection of two quadratic equations. Some applications of this framework are also discussed.

© 2010 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, 1965).
  2. E. Hecht, Optics (Addison-Wesley, 1987).
  3. L. Szirmay-Kalos, T. Umenhoffer, G. Patow, L. Szécsi, and M. Sbert, in Computer Graphics Forum (2009), pp. 1-31.
  4. K. Nielsen and N. Christensen, Journal of WSCG (2002), Vol. 10, pp. 91-98.
  5. P. Estalella, I. Martin, G. Drettakis, and D. Tost, in Eurographics Symposium on Rendering (2006), pp. 312-318.
  6. P. Estalella, I. Martin, G. Drettakis, D. Tost, O. Devillers, and F. Cazals, in Proceedings of Vision Modeling and Visualization (2005), pp. 471-478.
  7. D. Roger and N. Holzschuch, in Proceedings of Eurographics (2006), Vol. 25.
  8. J. Stolfi, Oriented Projective Geometry (Academic, 1991).
  9. J. Levin, Comput. Graph. Image Process. 11, 73 (1979).
    [CrossRef]
  10. L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).
    [CrossRef]
  11. L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 192 (2008).
    [CrossRef]
  12. L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 216 (2008).
    [CrossRef]

2008 (3)

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 192 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 216 (2008).
[CrossRef]

1979 (1)

J. Levin, Comput. Graph. Image Process. 11, 73 (1979).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1965).

Cazals, F.

P. Estalella, I. Martin, G. Drettakis, D. Tost, O. Devillers, and F. Cazals, in Proceedings of Vision Modeling and Visualization (2005), pp. 471-478.

Christensen, N.

K. Nielsen and N. Christensen, Journal of WSCG (2002), Vol. 10, pp. 91-98.

Devillers, O.

P. Estalella, I. Martin, G. Drettakis, D. Tost, O. Devillers, and F. Cazals, in Proceedings of Vision Modeling and Visualization (2005), pp. 471-478.

Drettakis, G.

P. Estalella, I. Martin, G. Drettakis, and D. Tost, in Eurographics Symposium on Rendering (2006), pp. 312-318.

P. Estalella, I. Martin, G. Drettakis, D. Tost, O. Devillers, and F. Cazals, in Proceedings of Vision Modeling and Visualization (2005), pp. 471-478.

Dupont, L.

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 192 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 216 (2008).
[CrossRef]

Estalella, P.

P. Estalella, I. Martin, G. Drettakis, D. Tost, O. Devillers, and F. Cazals, in Proceedings of Vision Modeling and Visualization (2005), pp. 471-478.

P. Estalella, I. Martin, G. Drettakis, and D. Tost, in Eurographics Symposium on Rendering (2006), pp. 312-318.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1987).

Holzschuch, N.

D. Roger and N. Holzschuch, in Proceedings of Eurographics (2006), Vol. 25.

Lazard, D.

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 216 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 192 (2008).
[CrossRef]

Lazard, S.

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 192 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 216 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).
[CrossRef]

Levin, J.

J. Levin, Comput. Graph. Image Process. 11, 73 (1979).
[CrossRef]

Martin, I.

P. Estalella, I. Martin, G. Drettakis, and D. Tost, in Eurographics Symposium on Rendering (2006), pp. 312-318.

P. Estalella, I. Martin, G. Drettakis, D. Tost, O. Devillers, and F. Cazals, in Proceedings of Vision Modeling and Visualization (2005), pp. 471-478.

Nielsen, K.

K. Nielsen and N. Christensen, Journal of WSCG (2002), Vol. 10, pp. 91-98.

Patow, G.

L. Szirmay-Kalos, T. Umenhoffer, G. Patow, L. Szécsi, and M. Sbert, in Computer Graphics Forum (2009), pp. 1-31.

Petitjean, S.

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 216 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 192 (2008).
[CrossRef]

Roger, D.

D. Roger and N. Holzschuch, in Proceedings of Eurographics (2006), Vol. 25.

Sbert, M.

L. Szirmay-Kalos, T. Umenhoffer, G. Patow, L. Szécsi, and M. Sbert, in Computer Graphics Forum (2009), pp. 1-31.

Stolfi, J.

J. Stolfi, Oriented Projective Geometry (Academic, 1991).

Szécsi, L.

L. Szirmay-Kalos, T. Umenhoffer, G. Patow, L. Szécsi, and M. Sbert, in Computer Graphics Forum (2009), pp. 1-31.

Szirmay-Kalos, L.

L. Szirmay-Kalos, T. Umenhoffer, G. Patow, L. Szécsi, and M. Sbert, in Computer Graphics Forum (2009), pp. 1-31.

Tost, D.

P. Estalella, I. Martin, G. Drettakis, D. Tost, O. Devillers, and F. Cazals, in Proceedings of Vision Modeling and Visualization (2005), pp. 471-478.

P. Estalella, I. Martin, G. Drettakis, and D. Tost, in Eurographics Symposium on Rendering (2006), pp. 312-318.

Umenhoffer, T.

L. Szirmay-Kalos, T. Umenhoffer, G. Patow, L. Szécsi, and M. Sbert, in Computer Graphics Forum (2009), pp. 1-31.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1965).

Comput. Graph. Image Process. (1)

J. Levin, Comput. Graph. Image Process. 11, 73 (1979).
[CrossRef]

J. Symb. Comput. (3)

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 192 (2008).
[CrossRef]

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 216 (2008).
[CrossRef]

Other (8)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1965).

E. Hecht, Optics (Addison-Wesley, 1987).

L. Szirmay-Kalos, T. Umenhoffer, G. Patow, L. Szécsi, and M. Sbert, in Computer Graphics Forum (2009), pp. 1-31.

K. Nielsen and N. Christensen, Journal of WSCG (2002), Vol. 10, pp. 91-98.

P. Estalella, I. Martin, G. Drettakis, and D. Tost, in Eurographics Symposium on Rendering (2006), pp. 312-318.

P. Estalella, I. Martin, G. Drettakis, D. Tost, O. Devillers, and F. Cazals, in Proceedings of Vision Modeling and Visualization (2005), pp. 471-478.

D. Roger and N. Holzschuch, in Proceedings of Eurographics (2006), Vol. 25.

J. Stolfi, Oriented Projective Geometry (Academic, 1991).

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Figures (2)

Fig. 1
Fig. 1

Light ray reflection by a quadratic surface.

Fig. 2
Fig. 2

Performance versus accuracy using a hyperbolic mirror and random source and target points.

Equations (3)

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

M ( U , V ) = [ 0 u 3 v 4 u 4 v 3 u 2 v 4 + u 4 v 2 u 2 v 3 u 3 v 2 u 3 v 4 + u 4 v 3 0 u 1 v 4 u 4 v 1 u 1 v 3 + u 3 v 1 u 2 v 4 u 4 v 2 u 1 v 4 + u 4 v 1 0 u 1 v 2 u 2 v 1 u 2 v 3 + u 3 v 2 u 1 v 3 u 3 v 1 u 1 v 2 + u 2 v 1 0 ] .
R T M T Q Q R = 0 ,
cos ( ( s r ) T s r n ) = cos ( ( t r ) T t r n ) ,

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