Abstract

We discuss new effects related to relativistic aberration, which is the apparent distortion of objects moving at relativistic speeds relative to an idealized camera. Our analysis assumes that the camera lens is capable of stigmatic imaging of objects at rest with respect to the camera, and that each point on the shutter surface is transparent for one instant, but different points are not necessarily transparent synchronously. We pay special attention to the placement of the shutter. First, we find that a wide aperture requires the shutter to be placed in the detector plane to enable stigmatic images. Second, a Lorentz-transformation window [Proc. SPIE 9193, 91931K (2014) [CrossRef]  ] can correct for relativistic distortion. We illustrate our results, which are significant for future spaceships, with raytracing simulations.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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  1. A. Einstein, “Zur Elektrodynamik bewegter Körper,” Ann. Phys. 17, 891–921 (1905).
    [Crossref]
  2. G. F. FitzGerald, “The ether and the Earth’s atmosphere,” Science 13, 390 (1889).
  3. A. A. Michelson and E. Morley, “On the relative motion of the Earth and the luminiferous ether,” Am. J. Sci. s3-34, 333–345 (1887).
    [Crossref]
  4. A. Lampa, “Wie erscheint nach der Relativitätstheorie ein bewegter Stab einem ruhenden Beobachter?” Z. Phys. 27, 138–148 (1924).
    [Crossref]
  5. R. Penrose, “The apparent shape of a relativistically moving sphere,” Proc. Cambridge Philos. Soc. 55, 137–139 (1959).
    [Crossref]
  6. J. Terrell, “Invisibility of the Lorentz contraction,” Phys. Rev. 116, 1041–1045 (1959).
    [Crossref]
  7. M. L. Boas, “Apparent shape of large objects at relativistic speeds,” Am. J. Phys. 29, 283–286 (1961).
    [Crossref]
  8. G. D. Scott and M. R. Viner, “The geometrical appearance of large objects moving at relativistic speeds,” Am. J. Phys. 33, 534–536 (1965).
    [Crossref]
  9. M. A. Duguay and A. T. Mattick, “Ultrahigh speed photography of picosecond light pulses and echoes,” Appl. Opt. 10, 2162–2170 (1971).
    [Crossref]
  10. A. Howard, S. Dance, and L. Kitchen, “Relativistic ray-tracing: Simulating the visual appearance of rapidly moving objects,” (University of Melbourne, 1995).
  11. S. Oxburgh, N. Gray, M. Hendry, and J. Courtial, “Lorentz-transformation and Galileo-transformation windows,” Proc. SPIE 9193, 91931K (2014).
    [Crossref]
  12. S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
    [Crossref]
  13. T. Müller, S. Grottel, and D. Weiskopf, “Special relativistic visualization by local ray tracing,” IEEE Trans. Vis. Comput. Graphics 16, 1243–1250 (2010).
    [Crossref]
  14. D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
    [Crossref]
  15. C. M. Savage, A. Searle, and L. McCalman, “Real time relativity: exploratory learning of special relativity,” Am. J. Phys. 75, 791–798 (2007).
    [Crossref]
  16. G. Kortemeyer, P. Tan, and S. Schirra, “A slower speed of light: developing intuition about special relativity with games,” in Proceedings of the International Conference on the Foundations of Digital Games (FDG 2013) (ACM, 2013), pp. 400–402.
  17. N. M. Atakishiyev, W. Lassner, and K. B. Wolf, “The relativistic coma aberration. I. Geometrical optics,” J. Math. Phys. 30, 2457–2462 (1989).
    [Crossref]
  18. N. M. Atakishiyev, W. Lassner, and K. B. Wolf, “The relativistic coma aberration. II. Helmholtz wave optics,” J. Math. Phys. 30, 2463–2468 (1989).
    [Crossref]
  19. K. B. Wolf, “Relativistic aberration of optical phase space,” J. Opt. Soc. Am. A 10, 1925–1934 (1993).
    [Crossref]
  20. A. J. S. Hamilton and G. Polhemus, “Stereoscopic visualization in curved spacetime: seeing deep inside a black hole,” New J. Phys. 12, 123027 (2010).
    [Crossref]
  21. A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
    [Crossref]
  22. D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comp. Phys. Commun. 183, 711–732 (2012).
    [Crossref]
  23. W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), Chap. 2.2.
  24. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.3.3.
  25. W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), Chap. 3.2, p. 67ff.
  26. “Dr TIM, a highly scientific raytracer,” https://github.com/jkcuk/Dr-TIM .

2014 (2)

S. Oxburgh, N. Gray, M. Hendry, and J. Courtial, “Lorentz-transformation and Galileo-transformation windows,” Proc. SPIE 9193, 91931K (2014).
[Crossref]

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

2012 (1)

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comp. Phys. Commun. 183, 711–732 (2012).
[Crossref]

2010 (2)

T. Müller, S. Grottel, and D. Weiskopf, “Special relativistic visualization by local ray tracing,” IEEE Trans. Vis. Comput. Graphics 16, 1243–1250 (2010).
[Crossref]

A. J. S. Hamilton and G. Polhemus, “Stereoscopic visualization in curved spacetime: seeing deep inside a black hole,” New J. Phys. 12, 123027 (2010).
[Crossref]

2009 (1)

A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
[Crossref]

2007 (1)

C. M. Savage, A. Searle, and L. McCalman, “Real time relativity: exploratory learning of special relativity,” Am. J. Phys. 75, 791–798 (2007).
[Crossref]

2006 (1)

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

1993 (1)

1989 (2)

N. M. Atakishiyev, W. Lassner, and K. B. Wolf, “The relativistic coma aberration. I. Geometrical optics,” J. Math. Phys. 30, 2457–2462 (1989).
[Crossref]

N. M. Atakishiyev, W. Lassner, and K. B. Wolf, “The relativistic coma aberration. II. Helmholtz wave optics,” J. Math. Phys. 30, 2463–2468 (1989).
[Crossref]

1971 (1)

1965 (1)

G. D. Scott and M. R. Viner, “The geometrical appearance of large objects moving at relativistic speeds,” Am. J. Phys. 33, 534–536 (1965).
[Crossref]

1961 (1)

M. L. Boas, “Apparent shape of large objects at relativistic speeds,” Am. J. Phys. 29, 283–286 (1961).
[Crossref]

1959 (2)

R. Penrose, “The apparent shape of a relativistically moving sphere,” Proc. Cambridge Philos. Soc. 55, 137–139 (1959).
[Crossref]

J. Terrell, “Invisibility of the Lorentz contraction,” Phys. Rev. 116, 1041–1045 (1959).
[Crossref]

1924 (1)

A. Lampa, “Wie erscheint nach der Relativitätstheorie ein bewegter Stab einem ruhenden Beobachter?” Z. Phys. 27, 138–148 (1924).
[Crossref]

1905 (1)

A. Einstein, “Zur Elektrodynamik bewegter Körper,” Ann. Phys. 17, 891–921 (1905).
[Crossref]

1889 (1)

G. F. FitzGerald, “The ether and the Earth’s atmosphere,” Science 13, 390 (1889).

1887 (1)

A. A. Michelson and E. Morley, “On the relative motion of the Earth and the luminiferous ether,” Am. J. Sci. s3-34, 333–345 (1887).
[Crossref]

Atakishiyev, N. M.

N. M. Atakishiyev, W. Lassner, and K. B. Wolf, “The relativistic coma aberration. I. Geometrical optics,” J. Math. Phys. 30, 2457–2462 (1989).
[Crossref]

N. M. Atakishiyev, W. Lassner, and K. B. Wolf, “The relativistic coma aberration. II. Helmholtz wave optics,” J. Math. Phys. 30, 2463–2468 (1989).
[Crossref]

Boas, M. L.

M. L. Boas, “Apparent shape of large objects at relativistic speeds,” Am. J. Phys. 29, 283–286 (1961).
[Crossref]

Borchers, M.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.3.3.

Constable, G.

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comp. Phys. Commun. 183, 711–732 (2012).
[Crossref]

Courtial, J.

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

S. Oxburgh, N. Gray, M. Hendry, and J. Courtial, “Lorentz-transformation and Galileo-transformation windows,” Proc. SPIE 9193, 91931K (2014).
[Crossref]

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comp. Phys. Commun. 183, 711–732 (2012).
[Crossref]

A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
[Crossref]

Dance, S.

A. Howard, S. Dance, and L. Kitchen, “Relativistic ray-tracing: Simulating the visual appearance of rapidly moving objects,” (University of Melbourne, 1995).

Duguay, M. A.

Einstein, A.

A. Einstein, “Zur Elektrodynamik bewegter Körper,” Ann. Phys. 17, 891–921 (1905).
[Crossref]

Ertl, T.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Falk, M.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Fechtig, O.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

FitzGerald, G. F.

G. F. FitzGerald, “The ether and the Earth’s atmosphere,” Science 13, 390 (1889).

Frank, R.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Grave, F.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Gray, N.

S. Oxburgh, N. Gray, M. Hendry, and J. Courtial, “Lorentz-transformation and Galileo-transformation windows,” Proc. SPIE 9193, 91931K (2014).
[Crossref]

Grottel, S.

T. Müller, S. Grottel, and D. Weiskopf, “Special relativistic visualization by local ray tracing,” IEEE Trans. Vis. Comput. Graphics 16, 1243–1250 (2010).
[Crossref]

Hamilton, A. C.

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comp. Phys. Commun. 183, 711–732 (2012).
[Crossref]

A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
[Crossref]

Hamilton, A. J. S.

A. J. S. Hamilton and G. Polhemus, “Stereoscopic visualization in curved spacetime: seeing deep inside a black hole,” New J. Phys. 12, 123027 (2010).
[Crossref]

Hendry, M.

S. Oxburgh, N. Gray, M. Hendry, and J. Courtial, “Lorentz-transformation and Galileo-transformation windows,” Proc. SPIE 9193, 91931K (2014).
[Crossref]

Howard, A.

A. Howard, S. Dance, and L. Kitchen, “Relativistic ray-tracing: Simulating the visual appearance of rapidly moving objects,” (University of Melbourne, 1995).

King, A.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Kitchen, L.

A. Howard, S. Dance, and L. Kitchen, “Relativistic ray-tracing: Simulating the visual appearance of rapidly moving objects,” (University of Melbourne, 1995).

Kortemeyer, G.

G. Kortemeyer, P. Tan, and S. Schirra, “A slower speed of light: developing intuition about special relativity with games,” in Proceedings of the International Conference on the Foundations of Digital Games (FDG 2013) (ACM, 2013), pp. 400–402.

Kraus, U.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Lambert, D.

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comp. Phys. Commun. 183, 711–732 (2012).
[Crossref]

Lampa, A.

A. Lampa, “Wie erscheint nach der Relativitätstheorie ein bewegter Stab einem ruhenden Beobachter?” Z. Phys. 27, 138–148 (1924).
[Crossref]

Lassner, W.

N. M. Atakishiyev, W. Lassner, and K. B. Wolf, “The relativistic coma aberration. II. Helmholtz wave optics,” J. Math. Phys. 30, 2463–2468 (1989).
[Crossref]

N. M. Atakishiyev, W. Lassner, and K. B. Wolf, “The relativistic coma aberration. I. Geometrical optics,” J. Math. Phys. 30, 2457–2462 (1989).
[Crossref]

Mattick, A. T.

McCalman, L.

C. M. Savage, A. Searle, and L. McCalman, “Real time relativity: exploratory learning of special relativity,” Am. J. Phys. 75, 791–798 (2007).
[Crossref]

Mendez, I. R.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Michelson, A. A.

A. A. Michelson and E. Morley, “On the relative motion of the Earth and the luminiferous ether,” Am. J. Sci. s3-34, 333–345 (1887).
[Crossref]

Morley, E.

A. A. Michelson and E. Morley, “On the relative motion of the Earth and the luminiferous ether,” Am. J. Sci. s3-34, 333–345 (1887).
[Crossref]

Müller, T.

T. Müller, S. Grottel, and D. Weiskopf, “Special relativistic visualization by local ray tracing,” IEEE Trans. Vis. Comput. Graphics 16, 1243–1250 (2010).
[Crossref]

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Nollert, H.-P.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Oxburgh, S.

S. Oxburgh, N. Gray, M. Hendry, and J. Courtial, “Lorentz-transformation and Galileo-transformation windows,” Proc. SPIE 9193, 91931K (2014).
[Crossref]

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

Penrose, R.

R. Penrose, “The apparent shape of a relativistically moving sphere,” Proc. Cambridge Philos. Soc. 55, 137–139 (1959).
[Crossref]

Polhemus, G.

A. J. S. Hamilton and G. Polhemus, “Stereoscopic visualization in curved spacetime: seeing deep inside a black hole,” New J. Phys. 12, 123027 (2010).
[Crossref]

Ruder, H.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Savage, C. M.

C. M. Savage, A. Searle, and L. McCalman, “Real time relativity: exploratory learning of special relativity,” Am. J. Phys. 75, 791–798 (2007).
[Crossref]

Schafhitzel, T.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Schär, S.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Schirra, S.

G. Kortemeyer, P. Tan, and S. Schirra, “A slower speed of light: developing intuition about special relativity with games,” in Proceedings of the International Conference on the Foundations of Digital Games (FDG 2013) (ACM, 2013), pp. 400–402.

Scott, G. D.

G. D. Scott and M. R. Viner, “The geometrical appearance of large objects moving at relativistic speeds,” Am. J. Phys. 33, 534–536 (1965).
[Crossref]

Searle, A.

C. M. Savage, A. Searle, and L. McCalman, “Real time relativity: exploratory learning of special relativity,” Am. J. Phys. 75, 791–798 (2007).
[Crossref]

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), Chap. 2.2.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), Chap. 3.2, p. 67ff.

Snehanshu, H.

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comp. Phys. Commun. 183, 711–732 (2012).
[Crossref]

Talati, S.

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comp. Phys. Commun. 183, 711–732 (2012).
[Crossref]

Tan, P.

G. Kortemeyer, P. Tan, and S. Schirra, “A slower speed of light: developing intuition about special relativity with games,” in Proceedings of the International Conference on the Foundations of Digital Games (FDG 2013) (ACM, 2013), pp. 400–402.

Terrell, J.

J. Terrell, “Invisibility of the Lorentz contraction,” Phys. Rev. 116, 1041–1045 (1959).
[Crossref]

Tyc, T.

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

Viner, M. R.

G. D. Scott and M. R. Viner, “The geometrical appearance of large objects moving at relativistic speeds,” Am. J. Phys. 33, 534–536 (1965).
[Crossref]

Weiskopf, D.

T. Müller, S. Grottel, and D. Weiskopf, “Special relativistic visualization by local ray tracing,” IEEE Trans. Vis. Comput. Graphics 16, 1243–1250 (2010).
[Crossref]

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.3.3.

Wolf, K. B.

K. B. Wolf, “Relativistic aberration of optical phase space,” J. Opt. Soc. Am. A 10, 1925–1934 (1993).
[Crossref]

N. M. Atakishiyev, W. Lassner, and K. B. Wolf, “The relativistic coma aberration. II. Helmholtz wave optics,” J. Math. Phys. 30, 2463–2468 (1989).
[Crossref]

N. M. Atakishiyev, W. Lassner, and K. B. Wolf, “The relativistic coma aberration. I. Geometrical optics,” J. Math. Phys. 30, 2457–2462 (1989).
[Crossref]

Zahn, C.

D. Weiskopf, M. Borchers, T. Ertl, M. Falk, O. Fechtig, R. Frank, F. Grave, A. King, U. Kraus, T. Müller, H.-P. Nollert, I. R. Mendez, H. Ruder, T. Schafhitzel, S. Schär, C. Zahn, and M. Zatloukal, “Explanatory and illustrative visualization of special and general relativity,” IEEE Trans. Vis. Comput. Graphics 12, 522–534 (2006).
[Crossref]

Zatloukal, M.

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Figures (12)

Fig. 1.
Fig. 1. Relativistic distortion, and comparison with the effect of time-of-flight effects only. The images show raytracing simulations of photos of a scene (a) taken with a camera moving at relativistic speed through the scene (b) and with an (unphysical) camera in which only time-of-flight effects are taken into account, but those due to special relativity are ignored (c). The image shown in (c) was simulated by using the Galilean transformation instead of the Lorentz transformation when transforming between reference frames [11]. All simulations were performed for a pinhole camera using the pinhole shutter model. In (b) and (c), the camera moves with velocity ${\boldsymbol\beta}c$, where ${\boldsymbol\beta}=(0.1c,0,0.99c)^{\intercal}$ in the left-handed coordinate system used by our raytracing software, in which the $x$, $y$, and $z$ directions represent the right, up, and into the page, respectively. The simulations were performed using the scientific raytracer Dr TIM [12].
Fig. 2.
Fig. 2. (a) Trajectory of a photo ray (a light ray contributing to a photo), viewed in the camera frame. The trajectory is shown as a red line with an arrow tip at its end. The camera is moving with (relativistic) velocity ${\boldsymbol\beta}c$ through a stationary scene. The camera’s lens stigmatically images the position ${\textbf{P}}$ to the position ${\textbf{D}}$ on the detector, which means it re-directs any ray from ${\textbf{P}}$ such that it subsequently passes through ${\textbf{D}}$. ${\textbf{L}}$ is the point where the ray intersects the idealized thin lens (vertical double-sided arrow). In the camera frame, the light ray passes ${\textbf{P}}$ at time ${t_{\text{P}}}$, enters ${\textbf{L}}$ at ${t_{\text{L}}}$, and reaches ${\textbf{D}}$ at ${t_{\text{D}}}$. (b) Several photo rays from ${\textbf{P}}$, shown in the camera frame. Due to the imaging properties of the lens, these rays intersect the same image position ${\textbf{D}}$ on the detector (not shown). Different rays intersect the camera lens at different positions, labeled ${{\textbf{L}}_1}$ to ${{\textbf{L}}_5}$. Each ray can be Lorentz-transformed into the scene frame by considering two events: for ray $i$, these are the times and positions of the ray passing through ${\textbf{P}}$ and ${{\textbf{L}}_i}$. (c) The same rays, shown in the scene frame. Ray $i$ ($i=1$ to 5) passes through the scene-frame positions ${\textbf{P}}_i^\prime$ and ${\textbf{L}}_i^\prime$, the positions of the events described above, Lorentz-transformed into the scene frame. These positions depend on the choice of shutter model and camera velocity ${\boldsymbol\beta}c$. In the scene frame, the rays do not necessarily intersect in a single point, which is the case in the example shown.
Fig. 3.
Fig. 3. Simulated photos taken with a camera that uses the aperture-plane shutter model. In (a), the camera is at rest; in (b), it is moving at $\beta\approx 99.5\%$ of the speed of light. The scene contains a $9\times 9$ array of small white spheres centered on the scene-frame surface on which the camera is focused when moving. The figure is calculated for ${\boldsymbol\beta}{=(0.1,0,0.99)^{\intercal}}$. The camera was focused on a plane a distance 10 (in units of floor-tile lengths) in the camera frame, which transforms into a curved surface in the scene frame. The horizontal angle of view is 120° in (a) and 20° in (b). In both cases, the simulated aperture radius is 0.05 (Dr TIM’s interactive version refers to this aperture size as “medium”).
Fig. 4.
Fig. 4. Simulated photo taken with a camera that uses the aperture-plane shutter model. The simulation parameters differ from those used to create Fig. 3(b) only in the aperture radius, which is 0.2 (“huge”).
Fig. 5.
Fig. 5. Simulated photo taken with a moving camera that uses the detector-plane shutter model and an ideal thin lens as imaging element. AS in Fig. 3, the scene contains a $9\times 9$ array of small white spheres centered on the scene-frame surface on which the camera is focused when moving, but note that the scene-frame surface on which the cameras are focused is different from that on which the camera in Fig. 3 is focused. The spheres can be seen to be in sharp focus. The remainder of the scene and the camera velocity are identical to those used to calculate Fig. 3.
Fig. 6.
Fig. 6. Simulated photos taken with a camera that uses the detector-plane shutter model in combination with an ideal thin lens (a) and a phase hologram of a thin lens (b). The simulated aperture radius is 0.2 (called “huge” in Dr TIM’s interactive version) to make the blurring visible. In (a), which is calculated for parameters that differ from those used to calculate Fig. 5 only in the increased aperture size, the spheres are still in focus. In (b), which is calculated for parameters that differ from those used to calculate (a) only in the time delay introduced by the imaging element, the spheres are clearly blurred.
Fig. 7.
Fig. 7. Simulated photos taken with cameras that use the focus-surface shutter model. The simulated aperture radius is 0.05 (“medium”).
Fig. 8.
Fig. 8. Raytracing simulations illustrating the fixed-point-surface shutter model. In (a), the camera is at rest; in (b), it is moving with velocity ${\boldsymbol\beta}c=(0.1c,0,0.99c)^{\intercal}$ in the scene frame. The camera is focused on the plane $z=8$ in the camera frame. Because of the timing of the photo rays, objects in this plane appear identical in both photos. Objects outside this plane appear distorted; those close to the focusing plane, like the mantle of the cylinders centered in the focusing plane, are distorted only slightly.
Fig. 9.
Fig. 9. Relativistic blurring and distortion and its cancellation with a Lorentz window. The frames show simulated photos of a portrait scene, with the camera focused on the same plane in front of the camera. (a) The camera is stationary in the scene frame. The subject’s eyes are in the camera’s focusing plane. (b) The camera is moving with relativistic velocity ${\boldsymbol\beta}c$ with respect to the scene frame, where ${\boldsymbol\beta}=(0.1,0,0.99)^{\intercal}$. The camera’s shutter was placed in a plane perpendicular to ${\boldsymbol\beta}$ that passed through the position ${(0.111,0,1.1)^{\intercal}}$; the shutter opening time was ${t_{\text{S}}}=-1$. The scene appears distorted and out-of-focus. (c) Like (b), but with a Lorentz window placed in the shutter plane, which makes the photo look identical to that taken with the camera at rest. Throughout, the speed of light was set to $c=1$. All simulations were performed with an extended version of Dr TIM [12].
Fig. 10.
Fig. 10. Simulated photos taken through a Lorentz window with a camera that uses the detector-plane shutter model. (a) Without Lorentz window; (b) with Lorentz window. In (b), a slight distortion of the scene can be seen (floor tiles). The shutter-opening time was calculated such that the photo ray through the aperture center in the forward direction has the same timing as in the arbitrary-plane-shutter model.
Fig. 11.
Fig. 11. Anaglyphs made from simulated stereo pairs taken with a camera at rest (a), a camera moving at relativistic velocity ${\boldsymbol\beta}c=(0.1,0,0.99)^{\intercal}c$ relative to the scene (b), and the same relativistically moving camera but with a Lorentz window placed in the plane of spatial fixed points of the Lorentz transformation for the shutter-opening time (c). In (b) and (c), the camera’s shutter is located in a plane perpendicular to ${\boldsymbol\beta}$ through the position $(0.111,0,1.1)^{\intercal}$, and the shutter opening time is ${t_{\text{S}}}=-1$, parameters chosen such that the shutter plane is a plane of spatial fixed points of the Lorentz transformation for ${t_{\text{S}}}$. Eye separation 0.4 in the horizontal direction; camera directions are chosen such that a plane a distance 5 in front of the camera appears in the paper/monitor plane.
Fig. 12.
Fig. 12. (a) Photo rays from a point light source at position ${{\boldsymbol P}^\prime}$ in the scene frame pass through different positions ${\boldsymbol L}_i^\prime$ on the lens, indicated as a thick double-sided arrow. In general, the events of different photo rays passing through ${{\boldsymbol P}^\prime}$ occur at different times. (b) In the scene frame, the positions ${{\boldsymbol P}_i}$ of these events are spread out along a line parallel to ${\boldsymbol\beta}$. ${\boldsymbol \Delta}{\boldsymbol P}$ points between two of the outermost positions ${{\boldsymbol P}_i}$. In the small-aperture limit, from the perspective of the (small) aperture, the positions ${{\boldsymbol P}_i}$ are distributed over an angular range of size $\alpha$. (c) The dotted line indicates the surface that is best imaged into the detector plane.

Equations (12)

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

x = x + ( γ 1 ) ( β x ) β β 2 + γ β c t ,
d = d ^ + ( γ 1 ) ( β ^ d ^ ) β ^ + γ β ,
l ( r ) = l ( 0 ) + O ( r 2 ) ,
t = ( 1 γ ) ( β x ) c γ β 2 .
( γ 1 ) ( β x ) β 2 γ c t = 0.
β x = γ β 2 c t S γ 1 = γ + 1 γ c t S .
α Δ P sin ν d ,
t L , in = t S + a c ,
S = L a d ^ .
( S P ) n = 0.
a = ( P L ) n d ^ n .
t L , in = t S + ( P L ) n c d ^ n .

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