Euclid's law of reflection, r = i, rests on three assumptions: Fermat's principle of least time, constancy of speed of light, and that the mirror is stationary. For a moving mirror, the corresponding law is known as the relativistic reflection law; however, it was first inferred prior to relativity. While modern derivations rely on Lorentz transformations or Maxwell's equations, the original derivation was an elegant application of geometric optics. In this paper, we develop a new geometric approach which shows that the relativistic reflection from a plane mirror is equivalent to the ordinary reflection from a certain hyperbolic mirror, and that the incident and reflected rays are the focal rays of the hyperbola. We demonstrate that incident concentric spherical waves reflect as non-concentric spherical waves, and thereby derive the Doppler effect formula. We conclude with a class project for students to derive a generalization of Snell's law of refraction for a particular type of a moving interface.

1.
R. K.
Luneburg
and
M.
Herzberger
,
Mathematical Theory of Optics
(
University of California Press
,
CA
,
1964
).
2.
O. N.
Stavroudis
,
The Mathematics of Geometrical and Physical Optics
(
Wiley-VCH Verlag
,
Weinheim
,
2006
).
3.
A. A.
Michelson
and
E. W.
Morley
, “
On the relative motion of the earth and the luminiferous ether
,”
Am. J. Sci.
34
,
333
345
(
1887
).
4.
W. M.
Hicks
, “
II. On the Michelson-Morley Experiment relating to the Drift of the Aether
,”
London, Edinburgh, Dublin Philos. Mag. J. Sci.
3
,
9
42
(
1902
).
5.
A.
Einstein
, “
Zur elektrodynamik bewegter korper
,”
Ann. Phys.
322
,
891
921
(
1905
).
6.
W.
Pauli
,
Theory of Relativity
(
Pergamon Press
,
London & New York
,
1958
).
7.
A.
Zangwill
,
Modern Electrodynamics
(
Cambridge U. P.
,
2015
).
8.
S. J.
Orfanidis
,
Electromagnetic Waves and Antennas
(
Rutgers University
,
New Brunswick, NJ
2002
).
9.
B. M.
Bolotovskii
and
S. N.
Stolyarov
, “
Reflection of light from a moving mirror and related problems
,”
Phys. Usp.
32
,
813
827
(
1989
).
10.
A.
Gjurchinovski
, “
Reflection of light from a uniformly moving mirror
,”
Am. J. Phys.
72
,
1316
1324
(
2004
).
11.
B. M.
Jaffe
, “
Forward reflection of light by a moving mirror
,”
Am. J. Phys.
41
,
577
578
(
1973
).
12.
Z. K.
Silagadze
, “
Test problems in mechanics and special relativity
,” preprint arXiv:physics/0605057 (
2006
).
13.
K. A.
Lurie
,
An Introduction to the Mathematical Theory of Dynamic Materials
(
Springer
,
New York
2007
).
14.
E.
Teixeira
, “
Reflection and transmission of a plane electromagnetic wave on a moving boundary between two dielectrics
,” M.S. thesis,
Worcester Polytechnic Institute
(
2006
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.