Abstract

A terminology limited to point objects is introduced to distinguish between finite and infinite, transparent and opaque mirrors. A basic image-producing operator is defined, and its properties are examined in detail. An operational form of the basic operator is derived, two general definitions are presented, and a formal expression for the images of a system of plane mirrors is written. Some simple systems are analyzed in detail to illustrate the applications of the theory. Finally, indication is given of possible directions for future development of the subject, and of its relationship to existing mathematical theories.

© 1967 Optical Society of America

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References

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  1. J. S. Beggs, J. Opt. Soc. Am. 50, 388 (1960).
    [CrossRef]
  2. S. Walles, R. E. Hopkins, Appl. Opt. 3, 1447 (1964).
    [CrossRef]
  3. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).
  4. W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, New York, 1964).
  5. R. J. Pegis, M. M. Rao, Appl. Opt. 2, 1271 (1963); A. Walther, Appl. Opt. 3, 543 (1964).
    [CrossRef]
  6. G. R. Rosendahi, J. Opt. Soc. Am. 50, 287 (1960); J. Opt. Soc. Am. 50, 859 (1960).
    [CrossRef]
  7. A. Messiah, Quantum Mechanics (North Holland Publishing Co., Amsterdam, 1962), p. 1080.
  8. B. Van der Pol, H. Bremmer, Operational Calculus (Cambridge Univ. Press, Cambridge, 1959).
  9. C. S. Hastings, New Methods in Geometrical Optics (Macmillan Co., New York, 1927), p. 25.
  10. Ray tracing in plane systems is discussed in R. K. Luneburg, op. cit., Appx. II.
  11. J. P. C. Southall, Mirrors, Prisms, and Lenses (Dover Publications, Inc., New York, 1964), p. 37. This result follows from (14), because every factor of T(α)T(β) adds 2α− 2βto θ, and a finite number of images implies that θ+ N(2α− 2β) = θ+ k2π.
  12. F. J. H. Dibdin, Essentials of Light (Cleaver–Hume Press Ltd., London, 1961), p. 15.
  13. Faceted mirrors are discussed by H. A. E. Keitz, Light Calculations and Measurements (Philips Technical Library, 1955), p. 206.

1964 (1)

1963 (1)

1960 (2)

Beggs, J. S.

Bremmer, H.

B. Van der Pol, H. Bremmer, Operational Calculus (Cambridge Univ. Press, Cambridge, 1959).

Brouwer, W.

W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, New York, 1964).

Dibdin, F. J. H.

F. J. H. Dibdin, Essentials of Light (Cleaver–Hume Press Ltd., London, 1961), p. 15.

Hastings, C. S.

C. S. Hastings, New Methods in Geometrical Optics (Macmillan Co., New York, 1927), p. 25.

Hopkins, R. E.

Keitz, H. A. E.

Faceted mirrors are discussed by H. A. E. Keitz, Light Calculations and Measurements (Philips Technical Library, 1955), p. 206.

Luneburg, R. K.

Ray tracing in plane systems is discussed in R. K. Luneburg, op. cit., Appx. II.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

Messiah, A.

A. Messiah, Quantum Mechanics (North Holland Publishing Co., Amsterdam, 1962), p. 1080.

Pegis, R. J.

Rao, M. M.

Rosendahi, G. R.

Southall, J. P. C.

J. P. C. Southall, Mirrors, Prisms, and Lenses (Dover Publications, Inc., New York, 1964), p. 37. This result follows from (14), because every factor of T(α)T(β) adds 2α− 2βto θ, and a finite number of images implies that θ+ N(2α− 2β) = θ+ k2π.

Van der Pol, B.

B. Van der Pol, H. Bremmer, Operational Calculus (Cambridge Univ. Press, Cambridge, 1959).

Walles, S.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Other (9)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, New York, 1964).

A. Messiah, Quantum Mechanics (North Holland Publishing Co., Amsterdam, 1962), p. 1080.

B. Van der Pol, H. Bremmer, Operational Calculus (Cambridge Univ. Press, Cambridge, 1959).

C. S. Hastings, New Methods in Geometrical Optics (Macmillan Co., New York, 1927), p. 25.

Ray tracing in plane systems is discussed in R. K. Luneburg, op. cit., Appx. II.

J. P. C. Southall, Mirrors, Prisms, and Lenses (Dover Publications, Inc., New York, 1964), p. 37. This result follows from (14), because every factor of T(α)T(β) adds 2α− 2βto θ, and a finite number of images implies that θ+ N(2α− 2β) = θ+ k2π.

F. J. H. Dibdin, Essentials of Light (Cleaver–Hume Press Ltd., London, 1961), p. 15.

Faceted mirrors are discussed by H. A. E. Keitz, Light Calculations and Measurements (Philips Technical Library, 1955), p. 206.

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Figures (5)

Fig. 1
Fig. 1

The simplest mirror systems and the mirror operator associated with each: (a) the T system; (b) the X0 system; (c) the Z2 system; (d) the X1 system.

Fig. 2
Fig. 2

Geometry involved in the definition of the basic mirror operator, and relevant to the analysis of the T system: (a) rectangular coordinates; (b) cylindrical coordinates.

Fig. 3
Fig. 3

The X0 system. The distances a, b, and x are measured from an arbitrary reference at O to mirrors A and B, and the object point P.

Fig. 4
Fig. 4

Geometry of the X1 system. Only the first two images of P are shown.

Fig. 5
Fig. 5

The Z2 system: (a) a simple geometry, sufficient for finding images, but inferior in other ways; (b) the geometry used to determine the images given in Eq. (17).

Equations (65)

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M ( ξ , η ) P = P .
T ( d ) f ( x , y ) = f ( 2 d - x , y ) ,
T ( α ) f ( r , θ ) = f ( r , 2 α - θ ) .
[ T ( d ) ] 2 f ( x , y ) = T ( d ) f ( 2 d - x , y ) = f ( x , y )
[ T ( α ) ] 2 f ( r , θ ) = f ( r , θ ) .
T ( d ) f ( x ) = κ f ( 2 d - x ) , T ( d ) f ( 2 d - x ) = κ 2 f ( x ) .
[ T ( d ) ] 2 N f ( x ) = κ 2 N f ( x ) , [ T ( d ) ] 2 N + 1 f ( x ) = κ 2 N + 1 f ( 2 d - x ) .
P ( n ) = { P ( n + 1 ) if P ( n + 1 ) is not already obtained . 0 if P ( n + 1 ) is already obtained .
[ T ( a ) T ( b ) ] f ( x ) = f ( 2 a - x ) f ( 2 b - x ) ,
T ( a + b ) f ( x ) = f ( 2 ( a + b ) - x ) f ( 2 a - x ) f ( 2 b - x ) .
T ( d ) [ A f ( x ) B g ( x ) ] = A f ( 2 d - x ) B g ( 2 d - x ) .
[ T ( a ) , T ( b ) ] f ( x ) = [ f ( 2 a - x ) , f ( 2 b - x ) ] , T ( d ) [ A f ( x ) , B g ( x ) ] = [ A f ( 2 d - x ) , B g ( 2 d - x ) ] ,
T ( a ) T ( b ) f ( x ) = T ( a ) f ( 2 b - x ) = f ( 2 a - 2 b + x ) , T ( b ) T ( a ) f ( x ) = T ( b ) f ( 2 a - x ) = f ( 2 b - 2 a + x ) .
T ( a ) T ( b ) f ( x + c ) = T ( b ) T ( a ) f ( x - c ) = f ( x ) ,
T ( a ) T ( a ) f ( x ) = f ( x ) .
[ T ( a ) T ( b ) T ( z ) ] - 1 = T ( z ) T ( b ) T ( a ) .
[ T ( a ) ] 2 f ( x ) = λ 2 f ( x ) = f ( x ) ,
T ( a ) g ( x ) = λ 1 g ( x ) = + g ( x ) , T ( a ) h ( x ) = λ 2 h ( x ) = - h ( x ) .
g ( 2 a - x ) = g ( x ) , h ( 2 a - x ) = - h ( x ) ,
- [ g ( x ) ] 2 d x = 2 0 [ g ( a + z ) ] 2 d z = 1             z = x - a
u ( x ) = c 1 g ( x ) + c 2 h ( x ) .
M u v = u M v = - u ( x ) M v ( x ) d x .
T = ( T g g T g h T h g T h h ) = ( 1 0 0 - 1 )
T = ( T u u T u v T v u T v v ) = ( 0 1 1 0 ) .
T ( a + b ) T ( a ) T ( b ) = T ( 2 b ) , T ( a ) T ( a + b ) T ( b ) = T ( 0 ) , T ( a ) T ( b ) T ( a + b ) = T ( 2 a ) .
T ( a - b ) T ( a ) T ( b ) = T ( 0 ) , T ( a ) T ( a - b ) T ( b ) = T ( 2 b ) , T ( a ) T ( b ) T ( a - b ) = T ( 2 a - 2 b ) .
T ( a + b ) T ( a ) T ( a - b ) = T ( a ) , T ( a + b ) T ( b ) T ( a - b ) = T ( 2 a - b ) ,
T ( a ± b ) T ( a b ) T ( a ± b ) = T ( a ± 3 b ) .
n = 1 2 N + 1 T ( a n ) = T [ n = 1 2 N + 1 ( - 1 ) n + 1 a n ] .
sinh T = T sinh 1 cosh T = cosh 1 exp T = cosh 1 + T sinh 1.
X 1 ( α , β ) P = P 1 P 2 , X 1 ( α , β ) f ( θ ) = f ( 2 α - θ ) f ( 2 β - θ ) = T ( α ) f ( θ ) T ( β ) f ( θ ) ,
X 1 ( α , β ) = κ 1 κ 1 + κ 2 T ( α ) κ 2 κ 1 + κ 2 I ( β ) ,
X 1 ( α , β ) = 1 2 [ T ( α ) T ( β ) ] ,
T ( b ) f ( x ) = f ( 2 b - x ) = D 1 P f ( x ) = P D 2 f ( x ) ,
D 1 f ( x ) = f ( x + 2 b ) , D 2 f ( x ) = f ( x - 2 b ) .
f ( x ± 2 b ) = f ( x ) + ( ± 2 b ) d d x f ( x ) + + ( ± 2 b ) n n ! d n d x n f ( x ) + = n = 0 ( ± 2 b ) n n ! d n d x n f ( x ) = e ± 2 b f ( x )             d / d x
T ( b ) = e 2 b P = P e - 2 b .
[ T ( a ) T ( b ) ] 2 = 2 2 cosh 2 ( a - b ) p , T ( a ) T ( b ) T ( b ) T ( a ) = 2 sinh 2 ( a - b ) p .
P ζ M ( ξ ) P ζ = - P ζ
M { P } = Σ j M j { P } = Σ i Σ j P i j = { P } .
M { P ( n - 1 ) } = Σ l M l { P ( n - 1 ) } = Σ i Σ j Σ l P i j l ( n ) = { P ( n ) } .
{ M P } = Σ n { P ( n ) }
= Σ i [ P i Σ j ( P i j { Σ i j k } ] ) ]
= M P MM P MMM P
{ M } = Σ n { M ( n ) }
= M MM MMM
= Σ i M i Σ j M j Σ i M i Σ k M k Σ j M j Σ i M i
= [ ( [ 1 ] Σ k M k 1 ) Σ j M j 1 ] Σ i M i
X o = n = o [ T ( n a - n b + b ) T ( n b - n a + a ) ] n = - [ T ( n a - n b + x ) ]
X o f ( x ) = { M P } = n = o [ f ( 2 n a - 2 n b + 2 b - x ) f ( 2 n b - 2 n a + 2 a - x ) ] n = - [ f ( 2 n a - 2 n b + x ) ] .
M = X 1 ( α , β ) = I ( α ) I ( β ) ,
X 1 ( α , β ) f ( θ ) = f ( 2 α - θ ) f ( 2 β - θ ) .
[ X 1 ( α , β ) ] 2 f ( θ ) = X 1 ( α , β ) [ f ( 2 α - θ ) f ( 2 β - θ ) ] = f ( θ ) f ( 2 β - 2 α + θ ) f ( 2 α - 2 β + θ ) f ( θ ) .
[ T ( α ) T ( β ) ] N f ( θ ) = f ( θ ) .
Z 2 ( α , β , l ) = X 1 ( α , β ) T ( l ) = T ( α ) T ( β ) T ( l ) .
Z 2 ( α , β , l ) f ( r , θ ) = f ( r , 2 α - θ ) f ( r , 2 β - θ ) f ( r 2 + l 2 - 2 r l cos θ , tan - 1 [ r sin θ 2 l - r cos θ ] ) .
P A = [ r cos ( 2 α - θ ) - d sin 2 α ] e ^ x + [ r sin ( 2 α - θ ) + d ( 1 + cos 2 α ) ] e ^ y ,
P B = [ r cos ( 2 β - θ ) - ( c - d ) sin 2 β ] e ^ x + [ r sin ( 2 β - θ ) - ( c - d ) ( 1 + cos 2 β ) ] e ^ y ,
P C = [ - r cos θ ] e ^ x + [ r sin θ ] e ^ y .
r P ( n ) = [ x ] 2 + [ y ] 2 , θ P ( n ) = tan - 1 [ y P ( n ) x P ( n ) ] ,
P A = P B = P C = - r cos θ e ^ x + r sin θ e ^ x . ,
P = f ( r , π - θ ) = T ( π / 2 ) f ( r , θ ) . ,
P A = r cos ( 2 α - θ ) e ^ x + r sin ( 2 α - θ ) e ^ y , P A = f ( r , 2 α - θ ) = T ( α ) f ( r , θ ) ,
P A = r cos θ e ^ x + ( 2 d - r sin θ ) e ^ y , P B = r cos θ e ^ x + ( 2 c - 2 d - r sin θ ) e ^ y , P C = - r cos θ e ^ x + r sin θ e ^ y ,
P A = f ( x , 2 d - y ) = T y ( d ) f ( x , y ) , P B = f ( x , 2 c - 2 d - y ) = T y ( c - d ) f ( x , y ) , P C = f ( - x , y ) = T x ( 0 ) f ( x , y ) .

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