Abstract

This paper shows a general matrix method for analyzing systems of gimbaled mirrors, prisms, and ray projectors. The method shows how to find the gimbal angles which are necessary to ensure that a ray traversing the system obeys some requirement of collimation or orientation. Gimbal angles are considered to be the important measurement parameters, and more meaningful than the usual direction cosines. The methods apply to a large class of optical systems, including those in which gimbals are not specifically used. The gimbal concept will be found very useful in describing the orientation of images in prism systems, such as the peculiar behavior of a penta prism. A gimbal operator is defined and used throughout. The appendix contains a formal development of the mirror as a second-rank tensor together with important properties of prisms as products of mirror matrices.

© 1967 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. E. Hopkins, in Applied Optics and Engineering, Vol. 3, R. Kingslake, Ed. (Academic Press Inc., New York, 1965).
  2. W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, Inc., New York, 1964).
  3. T. Smith, Trans. Opt. Soc. (London) 30, 68 (1927–1928).
    [CrossRef]
  4. T. Y. Baker, Trans. Opt. Soc. (London) 29, 49 (1927–1928).
    [CrossRef]
  5. F. Blottiau, Rev. Opt. 27, 341 (1948).
  6. F. Blottiau, Rev. Opt. 33, 339 (1954).
  7. R. J. Pegis and M. M. Rao, Appl. Opt. 2, 1271 (1963).
    [CrossRef]
  8. G. R. Rosendahl, J. Opt. Soc. Am. 50, 287 (1960).
    [CrossRef]
  9. G. R. Rosendahl, J. Opt. Soc. Am. 50, 859 (1960).
    [CrossRef]
  10. P. R. Yoder, J. Opt. Soc. Am. 48, 496 (1958).
    [CrossRef]
  11. J. S. Beggs, J. Opt. Soc. Am. 50, 388 (1960).
    [CrossRef]
  12. S. Walles and R. E. Hopkins, Appl. Opt. 3, 1447 (1964).
    [CrossRef]
  13. A. N. deJong, Opt. Acta 10, 115 (1963).
    [CrossRef]
  14. S. Walles, Appl. Opt. 4, 737 (1965).
    [CrossRef]
  15. H. Jeffries and B. S. Jeffries, Methods of Mathematical Physics (Cambridge University Press, Cambridge, England, 1965), p. 96.
  16. L. Brand, Vector and Tensor Analysis (John Wiley & Sons, New York, 1947), pp. 419, 420.

1965 (1)

1964 (1)

1963 (2)

1960 (3)

1958 (1)

1954 (1)

F. Blottiau, Rev. Opt. 33, 339 (1954).

1948 (1)

F. Blottiau, Rev. Opt. 27, 341 (1948).

Baker, T. Y.

T. Y. Baker, Trans. Opt. Soc. (London) 29, 49 (1927–1928).
[CrossRef]

Beggs, J. S.

Blottiau, F.

F. Blottiau, Rev. Opt. 33, 339 (1954).

F. Blottiau, Rev. Opt. 27, 341 (1948).

Brand, L.

L. Brand, Vector and Tensor Analysis (John Wiley & Sons, New York, 1947), pp. 419, 420.

Brouwer, W.

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

deJong, A. N.

A. N. deJong, Opt. Acta 10, 115 (1963).
[CrossRef]

Hopkins, R. E.

S. Walles and R. E. Hopkins, Appl. Opt. 3, 1447 (1964).
[CrossRef]

R. E. Hopkins, in Applied Optics and Engineering, Vol. 3, R. Kingslake, Ed. (Academic Press Inc., New York, 1965).

Jeffries, B. S.

H. Jeffries and B. S. Jeffries, Methods of Mathematical Physics (Cambridge University Press, Cambridge, England, 1965), p. 96.

Jeffries, H.

H. Jeffries and B. S. Jeffries, Methods of Mathematical Physics (Cambridge University Press, Cambridge, England, 1965), p. 96.

Pegis, R. J.

Rao, M. M.

Rosendahl, G. R.

Smith, T.

T. Smith, Trans. Opt. Soc. (London) 30, 68 (1927–1928).
[CrossRef]

Walles, S.

Yoder, P. R.

Appl. Opt. (3)

J. Opt. Soc. Am. (4)

Opt. Acta (1)

A. N. deJong, Opt. Acta 10, 115 (1963).
[CrossRef]

Rev. Opt. (2)

F. Blottiau, Rev. Opt. 27, 341 (1948).

F. Blottiau, Rev. Opt. 33, 339 (1954).

Trans. Opt. Soc. (London) (2)

T. Smith, Trans. Opt. Soc. (London) 30, 68 (1927–1928).
[CrossRef]

T. Y. Baker, Trans. Opt. Soc. (London) 29, 49 (1927–1928).
[CrossRef]

Other (4)

R. E. Hopkins, in Applied Optics and Engineering, Vol. 3, R. Kingslake, Ed. (Academic Press Inc., New York, 1965).

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

H. Jeffries and B. S. Jeffries, Methods of Mathematical Physics (Cambridge University Press, Cambridge, England, 1965), p. 96.

L. Brand, Vector and Tensor Analysis (John Wiley & Sons, New York, 1947), pp. 419, 420.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Right-hand co-ordinate system, showing unit row vectors and directions of positive rotation.

Fig. 2
Fig. 2

Two-pivot gimbal in its home position on general axes i, j, and k, and its operator equivalent Gij=θiϕj.

Fig. 3
Fig. 3

Reference set of 15 mirrors and their matrices, set at 0° and 45° to the base axes. There are only 9 distinct kinds; the following pairs are identical: 1 2 ¯=32; 12=3 2 ¯=; 13=2 3 ¯; 1 3 ¯=23; 31=2 1 ¯; 3 1 ¯=21.

Fig. 5
Fig. 5

Azimuth alignment using Porro prism affected by tilt T.

Fig. 6
Fig. 6

Seeker in aircraft continuously tracks a point on the ground (line of sight) thus generating varying gimbal angles S and E. The projected reticle vector −R″ must be made parallel to S so the pilot synthetically sees the ground point. The solutions E() and D(δ) are shown in the box connecting the seeker and reticle-projector gimbals.

Fig. 7
Fig. 7

Image projection problem and congruent “swivel-chair” gimbal (left) and corresponding congruent viewing gimbal (right).

Fig. 8
Fig. 8

Penta-prism effects. Dotted object F is seen as the solid F. Tilting the prism by ϕin and θin causes the image to gyrate on congruent gimbals G31 and G13.

Tables (1)

Tables Icon

Table I Reference matrices for Porro, Penta, and Wollaston prisms. The mirror products are given also, indicated by their indices (M20M30=2030).

Equations (88)

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

V = V A B C .
V = V θ i .
θ 1 = ( 1 0 0 0 cos θ sin θ 0 - sin θ cos θ )             θ 2 = ( cos θ 0 - sin θ 0 1 0 sin θ 0 cos θ ) θ 3 = ( cos θ sin θ 0 - sin θ cos θ 0 0 0 1 ) .
G i j = θ i ϕ j .
V θ i ϕ j = V = V G i j .
ϕ ˜ j θ ˜ i T θ i ϕ j = T = G ˜ i j T G i j .
G i j = λ ˜ k θ i λ ˜ k ϕ j .
Tr θ i = θ ˜ i = ( θ i ) - 1 = ( - θ ) i ,
θ i φ i = ( φ + θ ) i ,
θ ˜ i φ i = φ i θ ˜ i = ( φ - θ ) i ,
U i θ i = U i ; θ i Ũ i = Ũ i .
M i k = U i M i k Ũ k = ( U i M i j ) ( M j k U k ) ,
θ ˜ i M i 0 θ i = M i 0 = M i .
M ( N ) = I - 2 Ñ N ,
M M = M M ˜ = M ˜ M = M ˜ M ˜ = I .
P ˜ = P - 1
P P ˜ = M N O O ˜ Ñ M ˜ = M N Ñ M ˜ = M M ˜ = I .
θ ˜ 1 X 1 θ 1 = X 1 .
P o i = π i ;             P o 2 = π 2 ;             P o 2 = π 3 .
P e 1 + = W o 1 + = N 1 ;             P e 2 + = W o 2 + = N 2 ; P e 3 + = W o 3 + = N 3 ,
P e 1 - = W o 1 - = Ñ 1 ;             P e 2 - = W o 2 - = Ñ 2 ; P e 3 - = W o 3 - = Ñ 3 .
R i k = ( δ i k cos d + ( 1 - cos d ) n i n k - sin d i k m n m ) .
N o = ± N i P ,
V = U 3 L ˜ 1 A 2 .
W = V T ˜ 3 P o 1 T 3 .
W = U 3 L ˜ 1 A 2 ( T ˜ 3 P o 1 T 3 ) = - U 3 L ˜ 1 A 2 = - V .
U 3 L ˜ 1 A 2 T ˜ 3 P o 1 = - U 3 L ˜ 1 A 2 T ˜ 3 I ;
U 3 L ˜ 1 A 2 T ˜ 3 ( P o 1 + I ) = 0.
( 1 0 0 0 - 1 0 0 0 - 1 ) + ( 1 0 0 0 1 0 0 0 1 ) = ( 2 0 0 0 0 0 0 0 0 ) .
U 3 L ˜ 1 A 2 T ˜ 3 Ũ 1 = 0.
( 001 ) ( 1 0 0 0 cos L - sin L 0 sin L cos L ) ( cos A 0 - sin A 0 1 0 sin A 0 cos A ) ( cos T - sin T 0 sin T cos T 0 0 0 1 ) ( 1 0 0 ) = ( 0 sin L cos L ) ( cos A 0 - sin A 0 1 1 sin A 0 cos A ) ( cos T sin T 0 ) = ( 0 sin L cos L ) ( cos A cos T sin T sin A cos T ) = sin L sin T + cos L sin A cos T = 0.
sin A = - tan L tan T .
N = U 1 T 3 , Ñ = T ˜ 3 Ũ 1 ,
V = U 3 L ˜ 1 A 2 ;
V Ñ = U 3 L ˜ 1 A 2 T ˜ 3 Ũ 1 = 0.
S = - R             or             - S = R ,
- S = U 1 δ 3 2 = R = U 2 E ˜ 2 D ˜ 1 M 3 D 1 ˆ E 2 θ ˜ 2 M 3 θ 2 .
- S 3 = cos δ sin .
R 3 = U 2 P P U 3 = R C ,
R = U 2 P = U 2 E ˜ 2 D ˜ 1 M 3 D 1 ,
C = P Ũ 3 = E 2 θ ˜ 2 M 3 θ 2 Ũ 3 .
R C = ( 0 cos 2 D sin 2 D ) ( sin ( E - 2 θ ) 0 - cos ( E - 2 θ ) ) = - sin 2 D · cos ( E - 2 θ ) = R 3 .
- S 3 = R 3 = cos δ sin = - sin 2 D cos ( E - 2 θ ) ,
sin ( 90 - δ ) sin = - sin 2 D sin ( 90 - E + 2 θ ) .
90 - δ = - 2 D             and             = 90 - E + 2 θ ,
D = ( δ / 2 ) - 45             and             E = 90 + 2 θ - = 50 - ,
90 - δ = 2 D             and             = E - 2 θ - 90 = E - 50 ,
D = 45 - δ / 2             and             E = + 50.
O = O P = O G = O α i β j .
O = O ϕ ˜ P ϕ = O G = O α i β j .
G s = G 312 = T 3 E ˜ 1 A 2 .
S = S P ˜ = S G ;
S G = S φ ˜ P ˜ φ = S P ˜ G ,             φ ˜ P ˜ φ = P ˜ G ,
G = P φ ˜ P ˜ φ .
O P = O M 31 ( R ˜ 2 P ˜ 1 M 3 P 1 R 2 ) = W = O T 3 E ˜ 1 A 2 .
U 3 ( M 31 R ˜ 2 P ˜ 1 M 3 P 1 R 2 ) 32 Ũ 2 = W 2 = U 3 ( T 3 E ˜ 1 A 2 ) 32 Ũ 2 = U 3 E 1 Ũ 2 .
W 2 = cos 2 P - sin 2 P = cos 2 P = sin E = sin ( 90 - 2 P ) ,             or             E = 90 - 2 P , - 10° ,             for             P = + 50° .
W 1 = 2 sin P cos P sin R = sin 2 P sin R = cos E sin A ,             or             A = R = 90° .
W 2 = 2 sin P cos P cos R = sin 2 P cos R = cos T cos E ,             or             T = R , = 90° .
P e 2 + φ ˜ 1 P e 2 - φ 1 = N 2 φ ˜ 1 Ñ 2 φ 1 = ( N 2 φ ˜ 1 Ñ 2 ) φ 1 = φ ˜ 3 φ 1 = G 31 .
P e 2 + θ ˜ 3 P e 2 - θ 3 = N 2 θ ˜ 3 N 2 θ 3 = ( N 2 θ ˜ 3 Ñ 2 ) θ 3 = θ 1 θ 3 = G 13 .
P N = Ñ N .
P N = P ( U 3 ) = Ũ 3 U 3 = ( 0 0 0 0 0 0 0 0 1 ) .
M = ( I - P N ) - P N = I - 2 P N = I - 2 Ñ N .
M = ( 1 0 0 0 1 0 0 0 1 ) - 2 ( l m n ) ( l m n ) = ( 1 - 2 l 2 - 2 l m - 2 l n - 2 l m 1 - 2 m 2 - 2 m n - 2 l n - 2 m n 1 - 2 n 2 ) .
N = N L             and             Ñ = L ˜ Ñ .
M ¯ = I - 2 Ñ N = I - 2 L ˜ Ñ N L = L ˜ ( I - 2 Ñ N ) L = L ˜ M L .
m i j = δ i j - 2 n i n j .
P = ( I - 2 M ˜ M ) ( I - 2 Ñ N ) = I - 2 ( M ˜ M - 2 Ñ N + 4 M ˜ M Ñ N ) .
M = N , M Ñ = 1 , P = I - 2 M ˜ M - 2 M ˜ M + 4 M ˜ M = I .
M Ñ = 0 , P = I - 2 ( M ˜ M + Ñ N ) .
M M = M M ˜ = M ˜ M = I .
P P ˜ = M N O O ˜ Ñ M ˜ = M N Ñ M ˜ = M M ˜ = I = P P - 1 .
V ( u , v , w ) = i k m u i v k w m .
T = i k m u i 1 v k 2 w m 3 = V ( u , v , w ) .
T P = T .
V = T P = V P = i k m u i v k w m P = ɛ i k m u i v k w m .
ɛ i k m = - i k m .
V P = a V = V R ˜ P R             a = ± 1
V R ˜ P = a V R ˜ = W P = a W             if
V R ˜ = W .
I - 2 Ñ N = R ˜ ( I - 2 Ñ N ) R = I - 2 R ˜ Ñ N R .
R ˜ Ñ = Ñ ;             N R = N .
V ( I - 2 M ˜ M ) ( I - 2 Ñ N ) = a V = V             if             a = 1 V ( I - 2 M ˜ M ) = V ( I - 2 Ñ N )
V M ˜ M = V Ñ N = 0 ,
R = - L M Ñ + M N L ˜ - N L M ˜
Magnitude of R = sin 1 2 ϕ .
θ ˜ 1 P θ 1 = θ ˜ 1 ϕ 1 θ 1 = ϕ 1 = P ,