Abstract

A matrix [M] is derived for a system of plane first-surface mirrors which transforms a point or the direction cosines of a ray in the image space to the object space. If the general displacement of the mirror system is represented by the screw matrix [S], then the new position of a point image is given by the transformation [S][M][S]−1. This method facilitates designing the mirror supporting structure for minimum image shift due to structural deflections.

© 1960 Optical Society of America

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References

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  1. Michel Chasles, Bull. sci. math. phys. chim. 14, 321–326 (1830).
  2. Joseph Stiles Beggs, Ein Beitrag zur Analyse Räumlichen Mechanismen, Diss. Technische Hochschule, Hanover, Germany (1959).
  3. S. Falk, Z. angew. Math. Mech. 31, 152–153 (1951).
    [Crossref]
  4. H. Wagner, Optik 8, 456–472 (1951).
  5. T. Smith, Trans. Opt. Soc. 30, 68 (1928).
    [Crossref]
  6. See reference 5, p. 79.
  7. T. Y. Baker, Trans. Opt. Soc. 29, p. 49 (1927–1928).
    [Crossref]
  8. See reference 7, p. 187.
  9. Felicien Blottiau, Rev. opt. 33, 339 (1954).

1954 (1)

Felicien Blottiau, Rev. opt. 33, 339 (1954).

1951 (2)

S. Falk, Z. angew. Math. Mech. 31, 152–153 (1951).
[Crossref]

H. Wagner, Optik 8, 456–472 (1951).

1928 (1)

T. Smith, Trans. Opt. Soc. 30, 68 (1928).
[Crossref]

1830 (1)

Michel Chasles, Bull. sci. math. phys. chim. 14, 321–326 (1830).

Baker, T. Y.

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

Beggs, Joseph Stiles

Joseph Stiles Beggs, Ein Beitrag zur Analyse Räumlichen Mechanismen, Diss. Technische Hochschule, Hanover, Germany (1959).

Blottiau, Felicien

Felicien Blottiau, Rev. opt. 33, 339 (1954).

Chasles, Michel

Michel Chasles, Bull. sci. math. phys. chim. 14, 321–326 (1830).

Falk, S.

S. Falk, Z. angew. Math. Mech. 31, 152–153 (1951).
[Crossref]

Smith, T.

T. Smith, Trans. Opt. Soc. 30, 68 (1928).
[Crossref]

Wagner, H.

H. Wagner, Optik 8, 456–472 (1951).

Bull. sci. math. phys. chim. (1)

Michel Chasles, Bull. sci. math. phys. chim. 14, 321–326 (1830).

Optik (1)

H. Wagner, Optik 8, 456–472 (1951).

Rev. opt. (1)

Felicien Blottiau, Rev. opt. 33, 339 (1954).

Trans. Opt. Soc. (2)

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

T. Smith, Trans. Opt. Soc. 30, 68 (1928).
[Crossref]

Z. angew. Math. Mech. (1)

S. Falk, Z. angew. Math. Mech. 31, 152–153 (1951).
[Crossref]

Other (3)

See reference 7, p. 187.

See reference 5, p. 79.

Joseph Stiles Beggs, Ein Beitrag zur Analyse Räumlichen Mechanismen, Diss. Technische Hochschule, Hanover, Germany (1959).

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Figures (3)

Fig. 1
Fig. 1

Corner reflector.

Fig. 2
Fig. 2

A nut turning on a screw fixed in a reference system.

Fig. 3
Fig. 3

The optical square rotated about an axis normal to the paper and through the point x=a, z=b.

Tables (3)

Tables Icon

Table I Coefficients in the equations of the reflecting surfaces of the corner reflector in Fig. 1.

Tables Icon

Table II The screw matrix. S, C, and V are written for the sine, cosine, and versine (1-cosine), respectively. The screw axis passes through the point (0x,0y,0z) and has direction cosines (Cxs,Cys,Czs) relative to XYZ, a right-hand coordinate system. The screw advances the body on which it operates a distance s while rotating it through an angle σ. The determinant of the screw matrix equals 1.

Tables Icon

Table III Computation of [S][M][S]−1.

Equations (55)

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A x + B y + C z + D = 0 ,
d = A x 1 + B y 1 + C z 1 + D F ,
F = ( A 2 + B 2 + C 2 ) 1 2             ( F > 0 ) .
C x n = A / F             C y n = B / F             C z n = C / F .
x 1 = x 1 - 2 d A F = x 1 - 2 ( A x 1 + B y 1 + C z 1 + D ) A F 2 ,
( 1 x 1 y 1 z 1 ) = = ( 1 0 0 0 - 2 A D / F 2 1 - 2 A 2 / F 2 - 2 A B / F 2 - 2 A C / F 2 - 2 B D / F 2 - 2 A B / F 2 1 - 2 B 2 / F 2 - 2 B C / F 2 - 2 C D / F 2 - 2 A C / F 2 - 2 B C / F 2 1 - 2 C 2 / F 2 ) ( 1 x 1 y 1 z 1 ) .
[ M ] = [ M n ] [ M 3 ] [ M 2 ] [ M 1 ]
( 1 x 1 y 1 z 1 ) = [ M ] ( 1 x 1 y 1 z 1 ) .
C x r = x 2 - x 1 e ,             C y r = y 2 - y 1 e ,             C z r = z 2 - z 1 e ,
( C x r C y r C z r ) = ( 1 - 2 A 2 / F 2 - 2 A B / F 2 - 2 A C / F 2 - 2 A B / F 2 1 - 2 B 2 / F 2 - 2 B C / F 2 - 2 A C / F 2 - 2 B C / F 2 1 - 2 C 2 / F 2 ) ( C x r C y r C z r ) .
( 1 x y z ) = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 - 1 ) Mirror 3             ( 1 0 0 0 0 1 0 0 0 0 - 1 0 0 0 0 1 ) Mirror 2             ( 1 0 0 0 0 - 1 0 0 0 0 1 0 0 0 0 1 ) Mirror 1             ( 1 x y z ) .
( x y z ) = = ( - 1 0 0 0 - 1 0 0 0 - 1 ) ( 10 10 10 ) .
x = - 10 ,             y = - 10 ,             z = - 10.
( C x r C y r C z r ) = - [ 1 ] ( C x r C y r C z r ) ,
( 1 x y z ) = [ S ] ( 1 x y z ) .
( x 2 y 2 z 2 ) = ( x 1 y 1 z 1 )
[ T 2 ] ( 1 x y z ) = [ T 1 ] ( 1 x y z ) .
[ T 2 ] = [ T 12 ] [ T 1 ] .
[ T 2 ] - 1 = { [ T 12 ] [ T 1 ] } - 1 = [ T 1 ] - 1 [ T 12 ] - 1 = [ T 1 ] - 1 [ T 21 ] .
( 1 x y z ) = [ T 1 ] - 1 [ T 21 ] [ T 1 ] ( 1 x y z ) .
[ S ] = [ T 1 ] - 1 [ T 21 ] [ T 1 ] .
C σ = 1 2 ( S 22 + S 33 + S 44 - 1 ) ,
C x s = S 43 - S 34 2 S σ ,             C y s = S 24 - S 42 2 S σ ,             C z s = S 32 - S 23 2 S σ ,
s = S 21 C x s + S 31 C y s + S 41 C z s .
x 0 = 0
y 0 = S 21 ( - V σ + V σ C z s 2 ) - S 41 ( S σ C y s + V σ C x s C z s ) + s ( V σ C x s + S σ C y s C z s ) 2 V σ C x s C y s
z 0 = S 21 ( - V σ + V σ C y s 2 ) - S 31 ( - S σ C z s + V σ C x s C y s ) + s ( V σ C x s - S σ C y s C z s ) 2 V σ C x s C z s .
( 1 x y z ) = [ M ] [ S ] - 1 ( 1 x y z ) ,
( 1 x y z ) = [ M ] ( 1 x y z )
( 1 x y z ) = [ S ] [ M ] [ S ] - 1 ( 1 x y z ) .
[ D ] [ M ] = [ S ] [ M ] [ S ] - 1 .
[ D ] = [ S ] [ M ] [ S ] - 1 [ M ] - 1 .
M 1 = ( 1 0 0 0 0 C 2 α 0 S 2 α 0 0 1 0 0 S 2 α - C 2 α ) .
( 0 δ x δ y δ z ) = { [ S ] [ M ] [ S ] - 1 - [ M ] } ( 1 x y z ) .
δ x = a ( S σ + V σ ) - b ( S σ - V σ ) , δ y = 0 , δ z = a ( S σ - V σ ) + b ( S σ + V σ ) .
Δ x 1 = x 1 - x 1 , etc .
A = Δ y 1 ( Δ z 2 - Δ z 3 ) + Δ y 2 ( Δ z 3 - Δ z 1 ) + Δ y 3 ( Δ z 1 - Δ z 2 ) ,
B = Δ z 1 ( Δ x 2 - Δ x 3 ) + Δ z 2 ( Δ x 3 - Δ x 1 ) + Δ z 3 ( Δ x 1 - Δ x 2 ) ,
C = Δ x 1 ( Δ y 2 - Δ y 3 ) + Δ x 2 ( Δ y 3 - Δ y 1 ) + Δ x 3 ( Δ y 1 - Δ y 2 ) .
C x s = A ( A 2 + B 2 + C 2 ) 1 2 ,             C y s = B ( A 2 + B 2 + C 2 ) 1 2 ,             C z s = C ( A 2 + B 2 + C 2 ) 1 2 .
s = Δ x i C x s + Δ y i C y s + Δ z i C z s ,
C σ = A 1 A 2 + B 1 B 2 + C 1 C 2 ( A 1 2 + B 1 2 + C 1 2 ) 1 2 ( A 2 2 + B 2 2 + C 2 2 ) 1 2 ,
A 1 = [ C y s ( z 2 - z 1 ) - C z s ( y 2 - y 1 ) ] A 2 = [ C y s ( z 2 - z 1 ) - C z s ( y 2 - y 1 ) ]
B 1 = [ C z s ( x 2 - x 1 ) - C x s ( z 2 - z 1 ) ] B 2 = [ C z s ( x 2 - x 1 ) - C x s ( z 2 - z 1 ) ]
C 1 = [ C x s ( y 2 - y 1 ) - C y s ( x 2 - x 1 ) ] C 2 = [ C x s ( y 2 - y 1 ) - C y s ( x 2 - x 1 ) ] .
x 1 = x 1 + s C x s             y 1 = y 1 + s C y s             z 1 = z 1 + s C z s .
d = + [ ( x 1 - x 1 ) 2 + ( y 1 - y 1 ) 2 + ( z 1 - z 1 ) 2 ] 1 2 .
C x x 1 = x 1 - x 1 d ,             C x y 1 = y 1 - y 1 d ,             C x z 1 = z 1 - z 1 d .
C z x 1 = C x s             C z y 1 = C y s             C z z 1 = C z s .
C y x 1 = - C x y 1 C z s + C y s C x z 1 ,
C y y 1 = C x x 1 C z s - C x s C x z 1 ,
C y z 1 = - C x x 1 C y s + C x s C x y 1 .
x 0 = x 1 + x 1 2 + d C y x 1 2 tan ( σ / 2 ) ,
y 0 = y 1 + y 1 2 + d C y y 1 2 tan ( σ / 2 ) ,
z 0 = z 1 + z 1 2 + d C y z 1 2 tan ( σ / 2 ) .