Abstract

The introduction of the natural and the diapoint variables has simplified considerably the investigation of problems in the large, problems where we can give the eiconal in closed form, and therefore for each object ray the image ray. The author and H. Boegehold have, first together and then alone, investigated problems of this kind. The author sketches first once more his new method, and gives then a series of examples, such as the eiconal for the case where to each object point belongs a rotation symmetric caustic, the eiconal for sharp image formation of two surfaces, the eiconal in the case where there is perfect centric symmetry, and other examples which are discussed as far as the time limit permits.

© 1937 Optical Society of America

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References

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  1. W. R. Hamilton, third supplement to an essay on the theory of systems of rays. Trans. Irish Academy 17, 1–144 (1837); also in Mathematical Papers, Vol.  I (1931).
  2. H. Bruns, “Das Eikonal,” Leipzig Sitz. ber. 21, 321–436 (1895).
  3. M. Herzberger, Strahlenoptik (J. Springer, 1931), pp. 167–175.
    [Crossref]
  4. See for instance M. Herzberger, “On the Fundamental Optical Invariant,” J. O. S. A. 25, 295–304 (1935).
    [Crossref]
  5. M. Herzberger, “New Theory of Optical Image Formation,” J. O. S. A. 26, 197–204 (1936).
    [Crossref]
  6. H. Boegehold and M. Herzberger, “Kugelsymmetrische Systeme,” Zeits. für Ang. Math. u. Mech. 15, 157–178 (1935).
    [Crossref]
  7. C. Maxwell, On the general laws of optical instruments, Scientific Papers I (1858), pp. 271–285.
  8. H. Boegehold and M. Herzberger, “Kann man zwei verschiedene optische Flächen scharf abbilden?” Comp. Math. 1, 1–29 (1935).
  9. H. Boegehold, “Raumsymmetrische Abbildung,” Zeits. f. Instrumentenk. 56, 98–109 (1936).

1936 (2)

M. Herzberger, “New Theory of Optical Image Formation,” J. O. S. A. 26, 197–204 (1936).
[Crossref]

H. Boegehold, “Raumsymmetrische Abbildung,” Zeits. f. Instrumentenk. 56, 98–109 (1936).

1935 (3)

See for instance M. Herzberger, “On the Fundamental Optical Invariant,” J. O. S. A. 25, 295–304 (1935).
[Crossref]

H. Boegehold and M. Herzberger, “Kann man zwei verschiedene optische Flächen scharf abbilden?” Comp. Math. 1, 1–29 (1935).

H. Boegehold and M. Herzberger, “Kugelsymmetrische Systeme,” Zeits. für Ang. Math. u. Mech. 15, 157–178 (1935).
[Crossref]

1895 (1)

H. Bruns, “Das Eikonal,” Leipzig Sitz. ber. 21, 321–436 (1895).

1858 (1)

C. Maxwell, On the general laws of optical instruments, Scientific Papers I (1858), pp. 271–285.

1837 (1)

W. R. Hamilton, third supplement to an essay on the theory of systems of rays. Trans. Irish Academy 17, 1–144 (1837); also in Mathematical Papers, Vol.  I (1931).

Boegehold, H.

H. Boegehold, “Raumsymmetrische Abbildung,” Zeits. f. Instrumentenk. 56, 98–109 (1936).

H. Boegehold and M. Herzberger, “Kugelsymmetrische Systeme,” Zeits. für Ang. Math. u. Mech. 15, 157–178 (1935).
[Crossref]

H. Boegehold and M. Herzberger, “Kann man zwei verschiedene optische Flächen scharf abbilden?” Comp. Math. 1, 1–29 (1935).

Bruns, H.

H. Bruns, “Das Eikonal,” Leipzig Sitz. ber. 21, 321–436 (1895).

Hamilton, W. R.

W. R. Hamilton, third supplement to an essay on the theory of systems of rays. Trans. Irish Academy 17, 1–144 (1837); also in Mathematical Papers, Vol.  I (1931).

Herzberger, M.

M. Herzberger, “New Theory of Optical Image Formation,” J. O. S. A. 26, 197–204 (1936).
[Crossref]

See for instance M. Herzberger, “On the Fundamental Optical Invariant,” J. O. S. A. 25, 295–304 (1935).
[Crossref]

H. Boegehold and M. Herzberger, “Kann man zwei verschiedene optische Flächen scharf abbilden?” Comp. Math. 1, 1–29 (1935).

H. Boegehold and M. Herzberger, “Kugelsymmetrische Systeme,” Zeits. für Ang. Math. u. Mech. 15, 157–178 (1935).
[Crossref]

M. Herzberger, Strahlenoptik (J. Springer, 1931), pp. 167–175.
[Crossref]

Maxwell, C.

C. Maxwell, On the general laws of optical instruments, Scientific Papers I (1858), pp. 271–285.

Comp. Math. (1)

H. Boegehold and M. Herzberger, “Kann man zwei verschiedene optische Flächen scharf abbilden?” Comp. Math. 1, 1–29 (1935).

J. O. S. A. (2)

See for instance M. Herzberger, “On the Fundamental Optical Invariant,” J. O. S. A. 25, 295–304 (1935).
[Crossref]

M. Herzberger, “New Theory of Optical Image Formation,” J. O. S. A. 26, 197–204 (1936).
[Crossref]

Leipzig Sitz. ber. (1)

H. Bruns, “Das Eikonal,” Leipzig Sitz. ber. 21, 321–436 (1895).

Scientific Papers I (1)

C. Maxwell, On the general laws of optical instruments, Scientific Papers I (1858), pp. 271–285.

third supplement to an essay on the theory of systems of rays. Trans. Irish Academy (1)

W. R. Hamilton, third supplement to an essay on the theory of systems of rays. Trans. Irish Academy 17, 1–144 (1837); also in Mathematical Papers, Vol.  I (1931).

Zeits. f. Instrumentenk. (1)

H. Boegehold, “Raumsymmetrische Abbildung,” Zeits. f. Instrumentenk. 56, 98–109 (1936).

Zeits. für Ang. Math. u. Mech. (1)

H. Boegehold and M. Herzberger, “Kugelsymmetrische Systeme,” Zeits. für Ang. Math. u. Mech. 15, 157–178 (1935).
[Crossref]

Other (1)

M. Herzberger, Strahlenoptik (J. Springer, 1931), pp. 167–175.
[Crossref]

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Equations (23)

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V ( x , y , z ;             x , y , z ) W ( x , y , z ;             ξ , η , ζ ) T ( ξ , η , ζ ;             ξ , η , ζ )
n ξ = - V x , n ξ = + V x , n η = - V y , n η = + V y , n ζ = - V z , n ζ = + V z ,
ξ 2 + η 2 + ζ 2 = ξ 2 + η 2 + ζ 2 = 1.
V ( x , y ,             x , y ) W ( x , y ,             ξ , η ) T ( ξ , η ,             ξ , η )
n ξ = - V x n ξ = W x n x = T ξ n η = - V y n η = W y n y = T η or or n ξ = V x n ξ = W ξ n x = - T ξ n η = V y n η = W η n y = - T η .
V x x V y y - V x y V y x 0 W x ξ W y η - W x η W y ξ 0 T ξ ξ T η η - T ξ η T η ξ 0
W = f ( x y ) ξ + g ( x y ) η + h ( x y ) .
n x = f ( x y ) ,             n y = g ( x y ) .
W = f ( ξ , η ) x + g ( ξ , η ) y + h ( ξ , η ) .
n ξ = f ( ξ , η ) ,             n η = g ( ξ , η ) ,
V = f ( x y ) x + g ( x y ) y + h ( x y ) .
n ξ = - f ( x , y ) ,             n η = - g ( x , y ) .
W = n ( x ξ + y η ) .
ξ = ξ , n x = n x , η = η , n y = n y .
ξ 2 + η 2 ,             2 ( ξ ξ + η η ) ,             ξ 2 + η 2 .
x 2 + y 2 ,             2 ( x x + y y ) ,             x 2 + y 2 .
κ = ζ ,             ν = ξ ξ + η η + ζ ζ ,             μ = ζ .
n f + n f = 0.
T λ = n f = - n f ,
T = f ( λ ) .
T = R [ ( β 2 + 1 / β ) - 2 λ ] 1 2 = R [ ( β ξ - ξ ) 2 + ( β η - η ) 2 + ( β ζ - ζ ) 2 β ] 1 2 .
T = R [ ( β ξ - ξ ) 2 + ( β η - η ) 2 + 1 - β 2 1 - α 2 ( α ζ - ζ ) 2 β ] 1 2 .
T = R ( 1 - λ - μ ˜ κ ˜ / C ) 1 2 ,