Abstract

This paper presents a paraxial study of the various types of four-component mechanically compensated zoom lens in terms of Gaussian Brackets. The expressions which define the displacement of components at zooming, the extremum of displacement, and the singular point of displacement are derived. As a result of the study, a new classification of four-component mechanically compensated zoom lenses is given. Some numerical examples of the zooming locus are shown.

© 1982 Optical Society of America

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Errata

Kazuo Tanaka, "Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type. Errata," Appl. Opt. 21, 3805-3805 (1982)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-21-21-3805

References

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  1. L. Bergstein, J. Opt. Soc. Am. 48, 154 (1958).
    [CrossRef]
  2. L. Bergstein, L. Motz, J. Opt. Soc. Am. 52, 353 (1962).
    [CrossRef]
  3. L. Bergstein, L. Motz, J. Opt. Soc. Am. 52, 363 (1962).
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  4. L. Bergstein, L. Motz, J. Opt. Soc. Am. 52, 376 (1962).
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  5. G. Wooters, E. W. Silvertooth, J. Opt. Soc. Am. 55, 347 (1965).
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  6. G. H. Matter, E. T. Luszcz, Appl. Opt. 9, 844 (1970).
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  8. K. Tanaka, Opt. Commun. 29, 138 (1979).
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  9. W. Besenmatter, Optik 47, 153 (1977).
  10. W. Besenmatter, Optik 47, 381 (1977).
  11. W. Besenmatter, Optik 48, 289 (1977).
  12. W. Besenmatter, Optik 49, 1 (1977).
  13. W. Besenmatter, Optik 49, 325 (1977).
  14. W. Besenmatter, Optik 51, 147 (1978).
  15. W. Besenmatter, Optik 51, 385 (1978).
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  23. See, for example, T. Poston, I. N. Stewart, Taylor Expansions and Catastrophes (Pitman, London, 1979), p. 15.
  24. K. Tanaka, Japanese Laid Open Patent Application54-28658 (9Aug.1979).
  25. K. Tanaka, Japanese Laid Open Patent Application54-28659 (9Aug.1979).
  26. K. Tanaka, Japanese Laid Open Patent Application56-25710 (12Mar.1981).
  27. K. Tanaka, Deutsches Patent Offenlegungsschrift2911794 (4Oct.1979).
  28. H. H. Hopkins, U.S.Patent2,746,350 (22May1956).
  29. E. Takano, Japanese Laid Open Patent Application51-144246 (5June1976).
  30. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 460.

1980

W. Besenmatter, Optik 57, 123 (1980).

1979

K. Tanaka, Opt. Commun. 29, 138 (1979).
[CrossRef]

1978

W. Besenmatter, Optik 51, 147 (1978).

W. Besenmatter, Optik 51, 385 (1978).

1977

W. Besenmatter, Optik 47, 153 (1977).

W. Besenmatter, Optik 47, 381 (1977).

W. Besenmatter, Optik 48, 289 (1977).

W. Besenmatter, Optik 49, 1 (1977).

W. Besenmatter, Optik 49, 325 (1977).

1972

1970

1967

K. Yamaji, Prog. Opt. 6, 107 (1967).

1965

1963

1962

1958

1952

1943

Bergstein, L.

Besenmatter, W.

W. Besenmatter, Optik 57, 123 (1980).

W. Besenmatter, Optik 51, 147 (1978).

W. Besenmatter, Optik 51, 385 (1978).

W. Besenmatter, Optik 47, 153 (1977).

W. Besenmatter, Optik 47, 381 (1977).

W. Besenmatter, Optik 48, 289 (1977).

W. Besenmatter, Optik 49, 1 (1977).

W. Besenmatter, Optik 49, 325 (1977).

Delano, E.

Herzberger, M.

M. Herzberger, J. Opt. Soc. Am. 42, 637 (1952).
[CrossRef]

M. Herzberger, J. Opt. Soc. Am. 33, 651 (1943).
[CrossRef]

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 460.

Hopkins, H. H.

H. H. Hopkins, U.S.Patent2,746,350 (22May1956).

H. H. Hopkins, U.S. Patent2,741,155 (10Apr.1956).

Luszcz, E. T.

Matter, G. H.

Motz, L.

O'Neill, E. L.

See, for example, E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), p. 32.

Poston, T.

See, for example, T. Poston, I. N. Stewart, Taylor Expansions and Catastrophes (Pitman, London, 1979), p. 15.

Sands, P. J.

Silvertooth, E. W.

Stewart, I. N.

See, for example, T. Poston, I. N. Stewart, Taylor Expansions and Catastrophes (Pitman, London, 1979), p. 15.

Takano, E.

E. Takano, Japanese Laid Open Patent Application51-144246 (5June1976).

Tanaka, K.

K. Tanaka, Opt. Commun. 29, 138 (1979).
[CrossRef]

K. Tanaka, Japanese Laid Open Patent Application54-28658 (9Aug.1979).

K. Tanaka, Japanese Laid Open Patent Application54-28659 (9Aug.1979).

K. Tanaka, Japanese Laid Open Patent Application56-25710 (12Mar.1981).

K. Tanaka, Deutsches Patent Offenlegungsschrift2911794 (4Oct.1979).

Wooters, G.

Yamaji, K.

K. Yamaji, Prog. Opt. 6, 107 (1967).

Appl. Opt.

J. Opt. Soc. Am.

Opt. Commun.

K. Tanaka, Opt. Commun. 29, 138 (1979).
[CrossRef]

Optik

W. Besenmatter, Optik 47, 153 (1977).

W. Besenmatter, Optik 47, 381 (1977).

W. Besenmatter, Optik 48, 289 (1977).

W. Besenmatter, Optik 49, 1 (1977).

W. Besenmatter, Optik 49, 325 (1977).

W. Besenmatter, Optik 51, 147 (1978).

W. Besenmatter, Optik 51, 385 (1978).

W. Besenmatter, Optik 57, 123 (1980).

Prog. Opt.

K. Yamaji, Prog. Opt. 6, 107 (1967).

Other

H. H. Hopkins, U.S. Patent2,741,155 (10Apr.1956).

See, for example, E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), p. 32.

See, for example, T. Poston, I. N. Stewart, Taylor Expansions and Catastrophes (Pitman, London, 1979), p. 15.

K. Tanaka, Japanese Laid Open Patent Application54-28658 (9Aug.1979).

K. Tanaka, Japanese Laid Open Patent Application54-28659 (9Aug.1979).

K. Tanaka, Japanese Laid Open Patent Application56-25710 (12Mar.1981).

K. Tanaka, Deutsches Patent Offenlegungsschrift2911794 (4Oct.1979).

H. H. Hopkins, U.S.Patent2,746,350 (22May1956).

E. Takano, Japanese Laid Open Patent Application51-144246 (5June1976).

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 460.

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Figures (6)

Fig. 1
Fig. 1

Four-component mechanically compensated zoom lens: (a) original position; (b) zooming position.

Fig. 2
Fig. 2

Numerical example of zooming locus of Regular Type I. The lens data at the wide angle setting are as follows: ϕ1 = 0.0241395; ϕ2 = −0.0792124; ϕ3 = 0.0251038; ϕ4 = 0.0441316; e10 = 7.5148; e20 = 19.7235; e30 = 34.5771; and s F O = 22.6595 . This zoom lens is for a system capable of continuously covering a focal length range from 13.975 to 39.734 as the variator moves from 0.000 to 13.800.

Fig. 3
Fig. 3

Numerical example of zooming locus of Regular Type II. The lens data at the wide angle setting are as follows: ϕ1 = 0.0149391; ϕ2 = −0.0699221; ϕ3 = −0.0272540; ϕ4 = 0.0236637; e10 = 5.9555; e20 = 52.9703; e30 = 49.3304; and s F O = 99.2451 . This zoom lens is for a system capable of continuously covering a focal length range from 9.338 to 118.025 as the variator moves from 0.000 to 43.541.

Fig. 4
Fig. 4

Numerical example of zooming locus of Regular Type III. The lens data at symmetrical configuration are as follows: ϕ1 = ϕ4 = 0.0699530; ϕ2 = ϕ3 = −0.1249904; e10 = e30 = 6.2947; e20 = 0.7391; and. s F O = . This zoom lens is for an afocal system capable of continuously covering an angular magnification range from 0.4463 to 2.2301 as the variator moves from −4.167 to 1.860.

Fig. 5
Fig. 5

Numerical example of zooming locus of Singular Type I. The lens data at singular point are as follows: ϕ1 = 0.0059914; ϕ2 = −0.0270204; ϕ3 = 0.0184525; ϕ4 = 0.0113114; e10 = 92.8831; e20 = 34.3680; e30 = 126.9040; and s F O = 23.4162 . This zoom lens is for a system capable of continuously covering a focal length range from 16.362 to 275.126 as the variator moves from −111.000 to 5.000.

Fig. 6
Fig. 6

Numerical example of zooming locus of Singular Type III. The lens data at singular point are as follows: ϕ1 = 0.0100000; ϕ2 = 0.0500000; ϕ3 = 0.0400000; ϕ4 = 0.1500000; e10 = 100.0000; e20 = 50.0000; e30 = 70.0000; and s F O = 100.0000 . This zoom lens is for a system capable of continuously covering a focal length range from 1.920 to 285.200 as the variator moves from −90.000 to 12.000.

Tables (1)

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Table I Classification of Four-Component Mechanically Compensated Zoom Lens

Equations (94)

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s F O = A 40 1 C 40 1 ,
A 40 1 = [ ϕ 1 , e 10 , ϕ 2 , e 20 , ϕ 3 , e 30 ] , C 40 1 = [ ϕ 1 , e 10 , ϕ 2 , e 20 , ϕ 3 , e 30 , ϕ 4 ] . }
s F = A 4 1 C 4 1 ,
A 4 1 = [ ϕ 1 , e 1 , ϕ 2 , e 2 , ϕ 3 , e 3 ] , C 4 1 = [ ϕ 1 , e 1 , ϕ 2 , e 2 , ϕ 3 , e 3 , ϕ 4 ] , e 1 = e 10 + x 1 , e 2 = e 20 x 1 + x 2 , e 3 = e 30 x 2 . }
s F = s F O ,
Z ( x 1 , x 2 ) = A 4 1 C 4 1 s F O = 0 .
lim s F O Z ( x 1 , x 2 ) = C 4 1 = 0 .
X 1 x 1 2 + Y 1 x 1 + Z 1 = 0 ,
X 1 = I 1 , Y 1 = I 1 ( e 10 + e 20 + x 2 ) + J 1 K 1 , Z 1 = I 1 e 10 ( e 20 + x 2 ) + J 1 e 10 + K 1 ( e 20 + x 2 ) + L 1 , I 1 = ϕ 1 ϕ 2 3 A 5 , J 1 = ϕ 1 ( A 5 3 + ϕ 2 3 B 5 ) , K 1 = ( ϕ 1 + ϕ 2 ) 3 A 5 , L 1 = A 5 3 + ( ϕ 1 + ϕ 2 ) 3 B 5 , A 5 3 = [ ϕ 3 , e 3 , ϕ 4 , s F O ] , B 5 3 = [ e 3 , ϕ 4 , s F O ] , e 3 = e 30 x 2 . }
x 1 = Y 1 ± ( Y 1 2 4 X 1 Z 1 ) 2 X 1 .
A 5 3 = 0
x 1 = e 10 + ϕ 1 + ϕ 2 ϕ 1 ϕ 2 .
X 2 x 2 2 + Y 2 x 2 + Z 2 = 0 ,
X 2 = I 2 , Y 2 = I 2 ( e 20 + e 30 + x 1 ) + J 2 K 2 , Z 2 = I 2 ( e 20 x 1 ) e 30 + J 2 ( e 20 x 1 ) + K 2 e 30 + L 2 , I 2 = C 2 1 ϕ 3 A 5 4 , J 2 = C 2 1 ( A 5 4 + ϕ 3 B 5 4 ) , K 2 = ( A 2 1 ϕ 3 + C 2 1 ) A 5 4 , L 2 = A 2 1 ( A 5 4 + ϕ 3 B 5 4 ) + C 2 1 B 5 4 , A 2 1 = [ ϕ 1 , e 1 ] , C 2 1 = [ ϕ 1 , e 1 , ϕ 2 ] , A 5 4 = [ ϕ 4 , s F O ] , B 5 4 = [ s F O ] , e 1 = e 10 + x 1 . }
x 2 = Y 2 ± ( Y 2 2 4 X 2 Z 2 ) 2 X 2 .
C 2 1 = 0 ,
x 2 = e 30 1 ϕ 3 B 5 4 A 5 4 .
A 5 4 = 0 ,
x 2 = ( e 20 x 1 ) + A 2 1 C 1 1 + 1 ϕ 3 .
X 1 * x 1 2 + Y 1 * x 1 + Z 1 * = 0 ,
X 1 * = I 1 * , Y 1 * = I 1 * ( e 10 + e 20 + x 2 ) + J 1 * K 1 * , Z 1 * = I 1 * e 10 ( e 20 + x 2 ) + J 1 * e 10 + K 1 * ( e 20 + x 2 ) + L 2 * , I 1 * = ϕ 1 ϕ 2 C 4 3 , J 1 * = ϕ 1 ( C 4 3 + ϕ 2 D 4 3 ) , K 1 * = ( ϕ 1 + ϕ 2 ) C 4 3 , L 1 * = C 4 3 + ( ϕ 1 + ϕ 2 ) D 4 3 , C 4 3 = [ ϕ 3 , e 3 , ϕ 4 ] , D 4 3 = [ e 3 , ϕ 4 ] , e 3 = e 30 x 2 . }
X 2 * x 2 2 + Y 2 * x 2 + Z 2 * = 0 ,
X 2 * = I 2 * , Y 2 * = I 2 * ( e 20 + e 30 + x 1 ) + J 2 * K 2 * , Z 2 * = I 2 * ( e 20 x 1 ) e 30 + J 2 * ( e 20 x 1 ) + K 2 * e 30 + L 2 * , I 2 * = C 2 1 ϕ 3 ϕ 4 , J 2 * = C 2 1 ( ϕ 3 + ϕ 4 ) , K 2 * = ( A 2 1 ϕ 3 + C 2 1 ) ϕ 4 , L 2 * = A 2 1 ( ϕ 3 + ϕ 4 ) + C 2 1 . }
Z x 1 = ϕ 1 A 5 2 + C 2 1 A 5 3 , Z x 2 = C 2 1 A 5 3 + C 3 1 A 5 4 , 2 Z x 1 2 = 2 ϕ 1 ϕ 2 A 5 3 , 2 Z x 2 2 = 2 C 2 1 ϕ 3 A 5 4 , 2 Z x 1 x 2 = ϕ 1 ϕ 2 A 5 3 ϕ 1 C 3 2 A 5 4 + C 2 1 ϕ 3 A 5 4 , }
C 2 1 = [ ϕ 1 , e 1 , ϕ 2 ] , C 3 1 = [ ϕ 1 , e 1 , ϕ 2 , e 2 , ϕ 3 ] , A 5 2 = [ ϕ 2 , e 1 , ϕ 3 , e 3 , ϕ 4 , s F O ] , A 5 3 = [ ϕ 3 , e 3 , ϕ 4 s F O ] , A 5 4 = [ ϕ 4 , s F O ] , C 3 2 = [ ϕ 2 , e 2 , ϕ 3 ] , e 1 = e 10 + x 1 , e 2 = e 20 x 1 + x 2 , e 3 = e 30 x 2 . }
Z x 1 = ϕ 1 2 + ( C 2 1 ) 2 Φ , Z x 2 = ( C 2 1 ) 2 + ( C 3 1 ) 2 Φ , 2 Z x 1 2 = 2 ϕ 1 ϕ 2 C 2 1 Φ , 2 Z x 2 2 = 2 C 2 1 ϕ 3 C 3 1 Φ , 2 Z x 1 x 2 = ϕ 1 ϕ 2 C 2 1 ϕ 1 C 3 2 C 3 1 + C 2 1 ϕ 3 C 3 1 Φ . }
Φ = [ ϕ 1 e 1 , ϕ 2 , e 2 , ϕ 3 , e 3 , ϕ 4 ] .
Z x 1 = ϕ 1 C 4 2 + C 2 1 C 4 3 , Z x 2 = C 2 1 C 4 3 + C 3 1 ϕ 4 , 2 Z x 1 2 = 2 ϕ 1 ϕ 2 C 4 3 , 2 Z x 2 2 = 2 C 2 1 ϕ 3 ϕ 4 , 2 Z x 1 x 2 = ϕ 1 ϕ 2 C 4 3 ϕ 1 C 3 2 ϕ 4 + C 2 1 ϕ 3 ϕ 4 , }
C 4 2 = [ ϕ 2 , e 2 , ϕ 3 , e 3 , ϕ 4 ] , C 4 3 = [ ϕ 3 , e 3 , ϕ 4 ] , e 2 = e 20 x 1 + x 2 , e 3 = e 30 x 2 . }
d x 2 d x 1 = ( Z x 1 ) / ( Z x 2 ) = 1 β 2 2 1 β 3 2 β 3 2 ,
C i 1 C j 1 = β i + 1 β i + 2 β j , ( 1 i < j ) ,
C i 1 1 C i 1 = β i .
d x 2 d x 1 = 1 β 2 2 ,
d x 2 d x 1 = ( ϕ 2 ϕ 3 ) 2 .
| β 2 | = 1 ,
d 2 x 2 d x 1 2 = 2 Z x 1 2 ( Z x 2 ) 2 2 2 Z x 1 x 2 Z x 1 Z x 2 + 2 Z x 2 2 ( Z x 1 ) 2 ( Z x 2 ) 3 .
( d 2 x 2 d x 1 2 ) | β 2 | = 1 = 2 ϕ 2 β 3 2 1 β 3 2 β 2 .
( d 2 x 2 d x 1 2 ) | β 2 | = 1 = 2 ϕ 2 β 2 ,
extremum of x 2 ; d x 2 d x 1 = 0 { minimum d 2 x 2 d x 1 2 > 0 { β 2 = + 1 { ϕ 2 > 0 , | β 3 | < 1 , ϕ 2 < 0 , | β 3 | > 1 , β 2 = 1 { ϕ 2 > 0 , | β 3 | > 1 , ϕ 2 < 0 , | β 3 | < 1 , maximum d 2 x 2 d x 1 2 < 0 { β 2 = + 1 { ϕ 2 > 0 , | β 3 | > 1 , ϕ 2 < 0 , | β 3 | < 1 , β 2 = 1 { ϕ 2 > 0 , | β 3 | < 1 , ϕ 2 < 0 , | β 3 | > 1 .
d x 1 d x 2 = { 1 β 3 2 1 β 2 2 1 β 3 2 , 1 1 β 2 2 , when A 4 4 = 0 , ( ϕ 3 ϕ 2 ) 2 , when C 2 1 = A 5 3 = 0 . ( 40 )
| β 3 | = 1 ,
( d 2 x 1 d x 2 2 ) | β 3 | = 1 = 2 ϕ 3 1 1 β 2 2 1 β 3 ,
extremum of x 1 ; d x 1 d x 2 = 0 { minimum d 2 x 1 d x 2 2 > 0 { β 3 = + 1 { ϕ 3 > 0 , | β 2 | < 1 , ϕ 3 < 0 , | β 2 | > 1 , β 3 = 1 { ϕ 3 > 0 , | β 2 | > 1 , ϕ 3 < 0 , | β 2 | < 1 , maximum d 2 x 1 d x 2 2 < 0 { β 3 = + 1 { ϕ 3 > 0 , | β 2 | > 1 , ϕ 3 < 0 , | β 2 | < 1 , β 2 = 1 { ϕ 3 > 0 , | β 2 | < 1 , ϕ 3 < 0 , | β 2 | > 1 .
| β 2 | = | β 3 | = 1 ,
Z x 1 = Z x 2 = 0.
H = | 2 Z x 1 2 2 Z x 1 x 2 2 Z x 1 x 2 2 Z x 2 2 | .
H = 4 ϕ 1 ϕ 2 ϕ 3 C 2 1 A 5 3 A 5 4 ( ϕ 1 ϕ 2 A 5 3 ϕ 1 C 3 2 A 5 4 + C 2 1 ϕ 3 A 5 4 ) 2 .
H = 4 ϕ 1 ϕ 2 ϕ 3 ( C 2 1 ) 2 C 3 1 Φ 2 ( ϕ 1 ϕ 2 C 2 1 ϕ 1 C 3 2 C 3 1 + C 2 1 ϕ 3 C 3 1 ) 2 Φ 2 .
H = 4 ϕ 1 ϕ 2 ϕ 3 ϕ 4 C 2 1 C 4 3 ( ϕ 1 ϕ 2 C 4 3 ϕ 1 C 3 2 ϕ 4 + C 2 1 ϕ 3 ϕ 4 ) 2 .
β 2 = β 3 = + 1 , or β 2 = β 3 = 1 ,
H = 4 ( ϕ 1 2 Φ ) 2 ϕ 2 ϕ 3 ,
H = 4 ( ϕ 1 ϕ 4 ) 2 ϕ 2 ϕ 3 ,
Z ( x 1 , x 2 ) = 0 = 2 Z x 1 2 x 1 2 + 2 2 Z x 1 x 2 x 1 x 2 + 2 Z x 2 2 x 2 2 = ϕ 2 x 1 2 + ϕ 3 x 2 2 .
sgn ( ϕ 2 ) = sgn ( ϕ 3 )
H > 0 ,
sgn ( ϕ 2 ) sgn ( ϕ 3 )
H < 0 ,
( x 1 + | ϕ 3 ϕ 2 | x 2 ) ( x 1 | ϕ 3 ϕ 2 | x 2 ) = 0 .
d x 2 d x 1 = ± | ϕ 2 ϕ 3 | .
β 2 = ± 1 , β 3 = 1 ,
H = 4 ( ϕ 1 2 Φ ) 2 ϕ 2 ϕ 3 ,
H = 4 ( ϕ 1 ϕ 4 ) 2 ϕ 2 ϕ 3 ,
H < 0 .
sgn ( ϕ 2 ) = sgn ( ϕ 3 ) ,
( x 1 + ϕ 3 ϕ 2 x 2 ) ( x 1 ϕ 3 ϕ 2 x 2 ) = 0.
d x 2 d x 1 = ± | ϕ 2 ϕ 3 | .
β 2 = β 3 = 1 ,
ϕ 2 < 0 , ϕ 3 > 0 ,
( x 1 1.21 x 2 ) ( x 1 + 1.21 x 2 ) = 0 .
( x 1 1.12 x 2 ) ( x 1 + 1.12 x 2 ) = 0 .
{ Regular Singular
{ | β 2 , wide | < 1 | β 2 , wide | > 1
{ | β 3 , wide | > 1 | β 3 , wide | < 1
{ | β 3 , wide | < 1 | β 3 , wide | > 1
{ β 2 = 1 β 2 = + 1
{ β 3 = 1 β 3 = + 1
{ β 3 = 1 β 3 = + 1
G k i = G j 1 i G k j + 1 a j + G j 2 i G k j + 1 + G j 1 i G k j + 2 ,
G k i = [ a i , a i + 1 , , a k ] ,
d d a j G k i = G j 1 i G k j + 1
β i β i + 1 β 4 = A 5 i = ( C i 1 1 ) / ( C 4 1 ) ,
1 < i 4 .
d x 2 d x 1 = ϕ 1 A 5 2 C 3 1 A 5 4 .
e 1 = ϕ 1 + ϕ 2 ϕ 1 ϕ 1 ,
e 3 = A 5 4 + ϕ 3 B 5 4 ϕ 3 A 5 4 .
ϕ 1 ϕ 2 ϕ 3 A 5 4 ,
ϕ 1 ϕ 3 ϕ 2 A 5 4 ,
( C 1 1 ) 2 ( C 2 1 ) 2 = 0 .
2 ϕ 1 ϕ 2 C 2 1 { ( C 2 1 ) 2 + ( C 3 1 ) 2 } 2 ( C 4 1 ) 3 ,
( C 2 1 ) 2 + ( C 3 1 ) 2 ( C 4 1 ) 3 .
C 3 1 = 0 .
β 2 = β 3 = 1 ,
C 1 1 = ϕ 1 , C 2 1 = ϕ 1 , C 3 1 = ϕ 1 , C 3 2 = ( ϕ 2 + ϕ 3 ) , e 1 = ( 2 ϕ 1 + ϕ 2 ) / ( ϕ 1 ϕ 2 ) , e 2 = 2 ( ϕ 2 + ϕ 3 ) / ( ϕ 2 ϕ 3 ) . }
2 Z x 1 2 = 2 ( ϕ 1 ) 2 ϕ 2 / Φ , 2 Z x 2 2 = 2 ( ϕ 1 ) 2 ϕ 3 / Φ , 2 Z x 1 x 2 = 0 . }

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