Abstract

This paper presents a paraxial study of generalized Yamaji Type V mechanically compensated zoom lenses 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. Some numerical examples of the zooming locus are shown.

© 1982 Optical Society of America

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References

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  1. K. Tanaka, Appl. Opt. 21, 2174 (1982).
    [Crossref] [PubMed]
  2. K. Yamaji, Prog. Opt. 6, 107 (1967).
  3. R. Richter, U.S. Patent2,078,586 (27Apr.1937).
  4. M. Reymond, French Patent1,081,948 (23Dec.1954).
  5. H. H. Hopkins, U.S. Patent2,782,684 (26Feb.1957).
  6. K. Yamaji, Research Report No. 3, Canon Camera, Inc. (1964), p. 37.
  7. K. Yamaji, J. Inst. Telev. Eng. Jpn. 13, 352 (1959).
  8. K. Yamaji, U.S. Patent3,192,829 (6July1965).
  9. C. Tao, Kexue Tongbao 22, 207 (1977).
  10. M. G. Shpyakin, Sov. J. Opt. Technol. 39, 742 (1972).
  11. See, for example, E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), p. 32.
  12. M. Herzberger, J. Opt. Soc. Am. 33, 651 (1943).
    [Crossref]
  13. M. Herzberger, J. Opt. Soc. Am. 42, 637 (1952).
    [Crossref]
  14. See, for example, T. Poston, I. N. Stewart, Taylor Expansions and Catastrophes (Pitman, London, 1976), p. 15.
  15. K. Tanaka, Japanese Laid Open Patent Application 55-28051 (28Feb.1980).
  16. G. H. Cook, Japanese Patent48-6813 (1Mar.1973).
  17. A. Someya, Japanese Patent40-17429 (8Aug.1965).
  18. See, for example, O. Perron, Die Lehre von den Kettenbrüchen (Teubner, Leipzig, 1913), s. 18.

1982 (1)

1977 (1)

C. Tao, Kexue Tongbao 22, 207 (1977).

1972 (1)

M. G. Shpyakin, Sov. J. Opt. Technol. 39, 742 (1972).

1967 (1)

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

1959 (1)

K. Yamaji, J. Inst. Telev. Eng. Jpn. 13, 352 (1959).

1952 (1)

1943 (1)

Cook, G. H.

G. H. Cook, Japanese Patent48-6813 (1Mar.1973).

Herzberger, M.

Hopkins, H. H.

H. H. Hopkins, U.S. Patent2,782,684 (26Feb.1957).

O’Neill, E. L.

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

Perron, O.

See, for example, O. Perron, Die Lehre von den Kettenbrüchen (Teubner, Leipzig, 1913), s. 18.

Poston, T.

See, for example, T. Poston, I. N. Stewart, Taylor Expansions and Catastrophes (Pitman, London, 1976), p. 15.

Reymond, M.

M. Reymond, French Patent1,081,948 (23Dec.1954).

Richter, R.

R. Richter, U.S. Patent2,078,586 (27Apr.1937).

Shpyakin, M. G.

M. G. Shpyakin, Sov. J. Opt. Technol. 39, 742 (1972).

Someya, A.

A. Someya, Japanese Patent40-17429 (8Aug.1965).

Stewart, I. N.

See, for example, T. Poston, I. N. Stewart, Taylor Expansions and Catastrophes (Pitman, London, 1976), p. 15.

Tanaka, K.

K. Tanaka, Appl. Opt. 21, 2174 (1982).
[Crossref] [PubMed]

K. Tanaka, Japanese Laid Open Patent Application 55-28051 (28Feb.1980).

Tao, C.

C. Tao, Kexue Tongbao 22, 207 (1977).

Yamaji, K.

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

K. Yamaji, J. Inst. Telev. Eng. Jpn. 13, 352 (1959).

K. Yamaji, U.S. Patent3,192,829 (6July1965).

K. Yamaji, Research Report No. 3, Canon Camera, Inc. (1964), p. 37.

Appl. Opt. (1)

J. Inst. Telev. Eng. Jpn. (1)

K. Yamaji, J. Inst. Telev. Eng. Jpn. 13, 352 (1959).

J. Opt. Soc. Am. (2)

Kexue Tongbao (1)

C. Tao, Kexue Tongbao 22, 207 (1977).

Prog. Opt. (1)

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

Sov. J. Opt. Technol. (1)

M. G. Shpyakin, Sov. J. Opt. Technol. 39, 742 (1972).

Other (11)

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

K. Yamaji, U.S. Patent3,192,829 (6July1965).

R. Richter, U.S. Patent2,078,586 (27Apr.1937).

M. Reymond, French Patent1,081,948 (23Dec.1954).

H. H. Hopkins, U.S. Patent2,782,684 (26Feb.1957).

K. Yamaji, Research Report No. 3, Canon Camera, Inc. (1964), p. 37.

See, for example, T. Poston, I. N. Stewart, Taylor Expansions and Catastrophes (Pitman, London, 1976), p. 15.

K. Tanaka, Japanese Laid Open Patent Application 55-28051 (28Feb.1980).

G. H. Cook, Japanese Patent48-6813 (1Mar.1973).

A. Someya, Japanese Patent40-17429 (8Aug.1965).

See, for example, O. Perron, Die Lehre von den Kettenbrüchen (Teubner, Leipzig, 1913), s. 18.

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Figures (4)

Fig. 1
Fig. 1

Thin lens configuration of a generalized Yamaji Type V mechanically compensated zoom lens: (a) original position, (b) zooming position.

Fig. 2
Fig. 2

Numerical example of zooming loci of a generalized Yamaji Type V. The lens data at wide angle settings are ϕ 1 = - 0.1461337 , ϕ 4 = 0.2052365 , e 20 = 2.0600 , ϕ 2 = 0.1588918 , ϕ 5 = 0.1696492 , e 30 = 10.1604 , ϕ 3 = - 0.3164300 , e 10 = 6.1193 , e 40 = 4.9327.This zoom lens system covers the focal length range from 1.0276 to 10.6677 as the second and fourth components move from 0.0000 to −2.4000. The back focal distance is 1.6774.

Fig. 3
Fig. 3

Numerical example of the zooming loci of a generalized Yamaji Type V. This example has singularity at the point where β2 = β3 = β4 = −1 is simultaneously satisfied. The lens data at the singular point are ϕ 1 = - 0.0048359 , ϕ 4 = 0.0077161 , e 20 = 86.3635 , ϕ 2 = 0.0077168 , ϕ 5 = 0.0040830 , e 30 = 86.3874 , ϕ 3 = - 0.0115734 , e 10 = 52.3874 , e 40 = 135.8135.This zoom lens system continuously covers the focal length range from 55.4762 to 369.2872 as the third component moves from −39.0000 to 43.0000. The back focal distance is 82.0510.

Fig. 4
Fig. 4

Numerical example of zooming loci of a generalized Yamaji Type V. This example has singularity at the point where β2 = −0.80, β3 = −1.00, and β4 = −1.25 are simultaneously satisfied. The lens data at the singular point are ϕ 1 = - 0.1500000 , ϕ 4 = 0.2100000 , e 20 = 5.0000 , ϕ 2 = 0.1600000 , ϕ 5 = 0.1700000 , e 30 = 2.3214 , ϕ 3 = - 0.3200000 , e 10 = 7.3958 , e 40 = 7.0000.This zoom lens system continuously covers the focal length range from 0.8234 to 7.7810 as the third component moves from −2.0000 to 0.5000. The back focal distance is 2.2767.

Equations (75)

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s F O = A 1 5 , 0 C 1 5 , 0 ,
A 1 5 , 0 = [ ϕ 1 , - e 10 , ϕ 2 , - e 20 , ϕ 3 , - e 30 , ϕ 4 , - e 40 ] , C 1 5 , 0 = [ ϕ 1 , - e 10 , ϕ 2 , - e 20 , ϕ 3 , - e 30 , ϕ 4 , - e 40 , ϕ 5 ] . }
s F = A 1 5 C 1 5 ,
A 1 5 = [ ϕ 1 , - e 1 , ϕ 2 , - e 2 , ϕ 3 , - e 3 , ϕ 4 , - e 4 ] , C 1 5 = [ ϕ 1 , - e 1 , ϕ 2 , - e 2 , ϕ 3 , - e 3 , ϕ 4 , - e 4 , ϕ 5 ] , e 1 = e 10 + x 1 , e 2 = e 20 - x 1 + x 2 , e 3 = e 30 + x 1 - x 2 , e 4 = e 40 - x 1 . }
s F = s F O ,
Z ( x 1 , x 2 ) A 1 5 - C 1 5 s F O = 0.
V 1 x 1 4 + W 1 x 1 3 + X 1 x 1 2 + Y 1 x 1 + Z 1 = 0 ,
V 1 = I 1 , W 1 = I 1 ( e 10 - e 20 + e 30 - e 40 - 2 x 2 ) - J 1 + J 2 - J 3 + J 4 , X 1 = I 1 { - e 10 ( e 20 + x 2 ) - e 40 ( e 30 - x 2 ) + ( - e 10 + e 20 + x 2 ) ( - e 30 + e 40 + x 2 ) } + J 1 ( - e 10 + e 20 - e 30 + 2 x 2 ) + J 2 ( e 10 - e 20 - e 40 - x 2 ) + J 3 ( - e 10 - e 30 + e 40 + x 2 ) + J 4 ( - e 20 + e 30 - e 40 - 2 x 2 ) + K 1 - K 2 + K 3 + K 4 - K 5 + K 6 , Y 1 = I 1 { e 10 ( e 20 + x 2 ) ( - e 30 + e 40 + x 2 ) + e 40 ( e 30 - x 2 ) ( - e 10 + e 20 + x 2 ) } + J 1 { e 10 ( e 20 - e 30 + 2 x 2 ) + ( e 20 + x 2 ) ( e 30 - x 2 ) } + J 2 { e 40 ( - e 10 + e 20 + x 2 ) - e 10 ( e 20 + x 2 ) } + J 3 { e 40 ( e 10 + e 30 - x 2 ) - e 10 ( e 30 - x 2 ) } + J 4 { e 40 ( e 20 - e 30 + 2 x 2 ) - ( e 20 + x 2 ) ( e 30 - x 2 ) } - K 1 ( - e 10 + e 20 + x 2 ) - K 2 ( e 10 + e 30 - x 2 ) - K 3 ( - e 10 + e 40 ) - K 4 ( e 20 - e 30 + 2 x 2 ) - K 5 ( - e 20 - e 40 - x 2 ) - K 6 ( - e 30 + e 40 + x 2 ) + L 1 - L 2 + L 3 - L 4 , Z 1 = I 1 e 10 e 40 ( e 20 + x 2 ) ( e 30 - x 2 ) + J 1 e 10 ( e 20 + x 2 ) ( e 30 - x 2 ) + J 2 e 10 e 40 ( e 20 + x 2 ) + J 3 e 10 e 40 ( e 30 - x 2 ) + J 4 e 40 ( e 20 + x 2 ) ( e 30 - x 2 ) - K 1 e 10 ( e 20 + x 2 ) - K 2 e 10 ( e 30 - x 2 ) - K 3 e 10 e 40 - K 4 ( e 20 + x 2 ) ( e 30 - x 2 ) - K 5 e 40 ( e 20 + x 2 ) - K 6 e 40 ( e 30 - x 2 ) + L 1 e 10 + L 2 ( e 20 + x 2 ) + L 3 ( e 30 - x 2 ) + L 4 e 40 + M 1 , I 1 = ϕ 1 ϕ 2 ϕ 3 ϕ 4 ( 1 - ϕ 5 s F O ) , J 1 = - ϕ 1 ϕ 2 ϕ 3 ( 1 - ϕ 5 s F O ) - ϕ 4 s F O } , J 2 = - ϕ 1 ϕ 2 ( ϕ 3 + ϕ 4 ) ( 1 - ϕ 5 s F O ) , J 3 = - ϕ 1 ϕ 4 ( ϕ 2 + ϕ 3 ) ( 1 - ϕ 5 s F O ) , J 4 = - ϕ 3 ϕ 4 ( ϕ 1 + ϕ 2 ) ( 1 - ϕ 5 s F O ) , K 1 = ϕ 1 ϕ 2 { s F O ( ϕ 3 + ϕ 4 ) - ( 1 - ϕ 5 s F O ) } K 2 = ϕ 1 { ϕ 4 s F O ( ϕ 2 + ϕ 3 ) - ( ϕ 2 + ϕ 3 ) ( 1 - ϕ 5 s F O ) } K 3 = - ϕ 1 ( ϕ 2 + ϕ 3 + ϕ 4 ) ( 1 - ϕ 5 s F O ) , K 4 = ϕ 3 { ϕ 4 s F O ( ϕ 1 + ϕ 2 ) - ( ϕ 1 + ϕ 2 ) ( 1 - ϕ 5 s F O ) } , K 5 = - ( ϕ 1 + ϕ 2 ) ( ϕ 3 + ϕ 4 ) ( 1 - ϕ 5 s F O ) , K 6 = - ϕ 4 ( ϕ 1 + ϕ 2 + ϕ 3 ) ( 1 - ϕ 5 s F O ) , L 1 = ϕ 1 { s F O ( ϕ 2 + ϕ 3 + ϕ 4 + ϕ 5 ) - 1 } , L 2 = ( ϕ 1 + ϕ 2 ) { s F O ( ϕ 3 + ϕ 4 + ϕ 5 ) - 1 } , L 3 = ( ϕ 1 + ϕ 2 + ϕ 3 ) { s F O ( ϕ 4 + ϕ 5 ) - 1 } , L 4 = ( ϕ 1 + ϕ 2 + ϕ 3 + ϕ 4 ) ( ϕ 5 s F O - 1 ) , M 1 = 1 - s F O ( ϕ 1 + ϕ 2 + ϕ 3 + ϕ 4 + ϕ 5 ) .
1 - ϕ 5 s F O = 0 ,
W 1 * x 1 3 + X 1 * x 1 2 + Y 1 * x 1 + Z 1 * = 0 ,
W 1 * = - J 1 * , X 1 * = J 1 * ( - e 10 + e 20 - e 30 + 2 x 2 ) + K 1 * - K 2 * + K 4 * Y 1 * = J 1 * ( e 10 ( e 20 - e 30 + 2 x 2 ) + ( e 20 + x 2 ) ( e 30 - x 2 ) } - K 1 * ( - e 10 + e 20 + x 2 ) - K 2 * ( e 10 + e 30 - x 2 ) + L 1 * - L 2 * + L 3 * , Z 1 * = J 1 * e 10 ( e 20 + x 2 ) ( e 30 - x 2 ) - K 1 * e 10 ( e 20 + x 2 ) - K 2 * e 10 ( e 30 - x 2 ) + L 1 * e 10 + L 2 * ( e 20 + x 2 ) + L 3 * ( e 30 - x 2 ) + M 1 * , J 1 * = ϕ 1 ϕ 2 ϕ 3 ϕ 4 s F O , K 1 * = ϕ 1 ϕ 2 ( ϕ 3 + ϕ 4 ) s F O , K 2 * = ϕ 1 ( ϕ 2 + ϕ 3 ) ϕ 4 s F O , K 4 * = ( ϕ 1 + ϕ 2 ) ϕ 3 ϕ 4 s F O , L 1 * = ϕ 1 ( ϕ 2 + ϕ 3 + ϕ 4 ) s F O , L 2 * = ( ϕ 1 + ϕ 2 ) ϕ 3 ϕ 4 s F O , L 3 * = ( ϕ 1 + ϕ 2 + ϕ 3 ) ϕ 4 s F O , M 1 * = - ( ϕ 1 + ϕ 2 + ϕ 3 + ϕ 4 ) s F O . }
X 2 x 2 2 + Y 2 x 2 + Z 2 = 0 ,
X 2 = - I 2 , Y 2 = I 2 ( - e 20 + e 30 + 2 x 1 ) + J 2 - K 2 , Z 2 = I 2 ( e 20 - x 1 ) ( e 30 + x 1 ) + J 2 ( e 20 - x 1 ) + K 2 ( e 30 + x 1 ) + L 2 , I 2 = C 1 2 ϕ 3 A 4 6 , J 2 = - C 1 2 ( A 4 6 + ϕ 3 B 4 6 ) , K 2 = - A 4 6 ( A 1 2 ϕ 3 + C 1 2 ) , L 2 = A 1 2 ( A 4 6 + ϕ 3 B 4 6 ) + C 1 2 B 4 6 , A 1 2 = [ ϕ 1 , - e 1 ] , C 1 2 = [ ϕ 1 , - e 1 , ϕ 2 ] , A 4 6 = [ ϕ 4 , - e 4 , ϕ 5 , - s F O ] , B 4 6 = [ - e 4 , ϕ 5 , - s F O ] , e 1 = e 10 + x 1 , e 4 = e 40 - x 1 . }
x 2 = - Y 2 ± Y 2 2 - 4 X 2 Z 2 2 X 2 .
C 1 2 = 0 ,
x 2 = ( e 30 + x 1 ) - 1 / ϕ 3 - B 4 6 / A 4 6 .
x 1 = ( ϕ 1 + ϕ 2 ) / ( ϕ 1 ϕ 2 ) - e 10 .
A 4 6 = 0 ,
x 2 = - ( e 20 - x 1 ) + A 1 2 / C 1 2 + 1 / ϕ 3 .
Z x 1 = - ϕ 1 A 2 6 + C 1 2 A 3 6 - C 1 3 A 4 6 + C 1 4 A 5 6 , Z x 2 = - C 1 2 A 3 6 + C 1 3 A 4 6 , 2 Z x 1 2 = - 2 ϕ 1 ϕ 2 A 3 6 + 2 ( ϕ 1 2 C 3 - C 1 2 ϕ 3 ) A 4 6 + 2 ( - ϕ 1 2 C 4 + C 1 2 3 C 4 - C 1 3 ϕ 4 ) A 5 6 , 2 Z x 2 2 = - 2 C 1 2 ϕ 3 A 4 6 , 2 Z x 1 x 2 = ϕ 1 ϕ 2 A 3 6 + ( - ϕ 1 2 C 3 + 2 C 1 2 ϕ 3 ) A 4 6 + ( - C 1 2 C 3 4 + C 1 3 ϕ 4 ) A 5 6 , }
C 1 2 = [ ϕ 1 , - e 1 , ϕ 2 ] , C 1 3 = [ ϕ 1 , - e 1 , ϕ 2 , - e 2 , ϕ 3 ] , C 1 4 = [ ϕ 1 , - e 1 , ϕ 2 , - e 2 , ϕ 3 , - e 3 , ϕ 4 ] A 2 6 = [ ϕ 2 , - e 2 , ϕ 3 , - e 3 , ϕ 4 , - e 4 , ϕ 5 - s F O ] , A 3 6 = [ ϕ 3 , - e 3 , ϕ 4 , - e 4 , ϕ 5 , - s F O ] , A 4 6 = [ ϕ 4 , - e 4 , ϕ 5 , - s F O ] , A 5 6 = [ ϕ 5 , - s F O ] , C 2 3 = [ ϕ 2 , - e 2 , ϕ 3 ] , C 2 4 = [ ϕ 2 , - e 2 , ϕ 3 , - e 3 , ϕ 4 ] , C 3 4 = [ 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 1 - x 2 , e 4 = e 40 - x 1 . }
Z x 1 = { - ϕ 1 2 + ( C 1 2 ) 2 - ( C 1 3 ) 2 + ( C 1 4 ) 2 } / Φ , Z x 2 = { - ( C 1 2 ) 2 + ( C 1 3 ) 2 } / Φ , 2 Z x 1 2 = - 2 { ϕ 1 ϕ 2 C 1 2 - ( ϕ 1 C 2 3 - C 1 2 ϕ 3 ) C 1 3 } - ( - ϕ 1 C 2 4 + C 1 2 3 C 4 - C 1 3 ϕ 4 ) C 1 4 } / Φ , 2 Z x 2 2 = - 2 { C 1 2 ϕ 3 C 1 3 } / Φ , 2 Z x 1 x 2 = { ϕ 1 ϕ 2 C 1 2 + ( - ϕ 1 C 2 3 + 2 C 1 2 ϕ 3 ) C 1 3 + ( - C 1 2 C 3 4 + C 1 3 ϕ 4 ) C 1 4 } / Φ . }
Φ = [ ϕ 1 , - e 1 , ϕ 2 , - e 2 , ϕ 3 , - e 3 , ϕ 4 , - e 4 , ϕ 5 ] .
C i j = [ ϕ i , - e i , ϕ i + 1 , , ϕ j ]
d x 1 d x 2 = - ( Z x 2 ) / ( Z x 1 ) = - - ( C 1 2 ) 2 + ( C 1 3 ) 2 - ϕ 1 2 + ( C 1 2 ) 2 - ( C 1 3 ) 2 + ( C 1 4 ) 2 , = ( β 3 2 - 1 ) β 4 2 - ( β 2 β 3 β 4 ) 2 + ( β 3 β 4 ) 2 - β 4 2 + 1 }
C 1 i C 1 j = β i + 1 β i + 2 β j ,             ( 1 i < j ) .
d x 1 d x 2 = - - ( C 1 2 ) 2 + ( C 1 3 ) 2 - ϕ 1 2 + ( C 1 2 ) 2 - ( C 1 3 ) 2 = β 3 2 - 1 - ( β 2 β 3 ) 2 + β 3 2 - 1 }
β 3 = 1 ,
β 2 β 4 1 ,
- ( 2 Z x 2 2 ) / ( Z x 1 ) .
- ( 2 Z x 2 2 ) / ( Z x 1 ) = 2 β 3 β 4 2 1 - ( β 2 β 4 ) 2 ϕ 3 .
- ( 2 Z x 2 2 ) / ( Z x 1 ) = - 2 β 3 β 2 2 ϕ 3 .
extremum of x 1 ; { minimum { ϕ 3 > 0 { β 3 = + 1 , 1 > β 2 β 4 , β 3 = - 1 , 1 < β 2 β 4 , ϕ 3 < 0 { β 3 = + 1 , 1 < β 2 β 4 , β 3 = - 1 , 1 > β 2 β 4 , maximum { ϕ 3 > 0 { β 3 = + 1 , 1 < β 2 β 4 , β 3 = - 1 , 1 > β 2 β 4 , ϕ 3 < 0 { β 3 = + 1 , 1 > β 2 β 4 , β 3 = - 1 , 1 < β 2 β 4 .
d x 2 d x 1 = { - ( β 2 β 3 β 4 ) 2 + ( β 3 β 4 ) 2 - β 4 2 + 1 ( β 3 2 - 1 ) β 4 2 1 - β 2 2 , when A 4 6 = 0.
- ( β 2 β 3 β 4 ) 2 + ( β 3 β 4 ) 2 - β 4 2 + 1 = 0 ,
- ( 2 Z x 1 2 ) / ( Z x 2 ) ,
β 3 = β 2 β 4 = 1
Z x 1 = Z x 2 = 0.
H = 2 Z 2 Z x 1 2 x 2 2 - ( 2 Z x 1 x 2 ) 2 ,
H < 0.
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 .
β 2 = m , β 3 = - 1 , β 4 = 1 / m . }
e 1 = ( m - 1 ) ϕ 1 + m ϕ 2 m ϕ 1 ϕ 2 , e 2 = 2 ϕ 2 - ( m - 1 ) ϕ 3 ϕ 2 ϕ 3 , e 3 = - ( m - 1 ) ϕ 3 + 2 ϕ 4 ϕ 3 ϕ 4 . }
C 1 2 = ϕ 1 / m , C 1 3 = - ϕ 1 / m , C 1 4 = - ϕ 1 , C 2 3 = - ϕ 2 + m ϕ 3 , C 2 4 = - m ( ϕ 2 - m ϕ 3 + ϕ 4 ) , C 3 4 = m ϕ 3 - ϕ 4 . }
2 Z x 1 2 = { 2 ( m 2 - 1 ) 2 m 2 ϕ 1 2 ϕ 3 - 2 m ϕ 1 2 ( ϕ 2 + ϕ 4 ) } / Φ , 2 Z x 1 x 2 = { 2 m 2 - 1 m 2 ϕ 1 2 ϕ 3 } / Φ , 2 Z x 2 2 = { 2 ϕ 1 2 ϕ 3 m 2 } / Φ . }
H = - 1 m ( 2 ϕ 1 2 Φ ) 2 ( ϕ 2 + ϕ 4 ) ϕ 3 .
1 m ( ϕ 2 + ϕ 4 ) ϕ 3 > 0.
Z ( x 1 , x 2 ) = { ( m 2 - 1 ) 2 m 2 ϕ 3 - m ( ϕ 2 + ϕ 4 ) } x 1 2 + 2 m 2 - 1 m 2 ϕ 3 x 1 x 2 + ϕ 3 m 2 x 2 2 = 0.
β 2 = β 3 = β 4 = - 1 ,
2 Z x 1 2 = { 2 ϕ 1 2 ( ϕ 2 + ϕ 4 ) } / Φ , 2 Z x 1 x 2 = 0 , 2 Z x 2 2 = { 2 ϕ 1 2 ϕ 3 } / Φ . }
H = ( 2 ϕ 1 2 Φ ) 2 ( ϕ 2 + ϕ 4 ) ϕ 3 .
sgn ( ϕ 2 + ϕ 4 ) sgn ( ϕ 3 ) .
( x 1 - | ϕ 3 ϕ 2 + ϕ 4 | x 2 ) ( x 1 + | ϕ 3 ϕ 2 + ϕ 4 | x 2 ) = 0.
β 2 β 4 = 1 and β 3 = - 1 ,
ϕ 2 + ϕ 4 > 0 and ϕ 3 < 0 ,
- ( β 2 β 3 β 4 ) 2 + ( β 3 β 4 ) 2 - β 4 2 + 1 = - { ( - 1.2542 ) × ( - 0.8798 ) × ( - 0.8323 ) } 2 + { ( - 0.8798 ) × ( - 0.8323 ) } 2 - ( - 0.8323 ) 2 + 1 = 0.
x 1 = - 1.6507 , x 2 = 3.7238. }
- ( β 2 β 3 β 4 ) 2 + ( β 3 β 4 ) 2 - β 4 2 + 1 = - { ( - 1.2075 ) × ( - 1.1371 ) × ( - 0.7925 ) } 2 + { ( - 1.1371 ) × ( - 0.7925 ) } 2 - ( - 0.7925 ) 2 + 1 = 0.
x 1 = - 1.4567 , x 2 = 4.4369. }
β 2 = β 3 = β 4 = - 1
H = - 7.4 × 10 - 9 < 0.
( x 1 - 0.8660 x 2 ) ( x 1 + 0.8660 x 2 ) = 0.
β 2 = - 0.80 , β 3 = - 1.00 , β 4 = - 1.25 }
H = - 5.0 × 10 - 3 < 0.
0.2312 x 1 2 + 0.3600 x 1 x 2 - 0.5000 x 2 2 = ( x 1 - 0.8854 x 2 ) ( x 1 + 2.4425 x 2 ) = 0.
ϕ 1 = - 0.1461337 , ϕ 4 = 0.2052365 , e 20 = 2.0600 , ϕ 2 = 0.1588918 , ϕ 5 = 0.1696492 , e 30 = 10.1604 , ϕ 3 = - 0.3164300 , e 10 = 6.1193 , e 40 = 4.9327.
ϕ 1 = - 0.0048359 , ϕ 4 = 0.0077161 , e 20 = 86.3635 , ϕ 2 = 0.0077168 , ϕ 5 = 0.0040830 , e 30 = 86.3874 , ϕ 3 = - 0.0115734 , e 10 = 52.3874 , e 40 = 135.8135.
ϕ 1 = - 0.1500000 , ϕ 4 = 0.2100000 , e 20 = 5.0000 , ϕ 2 = 0.1600000 , ϕ 5 = 0.1700000 , e 30 = 2.3214 , ϕ 3 = - 0.3200000 , e 10 = 7.3958 , e 40 = 7.0000.
A i 6 = [ ϕ i , - e i , , ϕ 5 , - s F O ]
= A i 5 - s F O C i 5 .
A i 6 = A i 5 C 1 5 - C i 5 A 1 5 Φ .
G i n G j m - G j n G i m = ( - 1 ) m - j + 1 i G j - 2 G m + 2 n ,
A i 6 = C 1 i - 1 Φ .
d 2 x 1 d x 2 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 1 ) 3 .
Z x 2 = 0.

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