Abstract

With the help of Gaussian Brackets, this paper presents a paraxial study of the mechanically compensated zoom lens which consists of a fixed front component, three independently movable components, and a fixed fifth component. The expressions, which define the displacement of components at zooming, the critical point of displacement, and the singular point of displacement are derived. Some numerical examples of the zooming locus are given.

© 1983 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. Tanaka, Appl. Opt. 21, 4045 (1982).
    [CrossRef] [PubMed]
  3. K. Yamaji, Prog. Opt. 6, 107 (1967).
  4. A. Warmisham, British Patent 398307 (14Sept.1933).
  5. W. P. M. M. van Gennip, Nederland Patent 282395 (28Dec.1964).
  6. D. S. Volosov, N. A. Gradoboeva, I. E. Isaeva, M. S. Stefanskii, Soviet Patent 342162 (14June1972).
  7. S. Nakamura, U.S. Patent3,771,853 (13Nov.1973).
  8. T. Sekiguchi, French Patent2,222,664 (22Mar.1974).
  9. K. Ikemori, Japanese Laid Open Patent Application50-126441 (4Oct.1975).
  10. E. Takano, Deutsches Patent Auslegeschrift 2533194 (15Sept.1977).
  11. I. Kawaguchi, M. Tai, Japanese Laid Open Patent Application53-34539 (31Mar.1978).
  12. F. Laurent, U.K. Patent Application2013926 (22Jan.1979).
  13. E. I. Betensky, Japanese Laid Open Patent Application55-62419 (10May1980).
  14. S. Suda, K. Tanaka, Japanese Laid Open Patent Application56-1009 (8Jan.1981).
  15. Y. Doi, K. Sado, U.S. Patent4,245,891 (20Jan.1981).
  16. M. S. Stefanskii, N. A. Gradoboeva, I. E. Isaeva, Sov. J. Opt. Technol. 44, 468 (1977).
  17. K. Macher, J. SMPTE 83, 39 (1974).
    [CrossRef]
  18. K. Macher, Fernseh Kino Tech. 28, 3 (1974).
  19. K. Tanaka, Jpn. J. Opt. 10, 356 (1981).
  20. K. Tanaka, Opt. Commun. 43, 315 (1982).
    [CrossRef]
  21. See, for example, E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), p. 32.
  22. M. Herzberger, J. Opt. Soc. Am. 33, 651 (1943).
    [CrossRef]
  23. M. Herzberger, J. Opt. Soc. Am. 42, 637 (1952).
    [CrossRef]
  24. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 457.
  25. K. Tanaka, Optik 62, 211 (1982).
  26. See, for example, Y. C. Lu, Singularity Theory and an Introduction to Catastrophe Theory (Springer, New York, 1976), p. 17.
  27. See, for example, I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958), p. 335.
  28. See, for example, T. Poston, I. N. Stewart, Taylor Expansions and Catastrophes (Pitman, London, 1979), p. 15.
  29. See, for example, S. Moriguchi, K. Udagawa, S. Hitotsumatsu, Mathematical Formulae I (Iwanami, Tokyo, 1968), p. 11.
  30. See, for example, Iwanami Encyclopedic Dictionary of Mathematics, Edited by the Mathematical Society of Japan (Iwanami, Tokyo, 1972), p. 510.

1982 (4)

1981 (1)

K. Tanaka, Jpn. J. Opt. 10, 356 (1981).

1977 (1)

M. S. Stefanskii, N. A. Gradoboeva, I. E. Isaeva, Sov. J. Opt. Technol. 44, 468 (1977).

1974 (2)

K. Macher, J. SMPTE 83, 39 (1974).
[CrossRef]

K. Macher, Fernseh Kino Tech. 28, 3 (1974).

1967 (1)

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

1952 (1)

1943 (1)

Betensky, E. I.

E. I. Betensky, Japanese Laid Open Patent Application55-62419 (10May1980).

Doi, Y.

Y. Doi, K. Sado, U.S. Patent4,245,891 (20Jan.1981).

Gradoboeva, N. A.

M. S. Stefanskii, N. A. Gradoboeva, I. E. Isaeva, Sov. J. Opt. Technol. 44, 468 (1977).

D. S. Volosov, N. A. Gradoboeva, I. E. Isaeva, M. S. Stefanskii, Soviet Patent 342162 (14June1972).

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. 457.

Hitotsumatsu, S.

See, for example, S. Moriguchi, K. Udagawa, S. Hitotsumatsu, Mathematical Formulae I (Iwanami, Tokyo, 1968), p. 11.

Ikemori, K.

K. Ikemori, Japanese Laid Open Patent Application50-126441 (4Oct.1975).

Isaeva, I. E.

M. S. Stefanskii, N. A. Gradoboeva, I. E. Isaeva, Sov. J. Opt. Technol. 44, 468 (1977).

D. S. Volosov, N. A. Gradoboeva, I. E. Isaeva, M. S. Stefanskii, Soviet Patent 342162 (14June1972).

Kawaguchi, I.

I. Kawaguchi, M. Tai, Japanese Laid Open Patent Application53-34539 (31Mar.1978).

Laurent, F.

F. Laurent, U.K. Patent Application2013926 (22Jan.1979).

Lu, Y. C.

See, for example, Y. C. Lu, Singularity Theory and an Introduction to Catastrophe Theory (Springer, New York, 1976), p. 17.

Macher, K.

K. Macher, J. SMPTE 83, 39 (1974).
[CrossRef]

K. Macher, Fernseh Kino Tech. 28, 3 (1974).

Moriguchi, S.

See, for example, S. Moriguchi, K. Udagawa, S. Hitotsumatsu, Mathematical Formulae I (Iwanami, Tokyo, 1968), p. 11.

Nakamura, S.

S. Nakamura, U.S. Patent3,771,853 (13Nov.1973).

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.

Redheffer, R. M.

See, for example, I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958), p. 335.

Sado, K.

Y. Doi, K. Sado, U.S. Patent4,245,891 (20Jan.1981).

Sekiguchi, T.

T. Sekiguchi, French Patent2,222,664 (22Mar.1974).

Sokolnikoff, I. S.

See, for example, I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958), p. 335.

Stefanskii, M. S.

M. S. Stefanskii, N. A. Gradoboeva, I. E. Isaeva, Sov. J. Opt. Technol. 44, 468 (1977).

D. S. Volosov, N. A. Gradoboeva, I. E. Isaeva, M. S. Stefanskii, Soviet Patent 342162 (14June1972).

Stewart, I. N.

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

Suda, S.

S. Suda, K. Tanaka, Japanese Laid Open Patent Application56-1009 (8Jan.1981).

Tai, M.

I. Kawaguchi, M. Tai, Japanese Laid Open Patent Application53-34539 (31Mar.1978).

Takano, E.

E. Takano, Deutsches Patent Auslegeschrift 2533194 (15Sept.1977).

Tanaka, K.

K. Tanaka, Opt. Commun. 43, 315 (1982).
[CrossRef]

K. Tanaka, Optik 62, 211 (1982).

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

K. Tanaka, Appl. Opt. 21, 4045 (1982).
[CrossRef] [PubMed]

K. Tanaka, Jpn. J. Opt. 10, 356 (1981).

S. Suda, K. Tanaka, Japanese Laid Open Patent Application56-1009 (8Jan.1981).

Udagawa, K.

See, for example, S. Moriguchi, K. Udagawa, S. Hitotsumatsu, Mathematical Formulae I (Iwanami, Tokyo, 1968), p. 11.

van Gennip, W. P. M. M.

W. P. M. M. van Gennip, Nederland Patent 282395 (28Dec.1964).

Volosov, D. S.

D. S. Volosov, N. A. Gradoboeva, I. E. Isaeva, M. S. Stefanskii, Soviet Patent 342162 (14June1972).

Warmisham, A.

A. Warmisham, British Patent 398307 (14Sept.1933).

Yamaji, K.

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

Appl. Opt. (2)

Fernseh Kino Tech. (1)

K. Macher, Fernseh Kino Tech. 28, 3 (1974).

J. Opt. Soc. Am. (2)

J. SMPTE (1)

K. Macher, J. SMPTE 83, 39 (1974).
[CrossRef]

Jpn. J. Opt. (1)

K. Tanaka, Jpn. J. Opt. 10, 356 (1981).

Opt. Commun. (1)

K. Tanaka, Opt. Commun. 43, 315 (1982).
[CrossRef]

Optik (1)

K. Tanaka, Optik 62, 211 (1982).

Prog. Opt. (1)

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

Sov. J. Opt. Technol. (1)

M. S. Stefanskii, N. A. Gradoboeva, I. E. Isaeva, Sov. J. Opt. Technol. 44, 468 (1977).

Other (19)

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 457.

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

A. Warmisham, British Patent 398307 (14Sept.1933).

W. P. M. M. van Gennip, Nederland Patent 282395 (28Dec.1964).

D. S. Volosov, N. A. Gradoboeva, I. E. Isaeva, M. S. Stefanskii, Soviet Patent 342162 (14June1972).

S. Nakamura, U.S. Patent3,771,853 (13Nov.1973).

T. Sekiguchi, French Patent2,222,664 (22Mar.1974).

K. Ikemori, Japanese Laid Open Patent Application50-126441 (4Oct.1975).

E. Takano, Deutsches Patent Auslegeschrift 2533194 (15Sept.1977).

I. Kawaguchi, M. Tai, Japanese Laid Open Patent Application53-34539 (31Mar.1978).

F. Laurent, U.K. Patent Application2013926 (22Jan.1979).

E. I. Betensky, Japanese Laid Open Patent Application55-62419 (10May1980).

S. Suda, K. Tanaka, Japanese Laid Open Patent Application56-1009 (8Jan.1981).

Y. Doi, K. Sado, U.S. Patent4,245,891 (20Jan.1981).

See, for example, Y. C. Lu, Singularity Theory and an Introduction to Catastrophe Theory (Springer, New York, 1976), p. 17.

See, for example, I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958), p. 335.

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

See, for example, S. Moriguchi, K. Udagawa, S. Hitotsumatsu, Mathematical Formulae I (Iwanami, Tokyo, 1968), p. 11.

See, for example, Iwanami Encyclopedic Dictionary of Mathematics, Edited by the Mathematical Society of Japan (Iwanami, Tokyo, 1972), p. 510.

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Figures (4)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Numerical example of the solution surface of the zooming locus. This example has a maximum of x1 at the point (x1,x2,x3) = (5.291, −1.840, 1.363). The lens construction data at the wide angle position are ϕ1 = 0.0000000; ϕ2 = −0.1953033; ϕ3 = 0.3600437; ϕ4 = −0.0963756; ϕ5 = 0.0000000; e10 = 0.0000; e20 = 7.5655; e30 = 2.3479; e40 = 0.0000; and s F O = 1.3673. The ranges of displacement are 0.000 ≦ x1 ≦ 5.291; −5.000 ≦ x2 ≦ 0.000; and −5.000 ≦ x3 ≦ 2.000. The focal lengths of the whole system at the wide angle position and at the maximum point of x1 are f′ = 1.624 and f′ = 5.124, respectively. The maximum focal length of this system is f′ = 10.460 and occurs at the point (x1,x2,x3) = (2.715, −5.000, −5.000).

Fig. 3
Fig. 3

Numerical example of the solution surface of the zooming locus. This example has a saddle point of x3 at (−3.908, 19.918, −6.501). The lens construction data at the wide angle position are ϕ1 = −0.0253236; ϕ2 = 0.0292371; ϕ3 = −0.0586756; ϕ4 = 0.0434877; ϕ5 = 0.0252336; e10 = 32.8240; e20 = 10.4997; e30 = 45.0680; e40 = 36.2947; and s F O = 2.1319. The ranges of displacement are −12.000 ≦ x1 ≦ 0.000; 0.000 ≦ x2 ≦ 32.000; and −11.300 ≦ x3 ≦ 3.654. The focal lengths of the whole system at the wide angle position and at the saddle point of x3 are f′ = 8.356 and f′ = 32.179, respectively. The maximum focal length of this system is f′ = 62.564; it occurs at (−12.000, 32.000, −0.284).

Fig. 4
Fig. 4

Numerical example of the solution surface of the zooming locus. This example has a singular point at (−5.000, 0.000, 8.726). The lens configuration data at the original position are ϕ1 = 0.0100000; ϕ2 = −0.0500000; ϕ3 = 0.0400000; ϕ4 = 0.0250000; ϕ5 = 0.0800000; e10 = 65.0000; e20 = 5.0000; e30 = 121.2745; e40 = 108.7255; and s F O = 33.3333. The ranges of displacement are −15.000 ≦ x1 ≦ −5.000; 0.000 ≦ x2 ≦ 7.613; −3.691 ≦ x3 ≦ 18.201; and −5.000 ≦ x1 ≦ 0.000; −7.342 ≦ x2 ≦ 0.000; 0.000 ≦ x3 ≦ 15.888. The focal lengths at the original position and the singular point are f′ = 253.778 and f′ = 166.667, respectively. The maximum and minimum focal lengths are f′ = 308.706 at (0.000, −6.241, 3.325) and f′ = 69.844 at (−15.000, 4.876, 15.325), respectively.

Equations (121)

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s F O = 1 A 5,0 1 C 5,0 ,
1 A 5,0 = [ ϕ 1 , e 10 , ϕ 2 , e 20 , ϕ 3 , e 30 , ϕ 4 , e 40 ] , 1 C 5,0 = [ ϕ 1 , e 10 , ϕ 2 , e 20 , ϕ 3 , e 30 , ϕ 4 , e 40 , ϕ 5 ] . }
s F = 1 A 5 1 C 5 ,
1 A 5 = [ ϕ 1 , e 1 , ϕ 2 , e 2 , ϕ 3 , e 3 , ϕ 4 , e 4 ] , 1 C 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 2 + x 3 , e 4 = e 40 x 3 . }
s F = s F O ,
Z ( x 1 , x 2 , x 3 ) 1 A 5 1 C 5 s F O = 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 6 , J 1 = ϕ 1 ( 3 A 6 + ϕ 2 3 B 6 ) , K 1 = ( ϕ 1 + ϕ 2 ) 3 A 6 , L 1 = 3 A 6 + ( ϕ 1 + ϕ 2 ) 3 B 6 , 3 A 6 = [ ϕ 3 , e 3 , ϕ 4 , e 4 , ϕ 5 , s F O ] , 3 B 6 = [ e 3 , ϕ 4 , e 4 , ϕ 5 , s F O ] , e 3 = e 30 x 2 + x 3 , e 4 = e 40 x 3 . }
x 1 = Y 1 ± Y 1 2 4 X 1 Z 1 2 X 1 .
3 A 6 = 0
x 1 = e 10 + ϕ 1 + ϕ 2 ϕ 1 ϕ 2 .
ϕ 1 = 0 ,
x 1 = ( e 20 + x 2 ) 1 ϕ 2 3 B 6 3 A 6 .
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 + x 3 ) + J 2 K 2 , Z 2 = I 2 ( e 20 x 1 ) ( e 30 + x 3 ) + J 2 ( e 20 x 1 ) + K 2 ( e 30 + x 3 ) + L 2 , I 2 = 1 C 2 ϕ 3 4 A 6 , J 2 = 1 C 2 ( 4 A 6 + ϕ 3 4 B 6 ) , K 2 = ( 1 A 2 ϕ 3 + 1 C 2 ) 4 A 6 , L 2 = 1 A 2 ( 4 A 6 + ϕ 3 4 B 6 ) + 1 C 2 4 B 6 , 1 A 2 = [ ϕ 1 , e 1 ] , 1 C 2 = [ ϕ 1 , e 1 , ϕ 2 ] , 4 A 6 = [ ϕ 4 e 4 , ϕ 5 , s F O ] , 4 B 6 = [ e 4 , ϕ 5 , s F O ] e 1 = e 10 + x 1 , e 4 = e 40 x 3 . }
x 2 = Y 2 ± Y 2 2 4 X 2 Z 2 2 X 2 .
1 C 2 = 0
x 2 = ( e 30 + x 3 ) 1 ϕ 3 4 B 6 4 A 6 .
4 A 6 = 0 ,
x 2 = ( e 20 x 1 ) + 1 A 2 1 C 2 + 1 ϕ 3 .
X 3 x 3 2 + Y 3 x 3 + Z 3 = 0,
X 3 = I 3 , Y 3 = I 3 ( e 30 + e 40 + x 2 ) + J 3 K 3 , Z 3 = I 3 ( e 30 x 2 ) e 40 + J 3 ( e 30 x 3 ) + K 3 e 40 + L 3 , I 3 = 1 C 3 ϕ 4 5 A 6 , J 3 = 1 C 3 ( 5 A 6 + ϕ 4 5 B 6 ) , K 3 = ( 1 A 3 ϕ 4 + 1 C 3 ) 5 A 6 , L 3 = 1 A 3 ( 5 A 6 + ϕ 4 5 B 6 ) + 1 C 3 5 B 6 , 1 A 3 = [ ϕ 1 , e 1 , ϕ 2 , e 2 ] , 1 C 3 = [ ϕ 1 , e 1 , ϕ 2 , e 2 , ϕ 3 ] , 5 A 6 = [ ϕ 5 , s F O ] , 5 B 6 = [ s F O ] , e 1 = e 10 + x 1 , e 2 = e 20 x 1 + x 2 . }
x 3 = Y 3 ± Y 3 2 4 X 3 Z 3 2 X 3 .
1 C 3 = 0 ,
x 3 = e 40 1 ϕ 4 5 B 6 5 A 6 .
5 A 6 = 0 ,
x 3 = ( e 30 x 2 ) + 1 A 3 1 C 3 + 1 ϕ 4 .
Z x 1 = ϕ 1 2 A 6 + 1 C 2 3 A 6 , Z x 2 = 1 C 2 3 A 6 + 1 C 3 4 A 6 , Z x 3 = 1 C 3 4 A 6 + 1 C 4 5 A 6 , }
1 C 2 = [ ϕ 1 , e 1 , ϕ 2 ] , 1 C 3 = [ ϕ 1 , e 1 , ϕ 2 , e 2 , ϕ 3 ] , 1 C 4 = [ ϕ 1 , e 1 , ϕ 2 , e 2 , ϕ 3 , e 3 , ϕ 4 ] , 2 A 6 = [ ϕ 2 , e 2 , ϕ 3 , e 3 , ϕ 4 , e 4 , ϕ 5 , s F O ] , 3 A 6 = [ ϕ 3 , e 3 , ϕ 4 , e 4 , ϕ 5 , s F O ] , 4 A 6 = [ ϕ 4 , e 4 , ϕ 5 , s F O ] , 5 A 6 = [ ϕ 5 , s F O ] . }
2 Z x 1 2 = 2 ϕ 1 ϕ 2 3 A 6 , 2 Z x 2 2 = 2 1 C 2 ϕ 3 4 A 6 , 2 Z x 3 2 = 2 1 C 3 ϕ 4 5 A 6 , 2 Z x 1 x 2 = ϕ 1 ϕ 2 3 A 6 + ( ϕ 1 2 C 3 + 1 C 2 ϕ 3 ) 4 A 6 , 2 Z x 1 x 3 = ( ϕ 1 2 C 3 1 C 2 ϕ 3 ) 4 A 6 + ( ϕ 1 2 C 4 + 1 C 2 3 C 4 ) 5 A 6 , 2 Z x 2 x 3 = 1 C 2 ϕ 3 4 A 6 + ( 1 C 2 3 C 4 + 1 C 3 ϕ 4 ) 5 A 6 , }
2 C 3 = [ ϕ 2 , e 2 , ϕ 3 ] , 2 C 4 = [ ϕ 2 , e 2 , ϕ 3 , e 3 , ϕ 4 ] , 3 C 4 = [ ϕ 3 , e 3 , ϕ 4 ] . }
Z x 1 = { ϕ 1 2 + ( 1 C 2 ) 2 } / Φ , Z x 2 = { ( 1 C 2 ) 2 + ( 1 C 3 ) 2 } / Φ , Z x 3 = { ( 1 C 3 ) 2 + ( 1 C 4 ) 2 } / Φ , }
2 Z x 1 2 = 2 ϕ 1 ϕ 2 1 C 2 / Φ , 2 Z x 2 2 = 2 1 C 2 ϕ 3 1 C 3 / Φ , 2 Z x 3 2 = 2 1 C 3 ϕ 4 1 C 4 / Φ , 2 Z x 1 x 2 = { ϕ 1 ϕ 2 1 C 2 + ( ϕ 1 2 C 3 + 1 C 2 ϕ 3 ) 1 C 3 } / Φ , 2 Z x 1 x 3 = { ( ϕ 1 2 C 3 1 C 2 ϕ 3 ) 1 C 3 + ( ϕ 1 2 C 4 + 1 C 2 3 C 4 ) 1 C 4 } / Φ , 2 Z x 2 x 3 = { 1 C 2 ϕ 3 1 C 3 + ( 1 C 2 3 C 4 + 1 C 3 ϕ 4 ) 1 C 4 } / Φ . }
Φ = [ ϕ 1 , e 1 , ϕ 2 , e 2 , ϕ 3 , e 3 , ϕ 4 , e 4 , ϕ 5 ] .
i C j = [ ϕ i , e i , ϕ i + 1 , , e j 1 , ϕ j ] ,
Z x 2 = ( β 3 2 + 1 ) ( β 4 β 5 ) Φ 0 ,
x 2 f 2 ( x 1 , x 3 ) .
f 2 x 1 = ( Z x 1 ) / ( Z x 2 ) = ( β 2 2 + 1 ) β 3 2 β 3 2 + 1 , f 2 x 3 = ( Z x 3 ) / ( Z x 2 ) = β 4 2 + 1 ( β 3 2 + 1 ) β 4 2 . }
| β 2 | = | β 4 | = 1 ,
f 2 x 1 = f 2 x 3 = 0.
2 f 2 x 1 2 = ( 2 Z x 1 2 ) / ( Z x 2 ) , 2 f 2 x 3 2 = ( 2 Z x 3 2 ) / ( Z x 2 ) , 2 f 2 x 1 x 3 = ( 2 Z x 1 x 3 ) / ( Z x 2 ) , }
H 2 = | 2 f 2 x 1 2 2 f 2 x 1 x 3 2 f 2 x 1 x 3 2 f 2 x 3 2 | = 1 ( Z x 2 ) 2 { ( 2 Z x 1 2 ) ( 2 Z x 3 2 ) ( 2 Z x 1 x 3 ) 2 } .
x 2 = f 2 ( x 1 , x 3 ) 1 2 ( 2 f 2 x 1 2 ) x 1 2 + ( 2 f 2 x 1 x 3 ) x 1 x 3 + 1 2 ( 2 f 2 x 3 2 ) x 3 2 = 1 ( Z x 2 ) { 1 2 ( 2 Z x 1 2 ) x 1 2 + 2 Z x 1 x 3 x 1 x 3 + 1 2 ( 2 Z x 3 2 ) x 3 2 } .
H 2 > 0 ,
2 f 2 x 1 2 > 0 ,
2 f 2 x 1 2 < 0.
H 2 < 0 ,
β 2 = 1 , | β 3 | 1 , β 4 = 1.
e 1 = 1 / ϕ 1 + 2 / ϕ 2 , e 2 = 2 / ϕ 2 + ( β 3 1 ) / ϕ 3 β 3 , e 3 = ( 1 β 3 ) / ϕ 3 + 2 / ϕ 4 , }
1 C 2 = ϕ 1 , 1 C 3 = ϕ 1 / β 3 , 1 C 4 = ϕ 1 / β 3 , 2 C 3 = ϕ 3 / β 3 ϕ 3 , 2 C 4 = ϕ 2 / β 3 + ϕ 3 ϕ 4 β 3 , 3 C 4 = ϕ 3 + ϕ 4 β 3 . }
2 Z x 1 2 = 2 ϕ 1 2 ϕ 2 / Φ , 2 Z x 3 2 = 2 ( ϕ 1 / β 3 ) 2 ϕ 4 / Φ , 2 Z x 1 x 3 = 0. }
Z x 2 = ϕ 1 2 ( β 3 2 + 1 ) / β 3 2 Φ .
H 2 = ( 2 β 3 β 3 2 + 1 ) 2 ϕ 2 ϕ 4 ,
2 f 2 x 1 2 = 2 ( β 3 2 β 3 2 + 1 ) ϕ 2 .
x 2 = f 2 ( x 1 , x 2 ) = β 3 2 β 3 2 + 1 ( ϕ 2 x 1 2 + 1 β 3 2 ϕ 4 x 3 2 ) .
critical point of x 2 ( β 2 = β 4 = 1 ) { extremum { maximum { ϕ 2 > 0 , ϕ 4 > 0 , | β 3 | < 1 , ϕ 2 < 0 , ϕ 4 < 0 , | β 3 | > 1 , minimum { ϕ 2 > 0 , ϕ 4 > 0 , | β 3 | > 1 , ϕ 2 < 0 , ϕ 4 < 0 , | β 3 | < 1 , saddle point { ϕ 2 > 0 , ϕ 4 < 0 , ϕ 2 < 0 , ϕ 4 > 0.
β 2 = 1 , | β 3 | 1 , β 4 = + 1 ,
H 2 = ( 2 β 3 β 3 2 + 1 ) 2 ϕ 2 ϕ 4 , 2 f 2 x 1 2 = 2 β 3 2 β 3 2 + 1 ϕ 2 , x 2 = f 2 ( x 1 , x 3 ) = β 3 2 β 1 2 + 1 ( ϕ 2 x 1 2 1 β 3 2 ϕ 4 x 3 2 ) , }
critical point of x 2 ( β 2 = 1 β 4 = + 1 ) { extremum { maximum { ϕ 2 > 0 , ϕ 4 < 0 , | β 3 | < 1 , ϕ 2 < 0 , ϕ 4 > 0 , | β 3 | > 1 , minimum { ϕ 2 > 0 , ϕ 4 < 0 , | β 3 | > 1 , ϕ 2 < 0 , ϕ 4 > 0 , | β 3 | < 1 , saddle point { ϕ 2 > 0 , ϕ 4 > 0 , ϕ 2 < 0 , ϕ 4 < 0.
β 2 = + 1 , | β 3 | 1 , β 4 = 1 ,
H 2 = ( 2 β 3 β 3 2 + 1 ) 2 ϕ 2 ϕ 4 , 2 f 2 x 1 2 = 2 β 3 2 β 3 2 + 1 ϕ 2 , x 2 = f 2 ( x 1 , x 3 ) = β 3 2 β 3 2 + 1 ( ϕ 2 2 x 1 2 1 β 3 2 ϕ 4 x 3 2 ) , }
critical point of x 2 ( β 2 = + 1 , β 4 = 1 ) { extremum { maximum { ϕ 2 > 0 , ϕ 4 < 0 , | β 3 | > 1 , ϕ 2 < 0 , ϕ 4 > 0 , | β 3 | < 1 , minimum { ϕ 2 > 0 , ϕ 4 < 0 , | β 3 | < 1 , ϕ 2 < 0 , ϕ 4 > 0 , | β 3 | > 1 , saddle point { ϕ 2 > 0 , ϕ 4 > 0 , ϕ 2 < 0 , ϕ 4 < 0.
β 2 = + 1 , | β 3 | 1 , β 4 = + 1 ,
H 2 = ( 2 β 3 β 3 2 + 1 ) 2 ϕ 2 ϕ 4 , 2 f 2 x 1 2 = 2 β 3 2 β 3 2 + 1 ϕ 2 , x 2 = f 2 ( x 1 , x 3 ) = β 3 2 β 3 2 + 1 ( ϕ 2 x 1 2 1 β 3 2 ϕ 4 x 3 2 ) , }
critical point of x 2 ( β 2 = β 4 = + 1 ) { extremum { maximum { ϕ 2 > 0 , ϕ 4 > 0 , | β 3 | > 1 , ϕ 2 < 0 , ϕ 4 < 0 , | β 3 | < 1 , minimum { ϕ 2 > 0 , ϕ 4 > 0 , | β 3 | > 1 , ϕ 2 < 0 , ϕ 4 < 0 , | β 3 | < 1 , saddle point { ϕ 2 > 0 , ϕ 4 < 0 , ϕ 2 < 0 , ϕ 4 > 0.
Z x 1 = ( β 2 2 + 1 ) ( β 3 β 4 β 5 ) 2 Φ 0
x 1 f 1 ( x 2 , x 3 ) ,
critical point of x 1 | β 3 | = | β 4 | = 1 Z x 2 = Z x 3 = 0 { extremum H 1 > 0 { maximum 2 f 1 x 3 2 < 0 { β 3 = 1 , β 4 = 1 { ϕ 3 > 0 , ϕ 4 > 0 , | β 2 | < 1 , ϕ 3 < 0 , ϕ 4 < 0 , | β 2 | > 1 , β 3 = 1 , β 4 = + 1 { ϕ 3 > 0 , ϕ 4 < 0 , | β 2 | < 1 , ϕ 3 < 0 , ϕ 4 > 0 , | β 2 | > 1 , β 3 = + 1 , β 4 = 1 { ϕ 3 > 0 , ϕ 4 > 0 , | β 2 | > 1 , ϕ 3 < 0 , ϕ 4 < 0 , | β 2 | < 1 , β 3 = + 1 , β 4 = + 1 { ϕ 3 > 0 , ϕ 4 > 0 , | β 2 | > 1 , ϕ 3 < 0 , ϕ 4 < 0 , | β 2 | < 1 , minimum 2 f 1 x 3 2 > 0 { β 3 = 1 , β 4 = 1 { ϕ 3 > 0 , ϕ 4 > 0 , | β 2 | > 1 , ϕ 3 < 0 , ϕ 4 < 0 , | β 2 | < 1 , β 3 = 1 , β 4 = + 1 { ϕ 3 > 0 , ϕ 4 < 0 , | β 2 | > 1 , ϕ 3 < 0 , ϕ 4 > 0 , | β 2 | < 1 , β 3 = + 1 , β 4 = 1 { ϕ 3 > 0 , ϕ 4 < 0 , | β 2 | < 1 , ϕ 3 < 0 , ϕ 4 > 0 , | β 2 | > 1 , β 3 = + 1 , β 4 = + 1 { ϕ 3 > 0 , ϕ 4 > 0 , | β 2 | < 1 , ϕ 3 < 0 , ϕ 4 < 0 , | β 2 | > 1 , saddle point H 1 < 0 { β 3 = ± 1 , β 4 = ± 1 { ϕ 3 > 0 , ϕ 4 < 0 , ϕ 3 < 0 , ϕ 4 > 0 , β 3 = ± 1 , β 4 = 1 { ϕ 3 > 0 , ϕ 4 > 0 , ϕ 3 < 0 , ϕ 4 < 0.
Z x 3 = ( β 4 2 + 1 ) β 5 2 Φ 0 ,
x 3 f 3 ( x 1 , x 2 ) ,
critical point of x 3 | β 2 | = | β 3 | = 1 Z x 1 = Z x 2 = 0 { extremum H 3 > 0 { maximum 2 f 3 x 2 2 < 0 { β 2 = 1 , β 3 = 1 { ϕ 2 > 0 , ϕ 3 > 0 , | β 4 | < 1 , ϕ 2 < 0 , ϕ 3 < 0 , | β 4 | > 1 , β 2 = 1 , β 3 = + 1 { ϕ 2 > 0 , ϕ 3 < 0 , | β 4 | < 1 , ϕ 2 < 0 , ϕ 3 > 0 , | β 4 | > 1 , β 2 = + 1 , β 3 = 1 { ϕ 2 > 0 , ϕ 3 < 0 , | β 4 | > 1 , ϕ 2 < 0 , ϕ 3 > 0 , | β 4 | < 1 , β 2 = + 1 , β 3 = + 1 { ϕ 2 > 0 , ϕ 3 > 0 , | β 4 | > 1 , ϕ 2 < 0 , ϕ 3 < 0 , | β 4 | < 1 , minimum 2 f 3 x 2 2 > 0 { β 2 = 1 , β 3 = 1 { ϕ 2 > 0 , ϕ 3 > 0 , | β 4 | > 1 , ϕ 2 < 0 , ϕ 3 < 0 , | β 4 | < 1 , β 2 = 1 , β 3 = + 1 { ϕ 2 > 0 , ϕ 3 < 0 , | β 4 | > 1 , ϕ 2 < 0 , ϕ 3 > 0 , | β 4 | < 1 , β 2 = + 1 , β 3 = 1 { ϕ 2 > 0 , ϕ 3 < 0 , | β 4 | < 1 , ϕ 2 < 0 , ϕ 3 > 0 , | β 4 | > 1 , β 2 = + 1 , β 3 = + 1 { ϕ 2 > 0 , ϕ 3 > 0 , | β 4 | < 1 , ϕ 2 < 0 , ϕ 3 < 0 , | β 4 | > 1 , saddle point H 3 < 0 { β 2 = ± 1 , β 3 = ± 1 { ϕ 2 > 0 , ϕ 3 < 0 , ϕ 2 < 0 , ϕ 3 > 0 , β 2 = ± 1 , β 3 = 1 { ϕ 2 > 0 , ϕ 3 > 0 , ϕ 2 < 0 , ϕ 3 < 0.
| β 2 | = | β 3 | = | β 4 | = 1 ,
Z x 1 = Z x 2 = Z x 3 = 0.
H = | 2 Z x 1 2 2 Z x 1 x 2 2 Z x 1 x 3 2 Z x 1 x 2 2 Z x 2 2 2 Z x 2 x 3 2 Z x 1 x 3 2 Z x 2 x 3 2 Z x 3 2 | 0
Z ( x 1 , x 2 , x 3 ) = 1 2 2 Z x 1 2 x 1 2 + 1 2 2 Z x 2 2 x 2 2 + 1 2 2 Z x 3 2 x 3 2 + 2 Z x 1 x 2 x 1 x 2 + 2 Z x 1 x 3 x 1 x 3 + 2 Z x 2 x 3 x 2 x 3 = 0.
β 2 = β 3 = β 4 = 1.
e 1 = ( 2 ϕ 1 + ϕ 2 ) / ϕ 1 ϕ 2 , e 2 = 2 ( ϕ 2 + ϕ 3 ) / ϕ 2 ϕ 3 , e 3 = 2 ( ϕ 3 + ϕ 4 ) / ϕ 3 ϕ 4 ,
1 C 2 = ϕ 1 , 1 C 3 = ϕ 1 , 1 C 4 = ϕ 1 , 2 C 3 = ( ϕ 2 + ϕ 3 ) , 2 C 4 = ϕ 2 + ϕ 3 + ϕ 4 , 3 C 4 = ( ϕ 3 + ϕ 4 ) . }
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 3 2 = 2 ϕ 1 2 ϕ 2 / Φ , 2 Z x 1 x 2 = 2 Z x 1 x 3 = 2 Z x 2 x 3 = 0. }
H = 8 ( ϕ 1 2 Φ ) ϕ 2 ϕ 3 ϕ 4 .
Z ( x 1 , x 2 x 3 ) = ϕ 2 x 1 2 + ϕ 3 x 2 2 + ϕ 4 x 3 2 = 0.
sgn ( ϕ 2 ) = sgn ( ϕ 3 ) = sgn ( ϕ 4 )
sgn ( ϕ 2 ) sgn ( ϕ 3 ) , sgn ( ϕ 4 ) , or sgn ( ϕ 3 ) sgn ( ϕ 2 ) , sgn ( ϕ 4 ) , or sgn ( ϕ 4 ) sgn ( ϕ 2 ) , sgn ( ϕ 3 ) , }
Z ( x 1 , x 2 , x 3 ) = ϕ 2 x 1 2 + ϕ 3 x 2 2 ϕ 4 x 3 2 = 0 , when β 2 = β 3 = 1 , β 4 = + 1 , Z ( x 1 , x 2 , x 3 ) = ϕ 2 x 1 2 ϕ 3 x 2 2 + ϕ 4 x 3 2 = 0 , when β 2 = 1 , β 3 = + 1 , β 4 = 1 , Z ( x 1 , x 2 , x 3 ) = ϕ 2 x 1 2 + ϕ 3 x 2 2 + ϕ 4 x 3 2 = 0 , when β 2 = + 1 , β 3 = β 4 = 1 , Z ( x 1 , x 2 , x 3 ) = ϕ 2 x 1 2 ϕ 3 x 2 2 ϕ 4 x 3 3 = 0 , when β 2 = 1 , β 3 = β 4 = + 1 , Z ( x 2 , x 2 , x 3 ) = ϕ 2 x 1 2 + ϕ 3 x 2 2 ϕ 4 x 3 2 = 0 , when β 2 = + 1 , β 3 = 1 , β 4 = + 1 , Z ( x 1 , x 2 , x 3 ) = ϕ 2 x 1 2 ϕ 3 x 2 2 + ϕ 4 x 3 2 = 0 , when β 2 = β 3 = + 1 , β 4 = 1 , Z ( x 1 , x 2 , x 3 ) = ϕ 2 x 1 2 + ϕ 3 x 2 2 + ϕ 4 x 3 2 = 0 , when β 2 = β 3 = β 4 = + 1. }
x 2 = 1.8403 , x 3 = 1.3629 , }
β 2 = 0.000 , β 3 = 1.000 , β 4 = + 1.000. }
H 1 = 0.1388 > 0.
2 f 1 x 2 2 = 0.7201 < 0.
x 1 = 5.2912.
x 1 = ( 0.1953 x 2 2 + 0.0964 x 3 2 ) .
x 1 = 3.9084 , x 2 = 19.9175 , }
β 2 = 1.0000 , β 3 = 1.0000 , β 4 = 0.7731. }
H 3 = 0.0151 < 0.
x 3 = 6.5009.
x 3 = ( 0.0434 x 1 2 0.0872 x 2 2 ) .
x 1 = 5.0000 , x 2 = 0.0000 , x 3 = 8.7255 , }
β 2 = β 3 = β 4 = 1.0000.
H = 1.8518 × 10 9 0 ,
z ( x 1 , x 2 , x 3 ) = 0.0500 x 1 2 + 0.0400 x 2 2 + 0.0250 x 3 2 = 0.
sgn ( ϕ 2 ) sgn ( ϕ 3 ) , sgn ( ϕ 4 ) ,
β i = 1 C i 1 / 1 C i .
r = f ( x 1 , x 2 , , x n ) ,
F ( x 1 , x 2 , , x n , y ) = 0.
2 y x i x k = 1 ( F y ) 3 | 0 F x i F y F x k 2 F x i x k 2 F x y x k F y 2 F x i y 2 F y 2 | i , k = 1,2,3 , , n . ,
F ( x , y , z ) = a x 2 + b y 2 + c z 2 + 2 f y z + 2 g z x + 2 h x y + 2 p x + 2 q y + 2 r z + d = 0.
D = | a h g h b f g f c | = 0
Δ = | a h g p h b f q g f c r p q r d | 0
e i = ( 1 β i ) / ϕ i + ( β i + 1 1 ) / ( ϕ i + 1 β i + 1 ) .
1 C 2 = ϕ 1 / β 2 , 1 C 3 = ϕ 1 / ( β 2 β 3 ) , 1 C 4 = ϕ 1 / ( β 2 β 3 β 4 ) , 2 C 3 = ϕ 2 / β 3 + ϕ 3 β 2 , 2 C 4 = ϕ 2 / ( β 3 β 4 ) + ϕ 3 ( β 2 / β 4 ) + ϕ 4 ( β 2 β 3 ) , 3 C 4 = ϕ 3 / β 4 + ϕ 4 β 3 . }
Z x 2 = Z x 3 = 0 ,
| β 3 | = | β 4 | = 1 ,
H 1 = ( 2 β 2 2 + 1 ) 2 β 3 β 4 ϕ 3 ϕ 4 ,
2 f 1 x 3 2 = 2 β 2 2 + 1 β 4 ϕ 4 ,
x 1 = f 1 ( x 2 , x 3 ) = 1 β 2 2 + 1 ( β 3 ϕ 3 x 2 2 + β 4 ϕ 4 x 3 2 ) ,
Z x 1 = Z x 2 = 0 ,
| β 2 | = | β 3 | = 1 ,
H 3 = ( 2 β 4 2 β 4 2 + 1 ) 2 β 2 β 3 ϕ 2 ϕ 3 ,
2 f 3 x 2 2 = 2 β 4 2 β 4 2 + 1 β 3 ϕ 3 ,
x 3 = f 3 ( x 1 , x 2 ) = β 4 2 β 4 2 + 1 ( β 2 ϕ 2 x 1 2 + β 3 ϕ 3 x 2 2 ) ,
D 0 ,
Δ = 0 ,

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