Abstract

The study of the smoothest reflectance functions under a given illuminant is extended to the domain outside the principal domain. The functions are identically zero or unity on certain intervals determined by their tristimulus values. The domains of validity are obtained. There result 12 additional types of reflectance function. An algorithm for the generation of the most important ones from a practical point of view is described.

© 1990 Optical Society of America

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References

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  1. C. van Trigt, “Smoothest reflectance functions. I. Definition and main results,” J. Opt. Soc. Am. A 7, 1891–1904 (1990).
    [CrossRef]
  2. E. Schrödinger, “Theorie der Pigmente von grosster Leuchtkraft,” Ann. Phys. (Leipzig) 62, 603–622 (1920).
  3. W. Ostwald, “Neue Forschungen zur Farbenlehre,” Phys. Z. 17, 322–332 (1916).
  4. D. MacAdam, Color Measurement: Theme and Variations (Springer-Verlag, New York, 1981).
  5. G. West, M. H. Brill, “Conditions under which Schrödinger object colors are optimal,” J. Opt. Soc. Am. 73, 1223–1225 (1983).
    [CrossRef]
  6. C. Fox, An Introduction to the Calculus of Variations (Oxford U. Press, London, 1954).
  7. G. Polya, G. Szegö, Problems and Theorems in Analysis (Springer-Verlag, Berlin, 1972).
  8. J. N. Lythgoe, The Ecology of Vision (Clarendon, Oxford, 1979).
  9. A. C. Aitken, Determinants and Matrices (Oliver and Boyd, Edinburgh, 1959).

1990 (1)

1983 (1)

1920 (1)

E. Schrödinger, “Theorie der Pigmente von grosster Leuchtkraft,” Ann. Phys. (Leipzig) 62, 603–622 (1920).

1916 (1)

W. Ostwald, “Neue Forschungen zur Farbenlehre,” Phys. Z. 17, 322–332 (1916).

Aitken, A. C.

A. C. Aitken, Determinants and Matrices (Oliver and Boyd, Edinburgh, 1959).

Brill, M. H.

Fox, C.

C. Fox, An Introduction to the Calculus of Variations (Oxford U. Press, London, 1954).

Lythgoe, J. N.

J. N. Lythgoe, The Ecology of Vision (Clarendon, Oxford, 1979).

MacAdam, D.

D. MacAdam, Color Measurement: Theme and Variations (Springer-Verlag, New York, 1981).

Ostwald, W.

W. Ostwald, “Neue Forschungen zur Farbenlehre,” Phys. Z. 17, 322–332 (1916).

Polya, G.

G. Polya, G. Szegö, Problems and Theorems in Analysis (Springer-Verlag, Berlin, 1972).

Schrödinger, E.

E. Schrödinger, “Theorie der Pigmente von grosster Leuchtkraft,” Ann. Phys. (Leipzig) 62, 603–622 (1920).

Szegö, G.

G. Polya, G. Szegö, Problems and Theorems in Analysis (Springer-Verlag, Berlin, 1972).

van Trigt, C.

West, G.

Ann. Phys. (Leipzig) (1)

E. Schrödinger, “Theorie der Pigmente von grosster Leuchtkraft,” Ann. Phys. (Leipzig) 62, 603–622 (1920).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Phys. Z. (1)

W. Ostwald, “Neue Forschungen zur Farbenlehre,” Phys. Z. 17, 322–332 (1916).

Other (5)

D. MacAdam, Color Measurement: Theme and Variations (Springer-Verlag, New York, 1981).

C. Fox, An Introduction to the Calculus of Variations (Oxford U. Press, London, 1954).

G. Polya, G. Szegö, Problems and Theorems in Analysis (Springer-Verlag, Berlin, 1972).

J. N. Lythgoe, The Ecology of Vision (Clarendon, Oxford, 1979).

A. C. Aitken, Determinants and Matrices (Oliver and Boyd, Edinburgh, 1959).

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Figures (8)

Fig. 1
Fig. 1

Reflectance functions of this paper and Ref. 1 considered as increasingly smoother versions of Schrödinger’s optimal filters.

Fig. 2
Fig. 2

Two reflectance functions of Classes I and III (λ1 = λb). 1, λ2 = 584 nm, x = 0.1495, y = 0.4736. 2, λ2 = 640 nm, x = 0.2548, y = 0.5271. The chromaticity points lie on the common boundary of Classes I and III. The light source is D65.

Fig. 3
Fig. 3

Regions containing the chromaticity points for which the reflectance functions belong to the classes indicated. The light source is A.

Fig. 4
Fig. 4

Like Fig. 3 but for light source D65. Also are indicated the chromaticity points of the CIE test colors 9 and 12.

Fig. 5
Fig. 5

Like Fig. 3 but for the equal-energy spectrum. The equations of the straight lines are 9.918x = 3.721y = 1, 0.5694x + 1.591y = 1.

Fig. 6
Fig. 6

Two reflectance functions of Classes II and III (λ2 = λe). 1, λ1 = 400 nm, x = 0.2787, y = 0.5249. 2, λ1 = 520 nm, x = 0.4588, y = 0.5376. The chromaticity points lie on the common boundary of Classes II and III. The light source is D65.

Fig. 7
Fig. 7

Two reflectance functions of Class IV. 1, λ2 = 640 nm, λ1 = λe, x = 0.1473, y = 0.1295. 2, λ2 = λb, λ1 = 520 nm; x = 0.6024, y = 0.3966. The first one also belongs to Class I (λ2 = 640 nm) and the second one to Class II (λ1 = 520 nm). The chromaticity points lie on the common boundaries of Classes I and IV and II and IV. The light source is D65.

Fig. 8
Fig. 8

CIE test colors (dashed curves) 9 (strong red) and 12 (strong blue) compared with the reflectance functions defined by Eqs. (1), (2), and (3) (solid curves) for D65. The function 9 is zero for 489.4 nm ≤ λ ≤ 545.3 nm and 12 for λ ≥ 632.6 nm. Compare also Fig. 4.

Equations (100)

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0 ρ ( λ ) 1 , λ ( λ b , λ e ) .
ρ ( λ ) S ( λ ) x ¯ ( λ ) d λ = X , ρ ( λ ) S ( λ ) y ¯ ( λ ) d λ = Y , ρ ( λ ) S ( λ ) z ¯ ( λ ) d λ = Z .
( d ρ d λ ) 2 d λ = minimal .
r ( λ ) { d 2 ρ d λ 2 + [ μ 1 X 0 x ¯ ( λ ) + μ 2 Y 0 y ¯ ( λ ) + μ 3 Z 0 z ¯ ( λ ) ] S ( λ ) } = 0.
r ( λ ) d ρ ( λ ) d λ = 0 at λ = λ b and λ = λ e .
d 2 ρ d λ 2 + [ μ 1 X 0 x ¯ ( λ ) + μ 2 Y 0 y ¯ ( λ ) + μ 3 Z 0 z ¯ ( λ ) ] S ( λ ) = 0 , λ ( λ 1 k , λ 2 k ) .
ρ ( λ ) = 0 at λ = λ 1 k λ b and λ = λ 2 k λ e .
d ρ d λ = 0 at λ = λ 1 1 = λ b or λ = λ 2 K = λ e
k = 1 K λ 1 k λ 2 k ρ ( λ ) S ( λ ) x ¯ ( λ ) d λ = X , k = 1 K λ 1 k λ 2 k ρ ( λ ) S ( λ ) y ¯ ( λ ) d λ = Y , k = 1 K λ 1 k λ 2 k ρ ( λ ) S ( λ ) z ¯ ( λ ) d λ = Z ,
k = 1 K λ 1 k λ 2 k ( d ρ d λ ) 2 d λ = μ 1 X X 0 + μ 2 Y Y 0 + μ 3 Z Z 0 .
d ρ d λ = 0 at λ = λ 1 k λ b and λ = λ 2 k λ e .
f 1 ( λ ) = 1 X 0 λ b λ S ( λ ) x ¯ ( λ ) d λ , f 2 ( λ ) = 1 Y 0 λ b λ S ( λ ) y ¯ ( λ ) d λ , f 3 ( λ ) = 1 Z 0 λ b λ S ( λ ) z ¯ ( λ ) d λ
d ρ ( λ b ) d λ = ρ ( λ 2 ) = 0.
d ρ d λ = j = 1 3 μ j f j ( λ ) , ρ ( λ ) = j = 1 3 μ j λ λ 2 f j ( λ ) d λ .
A [ μ 1 μ 2 μ 3 ] = [ X / X 0 Y / Y 0 Z / Z 0 ] ,
a i , j = λ λ 2 f i ( λ ) f j ( λ ) d λ .
d ρ ( λ 2 ) d λ = j = 1 3 μ j f j ( λ 2 ) = 0 .
ρ ( λ 1 ) = d ρ ( λ e ) d λ = 0.
d ρ d λ = j = 1 3 μ j g j ( λ ) , ρ ( λ ) = j = 1 3 μ j λ 1 λ g j ( λ ) d λ .
B [ μ 1 μ 2 μ 3 ] = [ X / X 0 Y / Y 0 Z / Z 0 ] ,
b i , j = λ 1 λ e g i ( λ ) g j ( λ ) d λ .
d ρ ( λ 1 ) d λ = j = 1 3 μ j g j ( λ 1 ) = 0 .
ρ ( λ 1 ) = ρ ( λ 2 ) = 0.
d ρ d λ = ρ ( λ 1 ) j = 1 3 μ j ϕ j ( λ ) , ρ ( λ ) = ( λ λ 2 ) ρ ( λ 1 ) + j = 1 3 μ j λ λ 2 ϕ j ( λ ) d λ ,
ρ ( λ 1 ) = j = 1 3 μ j ( λ 2 λ 1 ) λ λ 2 ϕ j ( λ ) d λ ,
C [ μ 1 μ 2 μ 3 ] = [ X / X 0 Y / Y 0 Z / Z 0 ] ,
c i , j = λ 1 λ 2 ψ i ( λ ) ψ j ( λ ) d λ ,
ψ i ( λ ) = ϕ i ( λ ) 1 λ 2 λ 1 λ 1 λ 2 ϕ i ( λ ) d λ .
( λ 2 λ 1 ) d ρ ( λ 1 ) d λ = j = 1 3 μ j λ 1 λ 2 ϕ j ( d λ = 0 , d ρ ( λ 2 ) d λ = j = 1 3 μ j ϕ j ( λ 2 ) = 0 .
d ρ ( λ b ) d λ = ρ ( λ 2 ) = ρ ( λ 1 ) = d ρ ( λ e ) d λ = 0.
( A + B ) [ μ 1 μ 2 μ 3 ] = [ X / X 0 Y / Y 0 Z / Z 0 ] ,
f j * ( λ ) = f j ( λ ) on ( λ b , λ 2 ) = 0 elsewhere , g j * ( λ ) = g j ( λ ) on ( λ 1 , λ e ) = 0 elsewhere .
[ f i * ( λ ) + g i * ( λ ) ] [ f i * ( λ ) + g i * ( λ ) ] d λ = λ b λ 2 f i ( λ ) f j ( λ ) d λ + λ 1 λ e g i ( λ ) g j ( λ ) d λ .
k = 1 K A ( k ) [ μ 1 μ 2 μ 3 ] = [ X / X 0 Y / Y 0 Z / Z 0 ] ,
μ 1 ( λ 2 ) λ 2 X X 0 + μ 2 ( λ 2 ) λ 2 Y Y 0 + μ 3 ( λ 2 ) λ 2 Z Z 0 = 0 ,
A ( λ 2 ) λ 2 [ μ 1 μ 2 μ 3 ] + k = 1 K A ( k ) [ μ 1 / λ 2 μ 2 / λ 2 μ 3 / λ 2 ] = [ 0 0 0 ] ,
[ μ 1 μ 2 μ 3 ] · A ( λ 2 ) λ 2 [ μ 1 μ 2 μ 3 ] + [ X / X 0 Y / Y 0 Z / Z 0 ] · [ μ 1 / λ 2 μ 2 / λ 2 μ 3 / λ 2 ] = 0.
( d ρ d λ ) 2 = 0 at λ = λ 2 λ e .
ρ ( λ ) = γ | f 1 ( λ 2 ) f 2 ( λ 2 ) f 3 ( λ 2 ) f 2,1 f 2,2 f 2,3 f 3,1 f 3,2 f 3,3 | with f 2 , i = λ b λ 2 f i ( λ ) d λ and f 3 , i = λ b λ 2 f i ( λ ) d λ .
ρ ( λ ) = γ | g 1 ( λ 1 ) g 2 ( λ 1 ) g 3 ( λ 1 ) g 2,1 g 2,2 g 2,3 g 3,1 g 3,2 g 3,3 | with g 2 , i = λ 1 λ e g i ( λ ) d λ and g 3 , i = λ 1 λ g i ( λ ) d λ .
ρ ( λ ) = γ | ϕ 1 ( λ 2 ) ϕ 2 ( λ 2 ) ϕ 3 ( λ 2 ) ϕ 2,1 ϕ 2,2 ϕ 2,3 ϕ 3,1 ϕ 3,2 ϕ 3,3 | with ϕ 2 , i = λ 1 λ 2 ϕ i ( λ ) d λ and ϕ 3 , i = λ 1 λ ϕ i ( λ ) d λ .
ρ ( λ ) = γ | f 1 ( λ 2 ) f 2 ( λ 2 ) f 3 ( λ 2 ) g 1 ( λ 1 ) g 2 ( λ 1 ) g 3 ( λ 1 ) f 3,1 f 3,2 f 3,3 | with f 3 , i = λ λ 2 f i ( λ ) d λ .
ρ ( λ ) = γ | f 1 ( λ 2 ) f 2 ( λ 2 ) f 3 ( λ 2 ) g 1 ( λ 1 ) g 2 ( λ 1 ) g 3 ( λ 1 ) g 3,1 g 3,2 g 3,3 | with g 3 , i = λ 1 λ g i ( λ ) d λ .
ρ ( λ ) = γ | f 1 ( λ 2 ) f 2 ( λ 2 ) f 3 ( λ 2 ) x ¯ ( λ e ) / X 0 y ¯ ( λ e ) / Y 0 z ¯ ( λ e ) / Z 0 f 3,1 f 3,2 f 3,3 | .
ρ ( λ ) = γ | x ¯ ( λ b ) / X 0 y ¯ ( λ b ) / Y 0 z ¯ ( λ b ) / Z 0 g 1 ( λ 1 ) g 2 ( λ 1 ) g 3 ( λ 1 ) g 3,1 g 3,2 g 3,3 | .
0 = j = 1 3 μ j ( λ 2 + Δ λ 2 ) f j ( λ 2 + Δ λ 2 ) = j = 1 3 μ j ( λ 2 ) f j ( λ 2 ) + Δ λ 2 j = 1 3 μ j ( λ 2 ) d f j ( λ 2 ) / d λ
0 = j = 1 3 μ j ( λ 1 + Δ λ 1 ) g j ( λ 1 + Δ λ 1 ) = j = 1 3 μ j ( λ 1 ) g j ( λ 1 ) + Δ λ 1 j = 1 3 μ j ( λ 1 ) d g j ( λ 1 ) / d λ
ρ ( λ ) = 1 + r 2 ( λ ) s 2 ( λ ) 2
r 2 ( λ ) + s 2 ( λ ) = 1.
r ( λ ) { d 2 ρ d λ 2 + [ μ 1 X 0 x ¯ ( λ ) + μ 2 Y 0 y ¯ ( λ ) + μ 3 Z 0 z ¯ ( λ ) ] S ( λ ) + μ ( λ ) } = 0 , s ( λ ) { d 2 ρ d λ 2 + [ μ 1 X 0 x ¯ ( λ ) + μ 2 Y 0 y ¯ ( λ ) + μ 3 Z 0 z ¯ ( λ ) ] S ( λ ) μ ( λ ) } = 0.
r ( λ ) s ( λ ) d ρ ( λ ) d ( λ ) = 0 at λ = λ b and λ = λ e .
d 2 ρ d λ 2 + [ μ 1 X 0 x ¯ ( λ ) + μ 2 Y 0 y ¯ ( λ ) + μ 3 Z 0 z ¯ ( λ ) ] S ( λ ) = 0 , λ ( λ 1 k , λ 2 k ) , μ ( λ ) = 0 .
ρ ( λ ) = 0 at λ = λ 1 k and λ = λ 2 k
ρ ( λ ) = 1 at λ = λ 1 k and λ = λ 2 k
d ρ d λ = 0 at λ = λ 1 k λ b and λ 2 k λ e .
ρ ( λ 1 ) = 1 , ρ ( λ 2 ) = 0.
d ρ d λ = 1 λ 2 λ 1 j = 1 3 μ j ψ j ( λ ) , ρ ( λ ) = λ 2 λ λ 2 λ 1 + j = 1 3 μ j λ λ 2 ψ j ( λ ) d λ .
A [ μ 1 μ 2 μ 3 ] = [ X / X 0 ξ 1 Y / Y 0 ξ 2 Z / Z 0 ξ 3 ] ,
a i , j = λ 1 λ 2 ψ i ( λ ) ψ j ( λ ) d λ
ξ j = ξ j A = 1 λ 2 λ 1 λ 1 λ 2 f j ( λ ) d λ .
j = 1 3 μ j λ 1 λ 2 ϕ j ( λ ) d λ = 1 , j = 1 3 μ j ϕ j ( λ 2 ) = 0.
ρ ( λ ) = γ | ϕ 1,1 ϕ 1,2 ϕ 1,3 0 ϕ 1 ( λ 2 ) ϕ 2 ( λ 2 ) ϕ 3 ( λ 2 ) 0 ϕ 3,1 ϕ 3,2 ϕ 3,3 1 0 0 Δ γ | with ϕ 1 , j = λ λ 2 ϕ i ( λ ) d λ and ϕ 3 , i = λ 1 λ 2 ϕ i ( λ ) d λ .
Δ = | ϕ 1 ( λ 2 ) ϕ 2 ( λ 2 ) ϕ 3,1 ϕ 3,2 | 1 .
ρ ( λ 1 ) = 0 , ρ ( λ 2 ) = 1.
d ρ d λ = 1 λ 2 λ 1 j = 1 3 μ j ψ j ( λ ) , ρ ( λ ) = λ λ 1 λ 2 λ 1 j = 1 3 μ j λ 1 λ ψ j ( λ ) d λ .
B [ μ 1 μ 2 μ 3 ] = [ X / X 0 ξ 1 Y / Y 0 ξ 2 Z / Z 0 ξ 3 ] ,
B = A
ξ j = ξ j B = 1 λ 2 λ 1 λ 1 λ 2 g j ( λ ) d λ .
j = 1 3 μ j λ 1 λ 2 ϕ j ( λ ) d λ = 1 , j = 1 3 μ j ϕ j ( λ 2 ) = 0 .
ϕ 3 , i = λ 1 λ ϕ i ( λ ) d λ .
λ 1 λ 2 ( d ρ d λ ) 2 d λ = [ μ 1 μ 2 μ 3 ] A [ μ 1 μ 2 μ 3 ] + 1 λ 2 λ 1 .
A [ μ 1 μ 2 μ 3 ] = [ X / X 0 + f 1 ( λ 1 ) ξ 1 Y / Y 0 + f 2 ( λ 1 ) ξ 2 Z / Z 0 + f 3 ( λ 1 ) ξ 3 ] ,
X = λ 1 λ 2 ρ ( λ ) S ( λ ) x ¯ ( λ ) d λ , Y = λ 1 λ 2 ρ ( λ ) S ( λ ) y ¯ ( λ ) d λ , Z = λ 1 λ 2 ρ ( λ ) S ( λ ) z ¯ ( λ ) d λ .
B [ μ 1 μ 2 μ 3 ] = [ X / X 0 + g 1 ( λ 2 ) ξ 1 Y / Y 0 + g 2 ( λ 2 ) ξ 2 Z / Z 0 + g 3 ( λ 2 ) ξ 3 ] ,
ρ ( λ 1 ) = ρ ( λ 2 ) = 0 , ρ ( λ 2 ) = ρ ( λ 1 ) = 1.
ρ ( λ ) = 1.
( B + A ) [ μ 1 μ 2 μ 3 ] = [ X / X 0 + 1 ξ 1 ξ 1 Y / Y 0 + 1 ξ 2 ξ 2 Z / Z 0 + 1 ξ 3 ξ 3 ] ,
j = 1 3 μ j λ 1 λ 2 ϕ j ( λ ) d λ = 1 , j = 1 3 μ j ϕ j ( λ 2 ) = 0 , j = 1 3 μ j λ 1 λ 2 ϕ j ( λ ) d λ = 1 , j = 1 3 μ j ϕ j ( λ 2 ) = 0.
[ ϕ 1,1 ϕ 1,2 ϕ 1,3 ϕ 1 ( λ 2 ) ϕ 2 ( λ 2 ) ϕ 3 ( λ 2 ) ϕ 1 ( λ 2 ) ϕ 2 ( λ 2 ) ϕ 3 ( λ 2 ) ] [ μ 1 μ 2 μ 3 ] = [ 1 0 0 ]
[ ϕ 1,1 ϕ 1,2 ϕ 1,3 ϕ 1 ( λ 2 ) ϕ 2 ( λ 2 ) ϕ 3 ( λ 2 ) ϕ 1 ( λ 2 ) ϕ 2 ( λ 2 ) ϕ 3 ( λ 2 ) ] [ μ 1 μ 2 μ 3 ] = [ 1 0 0 ]
ϕ 1 , j = λ 1 λ 2 ϕ j ( λ ) d λ , ϕ 1 , j = λ 1 λ 2 ϕ j ( λ ) d λ ,
Δ = ( 1 ) Δ .
ρ ( λ ) = 1 Δ | ϕ 1,1 ϕ 1,2 ϕ 1,3 ϕ 1 ( λ 2 ) ϕ 2 ( λ 2 ) ϕ 3 ( λ 2 ) ϕ 1 ( λ 2 ) ϕ 2 ( λ 2 ) ϕ 3 ( λ 2 ) | ,
ϕ 1 , j = λ 1 λ ϕ j ( λ ) d λ ;
ρ ( λ ) = 1 ;
ρ ( λ ) = 1 Δ | ϕ 1,1 ϕ 1,2 ϕ 1,3 ϕ 1 ( λ 2 ) ϕ 2 ( λ 2 ) ϕ 3 ( λ 2 ) ϕ 1 ( λ 2 ) ϕ 2 ( λ 2 ) ϕ 3 ( λ 2 ) | ,
ϕ 1 , j = λ λ 2 ϕ j ( λ ) d λ .
ρ ( λ 1 ) = ρ ( λ 2 ) = 1 , ρ ( λ 2 ) = ρ ( λ 1 ) = 0.
ρ ( λ ) = 0 ,
( A + B ) [ μ 1 μ 2 μ 3 ] = [ X / X 0 ξ 1 ξ 1 Y / Y 0 ξ 2 ξ 2 Z / Z 0 ξ 3 ξ 3 ] ,
k = 1 K λ 1 k λ 2 k ( d ρ d λ ) 2 d λ = k = 1 K [ μ 1 μ 2 μ 3 ] · A ( k ) [ μ 1 μ 2 μ 3 ] + ( λ 2 k λ 1 k ) 1 .
k = 1 K A ( k ) [ μ 1 μ 2 μ 3 ] = [ X / X 0 Ξ 1 Ξ 1 + N Y / Y 0 Ξ 2 Ξ 2 + N Z / Z 0 Ξ 3 Ξ 3 + N ] .
Ξ j = k ξ j A , Ξ j = k ξ j B ,
2 [ μ 1 μ 2 μ 3 ] · k = 1 K A ( k ) [ μ 1 / λ 2 μ 2 / λ 2 μ 3 / λ 2 ] + [ μ 1 μ 2 μ 3 ] · A ( λ 2 ) λ 2 [ μ 1 μ 2 μ 3 ] 1 ( λ 2 λ 1 ) 2 = 0.
k = 1 K A ( k ) [ μ 1 / λ 2 μ 2 / λ 2 μ 3 / λ 2 ] + A ( λ 2 ) λ 2 [ μ 1 μ 2 μ 3 ] = ± 1 λ 2 λ 1 [ ψ 1 ( λ 2 ) ψ 2 ( λ 2 ) ψ 3 ( λ 2 ) ] ,
[ μ 1 μ 2 μ 3 ] · A ( λ 2 ) λ 2 [ μ 1 μ 2 μ 3 ] ± 2 λ 2 λ 1 j = 1 3 μ j ψ j ( λ 2 ) 1 ( λ 2 λ 1 ) 2 = 0.
[ j = 1 3 μ j ψ j ( λ 2 ) 1 λ 2 λ 1 ] 2 = 0.
( d ρ d λ ) 2 = 0 , λ = λ 2 λ e .
j = 1 3 μ j λ 1 λ 2 ϕ j ( λ ) d λ = 0 , j = 1 3 μ j ϕ j ( λ 2 ) = 0 , j = 1 3 μ j g j ( λ 1 2 ) = 0.
| x ¯ ( λ 1 ) y ¯ ( λ 1 ) z ¯ ( λ 1 ) x ¯ ( λ 2 ) y ¯ ( λ 2 ) z ¯ ( λ 2 ) x ¯ ( λ 3 ) y ¯ ( λ 3 ) z ¯ ( λ 3 ) | < 0

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