Abstract

Anomalous diffraction by an arbitrarily oriented ellipsoid with three different axes is derived. From the resulting expression the relationship between the shape of the ellipsoid and the intensity pattern is immediately evident: The axial ratio of the elliptical isointensity curve equals the axial ratio of the elliptical projected area of the ellipsoid. A comparison of anomalous diffraction with calculations performed with the T-matrix method reveals that the anomalous diffraction approximation is highly accurate for single ellipsoidal red blood cells. Application of the expression for anomalous diffraction by ellipsoids to a population of red blood cells shows that, even in a red-cell suspension as examined in an ektacytometer, the axial ratio of the isointensity curves is equal to the mean axial ratio of the red blood cells in the population. In ektacytometry this relationship between cell shape and intensity pattern is commonly assumed to hold true without reference to the light-scattering properties of the cells. The results presented here show that this assumption is valid, and we offer a profound theoretical basis for it by considering in detail the light scattering by the red blood cells.

© 1994 Optical Society of America

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References

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  1. M. Bessis, N. Mohandas, “A diffractometric method for the measurement of cellular deformability,” Blood Cells 1, 307–313 (1975).
  2. S. R. Keller, R. Skalak, “Motion of a tank treading ellipsoidal particle in a shear flow,” J. Fluid Mech. 120, 27–47 (1982).
    [Crossref]
  3. M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
    [Crossref] [PubMed]
  4. J. H. F. I. van Breugel, “Hemorheology and its role in blood platelet adhesion under flow conditions,” Ph.D. dissertation (State University of Utrecht, Utrecht, The Netherlands, 1989).
  5. N. Mohandas, M. R. Clark, M. S. Jacobs, S. B. Shohet, “Analysis of factors regulating erythrocyte deformability,” J. Clin. Invest. 66, 563–573 (1980).
    [Crossref] [PubMed]
  6. G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof, R. M. Heethaar, “Light scattering by red blood cells in ektacytometry: Fraunhofer versus anomalous diffraction,” Appl. Opt. 32, 2266–2272 (1993).
    [Crossref] [PubMed]
  7. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 11, p. 183.
  8. P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
    [Crossref] [PubMed]
  9. P. W. Barber, S. C. Hill, Light Scattering by Small Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3, pp. 79–185.
    [Crossref]
  10. R. Tran-Son-Tay, S. P. Sutera, P. R. Rao, “Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion,” Biophys. J. 46, 65–72 (1984).
    [Crossref] [PubMed]
  11. T. M. Fischer, “On the energy dissipation in a tank-treading human red blood cell,” Biophys. J. 32, 863–868 (1980).
    [Crossref] [PubMed]
  12. P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
    [Crossref]
  13. P. Chylek, J. D. Klett, “Extinction cross sections of nonspherical particles in the anomalous diffraction approximation,” J. Opt. Soc. Am. A 8, 274–281 (1991).
    [Crossref]
  14. R. Tran-Son-Tay, S. P. Sutera, G. I. Zahalak, P. R. Rao, “Membrane stress and internal pressure in a red blood cell freely suspended in a shear flow,” Biophys. J. 51, 915–924 (1987).
    [Crossref] [PubMed]
  15. O. Linderkamp, H. J. Meiselman, “Geometric, osmotic, and membrane mechanical properties of density-separated human red cells,” Blood 59, 1121–1127 (1982).
    [PubMed]
  16. G. B. Nash, H. J. Meiselman, “Red cell and ghost viscoelasticity,” Biophys. J. 43, 63–73 (1983).
    [Crossref] [PubMed]
  17. J. Plasek, T. Marik, “Determination of undeformable erythrocytes in blood samples using laser light scattering,” Appl. Opt. 21, 4335–4338 (1982).
    [Crossref] [PubMed]
  18. F. Storzicky, V. Blazek, J. Muzik, “An improved diffractometric method for measurement of cellular deformability,” J. Biomech. 13, 417–421 (1980).
    [Crossref]
  19. M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC laser diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and Myrenne aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).
  20. G. I. Zahalak, S. P. Sutera, “Fraunhofer diffraction pattern of an oriented monodisperse system of prolate ellipsoids,” J. Colloid Interface Sci. 82, 423–429 (1981).
    [Crossref]
  21. G. R. Fournier, B. T. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30, 2042–2048 (1991).
    [Crossref] [PubMed]

1993 (1)

1991 (2)

1988 (1)

M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC laser diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and Myrenne aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).

1987 (2)

M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
[Crossref] [PubMed]

R. Tran-Son-Tay, S. P. Sutera, G. I. Zahalak, P. R. Rao, “Membrane stress and internal pressure in a red blood cell freely suspended in a shear flow,” Biophys. J. 51, 915–924 (1987).
[Crossref] [PubMed]

1984 (1)

R. Tran-Son-Tay, S. P. Sutera, P. R. Rao, “Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion,” Biophys. J. 46, 65–72 (1984).
[Crossref] [PubMed]

1983 (1)

G. B. Nash, H. J. Meiselman, “Red cell and ghost viscoelasticity,” Biophys. J. 43, 63–73 (1983).
[Crossref] [PubMed]

1982 (3)

J. Plasek, T. Marik, “Determination of undeformable erythrocytes in blood samples using laser light scattering,” Appl. Opt. 21, 4335–4338 (1982).
[Crossref] [PubMed]

O. Linderkamp, H. J. Meiselman, “Geometric, osmotic, and membrane mechanical properties of density-separated human red cells,” Blood 59, 1121–1127 (1982).
[PubMed]

S. R. Keller, R. Skalak, “Motion of a tank treading ellipsoidal particle in a shear flow,” J. Fluid Mech. 120, 27–47 (1982).
[Crossref]

1981 (1)

G. I. Zahalak, S. P. Sutera, “Fraunhofer diffraction pattern of an oriented monodisperse system of prolate ellipsoids,” J. Colloid Interface Sci. 82, 423–429 (1981).
[Crossref]

1980 (3)

F. Storzicky, V. Blazek, J. Muzik, “An improved diffractometric method for measurement of cellular deformability,” J. Biomech. 13, 417–421 (1980).
[Crossref]

N. Mohandas, M. R. Clark, M. S. Jacobs, S. B. Shohet, “Analysis of factors regulating erythrocyte deformability,” J. Clin. Invest. 66, 563–573 (1980).
[Crossref] [PubMed]

T. M. Fischer, “On the energy dissipation in a tank-treading human red blood cell,” Biophys. J. 32, 863–868 (1980).
[Crossref] [PubMed]

1975 (3)

P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
[Crossref]

M. Bessis, N. Mohandas, “A diffractometric method for the measurement of cellular deformability,” Blood Cells 1, 307–313 (1975).

P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[Crossref] [PubMed]

Barber, P.

Barber, P. W.

P. W. Barber, S. C. Hill, Light Scattering by Small Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3, pp. 79–185.
[Crossref]

Bauersachs, R. M.

M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC laser diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and Myrenne aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).

Bessis, M.

M. Bessis, N. Mohandas, “A diffractometric method for the measurement of cellular deformability,” Blood Cells 1, 307–313 (1975).

Blazek, V.

F. Storzicky, V. Blazek, J. Muzik, “An improved diffractometric method for measurement of cellular deformability,” J. Biomech. 13, 417–421 (1980).
[Crossref]

Breederveld, D.

M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
[Crossref] [PubMed]

Chylek, P.

Clark, M. R.

N. Mohandas, M. R. Clark, M. S. Jacobs, S. B. Shohet, “Analysis of factors regulating erythrocyte deformability,” J. Clin. Invest. 66, 563–573 (1980).
[Crossref] [PubMed]

Evans, B. T.

Fischer, T. M.

T. M. Fischer, “On the energy dissipation in a tank-treading human red blood cell,” Biophys. J. 32, 863–868 (1980).
[Crossref] [PubMed]

Fournier, G. R.

Goedhart, P.

M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
[Crossref] [PubMed]

Hardeman, M. R.

M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC laser diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and Myrenne aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).

M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
[Crossref] [PubMed]

Heethaar, R. M.

Hill, S. C.

P. W. Barber, S. C. Hill, Light Scattering by Small Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3, pp. 79–185.
[Crossref]

Hoekstra, A. G.

Jacobs, M. S.

N. Mohandas, M. R. Clark, M. S. Jacobs, S. B. Shohet, “Analysis of factors regulating erythrocyte deformability,” J. Clin. Invest. 66, 563–573 (1980).
[Crossref] [PubMed]

Keller, S. R.

S. R. Keller, R. Skalak, “Motion of a tank treading ellipsoidal particle in a shear flow,” J. Fluid Mech. 120, 27–47 (1982).
[Crossref]

Klett, J. D.

Latimer, P.

P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
[Crossref]

Linderkamp, O.

O. Linderkamp, H. J. Meiselman, “Geometric, osmotic, and membrane mechanical properties of density-separated human red cells,” Blood 59, 1121–1127 (1982).
[PubMed]

Marik, T.

Meiselman, H. J.

M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC laser diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and Myrenne aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).

G. B. Nash, H. J. Meiselman, “Red cell and ghost viscoelasticity,” Biophys. J. 43, 63–73 (1983).
[Crossref] [PubMed]

O. Linderkamp, H. J. Meiselman, “Geometric, osmotic, and membrane mechanical properties of density-separated human red cells,” Blood 59, 1121–1127 (1982).
[PubMed]

Mohandas, N.

N. Mohandas, M. R. Clark, M. S. Jacobs, S. B. Shohet, “Analysis of factors regulating erythrocyte deformability,” J. Clin. Invest. 66, 563–573 (1980).
[Crossref] [PubMed]

M. Bessis, N. Mohandas, “A diffractometric method for the measurement of cellular deformability,” Blood Cells 1, 307–313 (1975).

Muzik, J.

F. Storzicky, V. Blazek, J. Muzik, “An improved diffractometric method for measurement of cellular deformability,” J. Biomech. 13, 417–421 (1980).
[Crossref]

Nash, G. B.

G. B. Nash, H. J. Meiselman, “Red cell and ghost viscoelasticity,” Biophys. J. 43, 63–73 (1983).
[Crossref] [PubMed]

Nijhof, E. J.

Plasek, J.

Rao, P. R.

R. Tran-Son-Tay, S. P. Sutera, G. I. Zahalak, P. R. Rao, “Membrane stress and internal pressure in a red blood cell freely suspended in a shear flow,” Biophys. J. 51, 915–924 (1987).
[Crossref] [PubMed]

R. Tran-Son-Tay, S. P. Sutera, P. R. Rao, “Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion,” Biophys. J. 46, 65–72 (1984).
[Crossref] [PubMed]

Shohet, S. B.

N. Mohandas, M. R. Clark, M. S. Jacobs, S. B. Shohet, “Analysis of factors regulating erythrocyte deformability,” J. Clin. Invest. 66, 563–573 (1980).
[Crossref] [PubMed]

Skalak, R.

S. R. Keller, R. Skalak, “Motion of a tank treading ellipsoidal particle in a shear flow,” J. Fluid Mech. 120, 27–47 (1982).
[Crossref]

Storzicky, F.

F. Storzicky, V. Blazek, J. Muzik, “An improved diffractometric method for measurement of cellular deformability,” J. Biomech. 13, 417–421 (1980).
[Crossref]

Streekstra, G. J.

Sutera, S. P.

R. Tran-Son-Tay, S. P. Sutera, G. I. Zahalak, P. R. Rao, “Membrane stress and internal pressure in a red blood cell freely suspended in a shear flow,” Biophys. J. 51, 915–924 (1987).
[Crossref] [PubMed]

R. Tran-Son-Tay, S. P. Sutera, P. R. Rao, “Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion,” Biophys. J. 46, 65–72 (1984).
[Crossref] [PubMed]

G. I. Zahalak, S. P. Sutera, “Fraunhofer diffraction pattern of an oriented monodisperse system of prolate ellipsoids,” J. Colloid Interface Sci. 82, 423–429 (1981).
[Crossref]

Tran-Son-Tay, R.

R. Tran-Son-Tay, S. P. Sutera, G. I. Zahalak, P. R. Rao, “Membrane stress and internal pressure in a red blood cell freely suspended in a shear flow,” Biophys. J. 51, 915–924 (1987).
[Crossref] [PubMed]

R. Tran-Son-Tay, S. P. Sutera, P. R. Rao, “Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion,” Biophys. J. 46, 65–72 (1984).
[Crossref] [PubMed]

van Breugel, J. H. F. I.

J. H. F. I. van Breugel, “Hemorheology and its role in blood platelet adhesion under flow conditions,” Ph.D. dissertation (State University of Utrecht, Utrecht, The Netherlands, 1989).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 11, p. 183.

Yeh, C.

Zahalak, G. I.

R. Tran-Son-Tay, S. P. Sutera, G. I. Zahalak, P. R. Rao, “Membrane stress and internal pressure in a red blood cell freely suspended in a shear flow,” Biophys. J. 51, 915–924 (1987).
[Crossref] [PubMed]

G. I. Zahalak, S. P. Sutera, “Fraunhofer diffraction pattern of an oriented monodisperse system of prolate ellipsoids,” J. Colloid Interface Sci. 82, 423–429 (1981).
[Crossref]

Appl. Opt. (4)

Biophys. J. (4)

R. Tran-Son-Tay, S. P. Sutera, P. R. Rao, “Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion,” Biophys. J. 46, 65–72 (1984).
[Crossref] [PubMed]

T. M. Fischer, “On the energy dissipation in a tank-treading human red blood cell,” Biophys. J. 32, 863–868 (1980).
[Crossref] [PubMed]

R. Tran-Son-Tay, S. P. Sutera, G. I. Zahalak, P. R. Rao, “Membrane stress and internal pressure in a red blood cell freely suspended in a shear flow,” Biophys. J. 51, 915–924 (1987).
[Crossref] [PubMed]

G. B. Nash, H. J. Meiselman, “Red cell and ghost viscoelasticity,” Biophys. J. 43, 63–73 (1983).
[Crossref] [PubMed]

Blood (1)

O. Linderkamp, H. J. Meiselman, “Geometric, osmotic, and membrane mechanical properties of density-separated human red cells,” Blood 59, 1121–1127 (1982).
[PubMed]

Blood Cells (1)

M. Bessis, N. Mohandas, “A diffractometric method for the measurement of cellular deformability,” Blood Cells 1, 307–313 (1975).

Clin. Chim. Acta (1)

M. R. Hardeman, P. Goedhart, D. Breederveld, “Laser diffraction ellipsometry of erythrocytes under controlled shear stress using a rotational viscosimeter,” Clin. Chim. Acta 165, 227–234 (1987).
[Crossref] [PubMed]

Clin. Hemorheol. (1)

M. R. Hardeman, R. M. Bauersachs, H. J. Meiselman, “RBC laser diffractometry and RBC aggregometry with a rotational viscometer: comparison with rheoscope and Myrenne aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).

J. Biomech. (1)

F. Storzicky, V. Blazek, J. Muzik, “An improved diffractometric method for measurement of cellular deformability,” J. Biomech. 13, 417–421 (1980).
[Crossref]

J. Clin. Invest. (1)

N. Mohandas, M. R. Clark, M. S. Jacobs, S. B. Shohet, “Analysis of factors regulating erythrocyte deformability,” J. Clin. Invest. 66, 563–573 (1980).
[Crossref] [PubMed]

J. Colloid Interface Sci. (2)

P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
[Crossref]

G. I. Zahalak, S. P. Sutera, “Fraunhofer diffraction pattern of an oriented monodisperse system of prolate ellipsoids,” J. Colloid Interface Sci. 82, 423–429 (1981).
[Crossref]

J. Fluid Mech. (1)

S. R. Keller, R. Skalak, “Motion of a tank treading ellipsoidal particle in a shear flow,” J. Fluid Mech. 120, 27–47 (1982).
[Crossref]

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

Other (3)

J. H. F. I. van Breugel, “Hemorheology and its role in blood platelet adhesion under flow conditions,” Ph.D. dissertation (State University of Utrecht, Utrecht, The Netherlands, 1989).

P. W. Barber, S. C. Hill, Light Scattering by Small Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3, pp. 79–185.
[Crossref]

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 11, p. 183.

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Figures (7)

Fig. 1
Fig. 1

Measuring configuration of the ektacytometer.

Fig. 2
Fig. 2

Ellipsoidal particle with semiaxes a, b, and c illuminated by a plane wave of intensity I 0. The longest axis a is oriented with an angle ψ relative to the flow. The b axis is oriented along the y axis of the coordinate system, perpendicular to the x and z axes. Traversing the ellipsoid, the light is phase shifted with magnitude k |m − 1|l(∊, η) compared with the light traveling along the particle. The border of the projected area is defined by the coordinates (x b , y b ).

Fig. 3
Fig. 3

Anomalous diffraction (curves) and T-matrix calculations (markers) of oblate spheroids oriented with the symmetry axis parallel to the incident light (left) and a prolate spheroid (right) oriented along the x axis of the Cartesian coordinate system (○, parallel polarized incident light; +, perpendicular polarized incident light). Right, the long axis and the short axis denote the angular scattering with θ in the yz and xz planes, respectively.

Fig. 4
Fig. 4

Anomalous diffraction (curves) and T-matrix calculations (dots) for oblate spheroids (q = 1) with orientation angle ψ = 45° and parallel polarized incident light (top, α = 20, c/a = 0.4; bottom, α = 31.7, c/a = 0.6).

Fig. 5
Fig. 5

Angular scattering of a cell population (solid curve, V mean = 95 μm3; standard deviation, 15 μm3) and a single red blood cell (dashed curve, V = 95 μm3) calculated by use of anomalous diffraction. The long axis and the short axis denote the angular scattering with θ in the yz and xz plane, respectively. The horizontal dashed line is drawn at intensity I(θ) = I(0)/10.

Fig. 6
Fig. 6

Orientation of the particle frame [x′, y′, z′] relative to the laboratory frame [x, y, z]. The orientation of the particle is defined by a rotation ψ in the xz plane followed by a rotation ξ around the x′ axis.

Fig. 7
Fig. 7

Projected area of an ellipsoidal particle with semiaxes a, b, and c and orientation angles ψ, ξ, and Φ. The projected area is an ellipse with semiaxes a′ and b′ along the x″ and y″ axes. The x″ and y″ axes are rotated by an angle δ + Φ relative to the laboratory frame.

Equations (25)

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I A = I 0 ( 1 / k 2 r 2 ) S ( ν ) 2 ,
S ( ν ) = α 2 0 π / 2 [ 1 - exp ( - i ϕ max sin τ ) ] J 0 ( α ν cos τ ) × sin τ cos τ d τ , r = ( x 2 + y 2 + z 2 ) 1 / 2 , ν = 1 r [ ( x 2 / q ) + q y 2 ] 1 / 2 , α = k ( a b ) 1 / 2 = ( 2 π n med / λ 0 ) ( a b ) 1 / 2 , q = a / b , ϕ max = 2 k c m - 1 .
a = f a , f = [ cos 2 ψ + ( c a ) 2 sin 2 ψ ] 1 / 2 .
ϕ = k m - 1 l ( , η ) ,
l ( , η ) = 2 c [ 1 - ( a ) 2 - ( η b ) 2 ] 1 / 2 ,
A = 2 π ( 3 V 4 π q ) 2 / 3 [ 1 + q 2 ( q 2 - 1 ) 1 / 2 arctan ( q 2 - 1 ) 1 / 2 ] , V = 4 3 π a 3 q 2 , A = κ V ,
H b ( V ) = H b mean ( V V mean ) ,
m ( V ) = g p [ H b mean ( V V mean ) ] ,
I mean ( θ ) = V cells w t ( V ) I ( V , θ ) d V ,
w t ( V ) = 1 σ d ( 2 π ) 1 / 2 exp [ - 1 2 ( V - V mean σ d ) 2 ] ,
I A = I 0 S ( β , γ ) k 2 r 2 ,
S ( β , γ ) = k 2 2 π A s c { 1 - exp [ - i ϕ ( , η ) ] } × exp [ i k ( β + η γ ) ] d d η , r = ( x 2 + y 2 + z 2 ) 1 / 2 , β = x / r ,             γ = y / r .
ϕ ( , η ) = k m - 1 l ( , η ) .
( x y z ) = ( cos ψ 0 sin ψ sin ψ sin ξ cos ξ - cos ψ sin ξ - sin ψ cos ξ sin ξ cos ψ cos ξ ) ( x y z ) .
( x / a ) 2 + ( y / b ) 2 + ( z / c ) 2 = 1.
A z 2 + B z + C = 0 ,
A = [ sin 2 ψ a 2 + cos 2 ψ ( sin 2 ξ b 2 + cos 2 ξ c 2 ) ] , B = b 1 x + b 2 y , b 1 = 2 cos ψ sin ψ [ 1 a 2 - ( sin 2 ξ b 2 + cos 2 ξ c 2 ) ] , b 2 = 2 cos ψ cos ξ sin ξ ( 1 c 2 - 1 b 2 ) , C = c 1 x 2 + c 2 y 2 + c 3 x y - 1 , c 1 = [ cos 2 ψ a 2 + sin 2 ψ ( sin 2 ξ b 2 + cos 2 ξ c 2 ) ] , c 2 = ( cos 2 ξ b 2 + sin 2 ξ c 2 ) , c 3 = - 2 sin ψ cos ξ sin ξ ( 1 c 2 - 1 b 2 ) .
B 2 - 4 A C = 0.
D x b 2 + E y b 2 + F x b y b = 1 ,
D = ( c 1 - b 1 2 4 A ) ,             E = ( c 2 - b 2 2 4 A ) ,             F = ( c 3 - b 1 b 2 2 A ) .
δ = 1 2 arctan ( F D - E ) , a = ( 1 - tan 2 δ D - E tan 2 δ ) 1 / 2 , b = ( 1 - tan 2 δ E - D tan 2 δ ) 1 / 2 .
l = ( B 2 - 4 A C ) 1 / 2 A = 2 A ( 1 - D x 2 - E y 2 - F x y ) 1 / 2 .
l ( , η ) = 2 c [ 1 - ( a ) 2 - ( η b ) 2 ] 1 / 2 ,
( x y ) = [ cos ( δ + Φ ) sin ( δ + Φ ) - sin ( δ + Φ ) cos ( δ + Φ ) ] ( x y ) .
a = f a , c = c / f , f = [ cos 2 ψ + ( c a ) 2 sin 2 ψ ] 1 / 2 .

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