Abstract

The single-scattering properties of randomly oriented triaxial ellipsoids with size parameters from the Rayleigh to geometric-optics regimes are investigated. A combination of the discrete dipole approximation (DDA) technique and an improved geometric optics method (IGOM) is applied to the computation of ellipsoidal particle scattering properties for a complete range of size parameters. Edge effect con tributions to the extinction and absorption efficiencies are included in the present IGOM simulation. It is found that the extinction efficiency, single-scattering albedo, and asymmetry factor computed from the DDA method for small size parameters smoothly transition to those computed from the IGOM for moderate-to-large size parameters. The phase matrix elements computed from these two methods are also quite similar when size parameters are larger than 30. Thus, the optical properties of ellipsoidal particles can be computed by combing the DDA and the IGOM for small-to-large size parameters. Furthermore, we also examine the applicability of the ellipsoid model to the simulation of the scatter ing properties of realistic aerosol particles by comparing the theoretical and experimental results for feldspar aerosols. It is shown that the ellipsoid model is better than the commonly used spheroid model for simulating dust particle optical properties, particularly, their polarization characteristics, realistically.

© 2008 Optical Society of America

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2008 (2)

G. Chen, P. Yang, and G. W. Kattawar, “Application of the pseudospectral time-domain method to the scattering of light by nonspherical particles,” J. Opt. Soc. Am. A 25, 785-790 (2008).
[CrossRef]

G. Hong, P. Yang, F. Z. Weng, and Q. H. Liu, “Microwave scattering properties of sand particles: application to the simulation of microwave radiances over sandstorms,” J. Quant. Spectrosc. Radiat. Transfer. 109, 684-702 (2008).
[CrossRef]

2007 (2)

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558-589 (2007).
[CrossRef]

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust particles,” J. Aerosol. Sci. 38, 995-1014 (2007).
[CrossRef]

2006 (2)

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. I. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Munoz, B. Veihelmann, W. J. van der Zande, J. F. Leon, M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, doi:10.1029/2005JD006619(2006).
[CrossRef]

D. L. Mitchell, A. J. Baran, W. P. Arnott, and C. Schmitt, “Testing and comparing the modified anomalous diffraction approximation,” J. Atmos. Sci. 63, 2948-2962 (2006).
[CrossRef]

2005 (1)

O. V. Kalashnikova, R. Kahn, I. N. Sokolik, and W.-H. Li, “Ability of multiangle remote sensing observations to identify and distiguish mineral dust types: optical models and retrievals of optically thick plumes,” J. Geophys. Res. 110, D18S14, doi:10.1029/2004JD004550 (2005).
[CrossRef]

2003 (3)

J. Zhao and Y. Hu, “Bridging technique for calculating the extinction efficiency of arbitrary shaped particles,” Appl. Opt. 42, 4931-4945 (2003).
[CrossRef]

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79, 775-824 (2003).
[CrossRef]

T. Nousiainen and K. Vermeulen, “Comparison of measured single-scattering matrix of feldspar particles with T-matrix simulations using spheroids,” J. Quant. Spectrosc. Radiat. Transfer 79, 1031-1042 (2003).
[CrossRef]

2002 (4)

T. Wriedt, “Using the T-matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shaped particles,” Part. Part. Syst. Charact. 19, 256-268 (2002).
[CrossRef]

O. V. Kalashnikova and I. N. Sokolik, “Importance of shapes and compositions of wind-blown dust particles for remote sensing at solar wavelengths,” Geophys. Res. Lett. 29, doi:10.1029/2002GL014947 (2002).
[CrossRef]

F. M. Kahnert, J. J. Stamnes, and K. Stamnes, “Can simple particle shapes be used to model scalar optical properties of an ensemble of wavelength-sized particles with complex shapes?,” J. Opt. Soc. Am. A 19, 521-531 (2002).
[CrossRef]

O. Dubovik, B. N. Holben, T. Lapyonok, A. Sinyuk, M. I. Mishchenko, P. Yang, and I. Slutsker, “Non-spherical aerosol retrieval method employing light scattering by spheroids,” Geophys. Res. Lett. 29, doi:10.1029/2001GL014506 (2002).
[CrossRef]

2001 (2)

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375-17401 (2001).
[CrossRef]

D. L. Mitchell, W. P. Arnott, C. Schmitt, A. J. Baran, S. Havemann, and Q. Fu, “Contributions of photon tunneling to extinction in laboratory grown hexagonal columns,” J. Quant. Spectrosc. Radiat. Transfer 70, 761-776 (2001).
[CrossRef]

2000 (1)

1999 (2)

Y. Liu, W. P. Arnott, and J. Hallertt, “Particle size distribution retrieval from multispectral optical depth: influences of particle nonsphericity and refractive index,” J. Geophys. Res. 104, 31753-31762 (1999).
[CrossRef]

A. J. Baran and S. Havemann, “Rapid computation of the optical properties of hexagonal columns using complex angular momentum theory,” J. Quant. Spectrosc. Radiat. Transfer 63, 499-519 (1999).
[CrossRef]

1998 (1)

Q. H. Liu, “The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1044-1055 (1998).
[CrossRef]

1997 (3)

R. Kahn, R. West, D. McDonald, B. Rheingans, and M. I. Mishchenko, “Sensitivity of multiangle remote sensing observations to aerosol sphericity,” J. Geophys. Res. 102, 16861-16870 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike troposheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 831-16-847 (1997).
[CrossRef]

C. Levoni, M. Cervino, R. Guzzi, and F. Torricella, “Atmospheric aerosol optical properties: a database of radiative characteristics for different components and classes,” Appl. Opt. 36, 8031-8041 (1997).
[CrossRef]

1996 (3)

1995 (1)

M. I. Mishchenko, A. A. Lacis, B. E. Carlson, and L. D. Travis, “Nonsphericity of dust-like tropospheric aerosols: implications for aerosol remote sensing and climate modeling,” Geophys. Res. Lett. 22, 1077-1080 (1995).
[CrossRef]

1994 (2)

1991 (2)

G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30, 2042-2048 (1991).
[CrossRef] [PubMed]

H. M. Nussenzveig and W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093-2112 (1991).
[CrossRef] [PubMed]

1988 (3)

J. B. Schneider and I. C. Peden, “Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method,” IEEE Trans. Antennas Propag. 36, 1317-1321 (1988).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848-872 (1988).
[CrossRef]

G. H. Goedecke and S. G. O'Brien, “Scattering by irregular inhomogeneous particles via the digitized Green's function algorithm,” Appl. Opt. 27, 2431-2438 (1988).
[CrossRef] [PubMed]

1987 (1)

S. I. Ghobrial and S. M. Sharief, “Microwave attenuation and cross polarization in dust storms,” IEEE. Trans. Antennas. Propagat. 35, 418-425 (1987).
[CrossRef]

1980 (1)

H. M. Nussenzveig and W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490-1494 (1980).
[CrossRef]

1979 (1)

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705-714 (1973).
[CrossRef]

1967 (1)

B. D. Sleeman, “The scalar scattering of a plane wave by an ellipsoid,” J. Instr. Math. Appl. 3, 4-15 (1967).
[CrossRef]

1957 (2)

D. S. Jones, “Approximate methods in high-frequency scattering,” Proc. R. Soc. A 239, 338-348 (1957).
[CrossRef]

D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. Lond. A 240, 206-213 (1957).
[CrossRef]

1956 (1)

S. I. Rubinow and T. T. Wu, “First correction to the geometric-optics scattering cross section from cylinders and spheres,” J. Appl. Phys. 27, 1032-1039 (1956).
[CrossRef]

1953 (1)

A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer) wavelength,” J. Appl. Phys. 24, 1134-1142 (1953).
[CrossRef]

1927 (1)

F. Möglich, “Beugungerscheinungen an körpern von ellipsoidischer gestalt,” Ann. Phys. 83, 609 (1927).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Arnott, W. P.

D. L. Mitchell, A. J. Baran, W. P. Arnott, and C. Schmitt, “Testing and comparing the modified anomalous diffraction approximation,” J. Atmos. Sci. 63, 2948-2962 (2006).
[CrossRef]

D. L. Mitchell, W. P. Arnott, C. Schmitt, A. J. Baran, S. Havemann, and Q. Fu, “Contributions of photon tunneling to extinction in laboratory grown hexagonal columns,” J. Quant. Spectrosc. Radiat. Transfer 70, 761-776 (2001).
[CrossRef]

Y. Liu, W. P. Arnott, and J. Hallertt, “Particle size distribution retrieval from multispectral optical depth: influences of particle nonsphericity and refractive index,” J. Geophys. Res. 104, 31753-31762 (1999).
[CrossRef]

Baran, A. J.

D. L. Mitchell, A. J. Baran, W. P. Arnott, and C. Schmitt, “Testing and comparing the modified anomalous diffraction approximation,” J. Atmos. Sci. 63, 2948-2962 (2006).
[CrossRef]

D. L. Mitchell, W. P. Arnott, C. Schmitt, A. J. Baran, S. Havemann, and Q. Fu, “Contributions of photon tunneling to extinction in laboratory grown hexagonal columns,” J. Quant. Spectrosc. Radiat. Transfer 70, 761-776 (2001).
[CrossRef]

A. J. Baran and S. Havemann, “Rapid computation of the optical properties of hexagonal columns using complex angular momentum theory,” J. Quant. Spectrosc. Radiat. Transfer 63, 499-519 (1999).
[CrossRef]

Carlson, B. E.

M. I. Mishchenko, A. A. Lacis, B. E. Carlson, and L. D. Travis, “Nonsphericity of dust-like tropospheric aerosols: implications for aerosol remote sensing and climate modeling,” Geophys. Res. Lett. 22, 1077-1080 (1995).
[CrossRef]

Cervino, M.

Chen, G.

de Haan, J. F.

H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375-17401 (2001).
[CrossRef]

Draine, B. T.

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848-872 (1988).
[CrossRef]

Draine, G. T.

G. T. Draine and P. J. Flatau, “User Guide to the Discrete Dipole Approximation Code DDSCAT6.1,” http://arxiv.org/abs/astro-ph/0409262v2 (2004).

Dubovik, O.

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust particles,” J. Aerosol. Sci. 38, 995-1014 (2007).
[CrossRef]

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. I. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Munoz, B. Veihelmann, W. J. van der Zande, J. F. Leon, M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, doi:10.1029/2005JD006619(2006).
[CrossRef]

O. Dubovik, B. N. Holben, T. Lapyonok, A. Sinyuk, M. I. Mishchenko, P. Yang, and I. Slutsker, “Non-spherical aerosol retrieval method employing light scattering by spheroids,” Geophys. Res. Lett. 29, doi:10.1029/2001GL014506 (2002).
[CrossRef]

Eck, T. F.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. I. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Munoz, B. Veihelmann, W. J. van der Zande, J. F. Leon, M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, doi:10.1029/2005JD006619(2006).
[CrossRef]

Evans, B. T. N.

Feng, Q.

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust particles,” J. Aerosol. Sci. 38, 995-1014 (2007).
[CrossRef]

Flatau, P. J.

G. T. Draine and P. J. Flatau, “User Guide to the Discrete Dipole Approximation Code DDSCAT6.1,” http://arxiv.org/abs/astro-ph/0409262v2 (2004).

Fournier, G. R.

Fu, Q.

D. L. Mitchell, W. P. Arnott, C. Schmitt, A. J. Baran, S. Havemann, and Q. Fu, “Contributions of photon tunneling to extinction in laboratory grown hexagonal columns,” J. Quant. Spectrosc. Radiat. Transfer 70, 761-776 (2001).
[CrossRef]

Gao, B.-C.

Ghobrial, S. I.

S. I. Ghobrial and S. M. Sharief, “Microwave attenuation and cross polarization in dust storms,” IEEE. Trans. Antennas. Propagat. 35, 418-425 (1987).
[CrossRef]

Goedecke, G. H.

Guzzi, R.

Hagness, S. C.

A. Taflove and S. C. Hagness, Advances in Computational Electrodynamics: the Finite-Difference-Time-Domain Method, 3rd ed. (Artech House, 2005).

Hallertt, J.

Y. Liu, W. P. Arnott, and J. Hallertt, “Particle size distribution retrieval from multispectral optical depth: influences of particle nonsphericity and refractive index,” J. Geophys. Res. 104, 31753-31762 (1999).
[CrossRef]

Havemann, S.

D. L. Mitchell, W. P. Arnott, C. Schmitt, A. J. Baran, S. Havemann, and Q. Fu, “Contributions of photon tunneling to extinction in laboratory grown hexagonal columns,” J. Quant. Spectrosc. Radiat. Transfer 70, 761-776 (2001).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Geometry of a non-axially-symmetric ellipsoid in o x y z and o x y z coordinate systems.

Fig. 2
Fig. 2

Schematic geometry for the ray-tracing calculations involving a triaxial ellipsoid.

Fig. 3
Fig. 3

Diffraction of an ellipsoid with an elliptic projection. Semi axes a ¯ and b ¯ and rotation angle ω are defined in Eqs. (6, 7, 8).

Fig. 4
Fig. 4

Integrated single-scattering properties (extinction efficiency, absorption efficiency, single-scattering albedo, and asymmetry factor) of randomly oriented ellipsoids. The wavelength is 0.66 μm , the complex refractive index is 1.53 + 0.008 i , and the aspect ratios are a b c = 0.53 0.71 1.00 and a b c = 0.30 0.70 1.00 for left and right panels, respectively.

Fig. 5
Fig. 5

Same as Fig. 4 except that the wavelength is 12 μm . The complex refractive index is 1.5502 + 0.0916 i . The aspect ratios are (a)  a b c = 0.53 0.71 1.00 and (b)  a b c = 0.30 0.70 1.00 .

Fig. 6
Fig. 6

Comparison of the phase matrix of an ellipsoid computed from the IGOM and DDA method at a size parameter of 30. The aspect ratios are (a)  0.53 0.71 1.0 and (b)  0.30 0.70 1.0 .

Fig. 7
Fig. 7

Same as Fig. 6 except the wavelength is 12 μm . The aspect ratios are (a)  0.53 0.71 1.0 and (b)  0.30 0.70 1.0 .

Fig. 8
Fig. 8

Comparison of the bulk phase function from laboratory measurement [5] with the present simulations based on spherical, spheroidal, and ellipsoidal models.

Equations (55)

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E i inc = α i 1 P i j i G i j P j ,
x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 ,
( x y z ) = ( sin β cos β 0 cos β sin β 0 0 0 1 ) ( 1 0 0 0 cos θ sin θ 0 sin θ cos θ ) ( x y z ) = ( sin β cos θ cos β sin θ cos β cos β cos θ sin β sin θ sin β 0 sin θ cos θ ) ( x y z ) ,
D z = E ± a b c D ( A x 2 + B y 2 + C x y ) ,
A = ( a 2 b 2 ) cos 2 θ cos 2 β + b 2 cos 2 θ + c 2 sin 2 θ , B = a 2 sin 2 β + b 2 cos 2 β , C = ( a 2 b 2 ) sin ( 2 β ) cos θ , D = c 2 sin 2 θ ( a 2 sin 2 β + b 2 cos 2 β ) + a 2 b 2 cos 2 θ , E = ( a 2 b 2 ) c 2 sin β cos β sin θ x + [ a 2 ( c 2 b 2 ) + c 2 ( b 2 a 2 ) cos 2 β ] sin θ cos θ y .
A x 2 + B y 2 + C x y = D ; z = E / D .
a ¯ = D ( 1 tan 2 ω ) A B tan 2 ω ,
b ¯ = D ( 1 tan 2 ω ) B A tan 2 ω ,
ω = 1 2 arctan C B A .
S = π a ¯ b ¯ = π D .
x = cos ω x ¯ + sin ω y ¯ ,
y = sin ω x ¯ + cos ω y ¯ ,
x ¯ 2 a ¯ 2 + y ¯ 2 b ¯ 2 = 1 .
x = a ¯ ξ cos ω cos ( 2 π χ ) + b ¯ ξ sin ω sin ( 2 π χ ) ,
y = a ¯ ξ sin ω cos ( 2 π χ ) + b ¯ ξ cos ω sin ( 2 π χ ) ,
z = [ E a b c D ( A x 2 + B y 2 + C x y ) ] / D ,
n ^ = ( x 1 / a 2 , y 1 / b 2 , z 1 / c 2 ) / x 1 2 / a 4 + y 1 2 / b 4 + z 1 2 / c 4 .
( x 2 , y 2 , z 2 ) = ( x 1 , y 1 , z 1 ) + d ( v 1 , v 2 , v 3 ) ,
d = 2 x 1 v 1 / a 2 + 2 y 1 v 2 / b 2 + 2 z 1 v 3 / c 2 v 1 2 / a 2 + v 2 2 / b 2 + v 3 2 / c 2 ,
A dif = k 2 2 π I s [ ( cos θ s + cos 2 θ s ) / 2 0 0 ( 1 + cos θ s ) / 2 ] ,
I s = s exp ( i k r ^ · ξ ) d 2 ξ ,
I s = π a ¯ b ¯ 2 J 1 ( k sin θ s a ¯ 2 cos 2 ϕ + b ¯ 2 sin 2 ϕ ) k sin θ s a ¯ 2 cos 2 ϕ + b ¯ 2 sin 2 ϕ ,
Q e , edge = c 0 k 2 / 3 S R ( s ) 1 / 3 sin 1 / 3 α ( s ) d s ,
Q e , edge = 2 c 0 ( k R ) 2 / 3 ,
2 c 0 = c TM + c TE = 0.1322 ,
Q e , edge = 0 1 0 π / 2 Q e , edge D 1 / 2 d μ d β 0 1 0 π / 2 D 1 / 2 d μ d β ,
Q e = Q e , IGOM + Q e , edge ,
R 0 r R 0 [ 1 + c + ( 2 k R 0 ) 2 / 3 ] ( above edge ) ,
R 0 [ 1 c ( 2 k R 0 ) 2 / 3 ] r R 0 ( below edge ) ,
Q a = Q a , GOM + Q a , a . e . + Q a , b . e . ,
A sph = 2 ( c + + c ) π R 0 4 / 3 ( 2 k ) 2 / 3 .
A ell = ( c + + c ) R 1 / 3 ( 2 k ) 2 / 3 d s ,
R 0 = [ R 1 / 3 d s 2 π ] 3 / 4 .
f c = 1 exp ( k m i r e ) 1 exp ( k m i R 0 ) ,
r e = 2 a b c a 2 + b a 2 / c 2 a 2 F ( Ω , q ) + b c 2 a 2 E ( Ω , q ) ,
q = c b b 2 a 2 c 2 a 2 , Ω = arcsin [ 1 a 2 / c 2 ] .
F [ Ω , q ] = 0 Ω d t 1 q 2 sin 2 t , E [ Ω , q ] = 0 Ω 1 q 2 sin 2 t d t .
Q a ( a , b , c , k , m ) = Q a , IGOM ( a , b , c , k , m ) + 0 1 0 π / 2 ( Q a , a . e . [ R 0 ( μ , β ) , k , m ] + Q a , b . e . [ R 0 ( μ , β ) , k , m ] ) f c R 0 2 d μ d β 0 1 0 π / 2 D 1 / 2 d μ d β .
n ( r ) = N tot 2 π ln ( 10 ) log ( σ ) r exp { [ log ( r ) log ( R ) ] 2 2 [ log ( σ ) ] 2 } ,
P i j = k = 1 4 W k P i j k ( r ) σ sca k ( r ) n ( r ) d r k = 1 4 W k σ sca k ( r ) n ( r ) d r ,
x ( χ ) = a ¯ cos ω cos χ + b ¯ sin ω sin χ ,
y ( χ ) = a ¯ sin ω cos χ + b ¯ cos ω sin χ ,
z ( χ ) = E ( χ ) / D .
d s = a ¯ 2 sin 2 χ + b ¯ 2 cos 2 χ d χ .
r ˜ ( y ˜ ) = [ x ˜ ( y ˜ ) , y ˜ , z ˜ ( y ˜ ) ] .
R 1 = | d r ˜ d y ˜ | 3 / 2 / | d r ˜ d y ˜ × d 2 r ˜ d y ˜ 2 | ,
d r ˜ ( y ˜ ) / d y ˜ = ( d x ˜ ( y ˜ ) / d y ˜ , 1 , d z ˜ ( y ˜ ) / d y ˜ ) ,
d 2 r ˜ ( y ˜ ) / d y ˜ 2 = ( d 2 x ˜ ( y ˜ ) / d y ˜ 2 , 0 , d 2 z ˜ ( y ˜ ) / d y ˜ 2 ) ,
d x ˜ ( y ˜ ) / d y ˜ = [ x ˜ C ˜ + 2 B ˜ y ˜ ] / [ y ˜ C ˜ + 2 A ˜ x ˜ ] ,
d z ˜ ( y ˜ ) / d y ˜ = [ ( a 2 b 2 ) c 2 sin β ¯ cos β ¯ sin θ ¯ d x ˜ ( y ˜ ) / d y ˜
+ [ a 2 ( c 2 b 2 ) + c 2 ( b 2 a 2 ) cos 2 β ¯ ] sin θ ¯ cos θ ¯ ] / D ˜ .
τ ^ = γ ^ × α ^ ,
α ^ = d r ˜ ( y ˜ ) / d y ˜ / | d r ˜ ( y ˜ ) / d y ˜ | ,
γ ^ = [ d r ˜ ( y ˜ ) / d y ˜ × d 2 r ˜ ( y ˜ ) / d y ˜ 2 ] / | d r ˜ ( y ˜ ) / d y ˜ × d 2 r ˜ ( y ˜ ) / d y ˜ 2 | .
R = R 1 / | τ ^ · n ^ | ,

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