Abstract

If the stop lies on the right of the anamorphic lenses, the entrance pupils in the tangential plane and the sagittal plane might not coincide with each other. In this condition, the anamorphic ratio will change according to the position of the object, which is not preferred in filming and machine vision. In view of this, a systematic approach for the paraxial design of anamorphic lenses with common entrance pupils is provided. Two basic types of anamorphic attachment are presented: basic anamorphic attachment and basic anamorphic zoom attachment. The anamorphic ratio can be changed by changing the interval distances between lens components if anamorphic zoom attachments are adopted. Formulas defining the interval distances between lens elements, the object position, the stop position, and the anamorphic ratio are derived. Illustrating figures and examples are also provided for each lens type.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2015 (1)

M. Soskind and Y. G. Soskind, “Propagation invariant laser beams for optical metrology applications,” Proc. SPIE 9526, 95261O (2015).
[Crossref]

2014 (1)

J. Zhang, X. Chen, J. Xi, and Z. Wu, “Paraxial analysis of double-sided telecentric zoom lenses with four components,” Opt. Eng. 53(11), 115103 (2014).
[Crossref]

2012 (1)

2010 (2)

2009 (3)

2008 (1)

T. Kryszczyński, “Development of the double-sided telecentric three-component zoom systems by means of matrix optics,” Proc. SPIE 7141, 71411Y (2008).
[Crossref]

2006 (1)

2005 (1)

A. Neil, “Compound zoom lenses,” Proc. SPIE 5865, 586507 (2005).
[Crossref]

1997 (1)

J. A. B. Francois Blais, “Calibration of an anamorphic laser based 3-D range sensor,” Proc. SPIE 3174, 113–122 (1997).
[Crossref]

1992 (1)

Bannert, C.

A. Dodoc, C. Bannert, V. Blahnik, and H. Sehr, “Anamorphic Objective,” US 8858099 B2 (2014).

Barrett, H. H.

H. L. Durko, H. H. Barrett, and L. R. Furenlid, “High-Resolution Anamorphic SPECT Imaging,” in Proceedings of IEEE Transactions on Nuclear Science (IEEE, 2014), pp. 1126–1135.

Blahnik, V.

A. Dodoc, C. Bannert, V. Blahnik, and H. Sehr, “Anamorphic Objective,” US 8858099 B2 (2014).

Blais, J. A. B. Francois

J. A. B. Francois Blais, “Calibration of an anamorphic laser based 3-D range sensor,” Proc. SPIE 3174, 113–122 (1997).
[Crossref]

Boni, R.

J. Qiao, P. A. Jaanimagi, R. Boni, J. Bromage, and E. Hill, “Beam-homogenization and space-charge-broadening calibration for accurately measuring high-intensity laser pulses using a high-speed streak camera,” in Conference on Lasers and Electro-Optics (CLEO, 1–2 (2012).

Bowron, J. W.

J. W. Bowron and R. P. Jonas, “Optical System Including An Anamorphic Lens,” US 7289272 B2 (2007).

Bromage, J.

J. Qiao, P. A. Jaanimagi, R. Boni, J. Bromage, and E. Hill, “Beam-homogenization and space-charge-broadening calibration for accurately measuring high-intensity laser pulses using a high-speed streak camera,” in Conference on Lasers and Electro-Optics (CLEO, 1–2 (2012).

Campbell, H. I.

Chen, X.

J. Zhang, X. Chen, J. Xi, and Z. Wu, “Paraxial analysis of double-sided telecentric zoom lenses with four components,” Opt. Eng. 53(11), 115103 (2014).
[Crossref]

Cottrell, D. M.

Davis, J. A.

Dodoc, A.

A. Dodoc, C. Bannert, V. Blahnik, and H. Sehr, “Anamorphic Objective,” US 8858099 B2 (2014).

Durko, H. L.

H. L. Durko, H. H. Barrett, and L. R. Furenlid, “High-Resolution Anamorphic SPECT Imaging,” in Proceedings of IEEE Transactions on Nuclear Science (IEEE, 2014), pp. 1126–1135.

Forkner, J. F.

J. F. Forkner, “Anamorphic Prism for Beam Shaping,” US 4623225 (1986).

Furenlid, L. R.

H. L. Durko, H. H. Barrett, and L. R. Furenlid, “High-Resolution Anamorphic SPECT Imaging,” in Proceedings of IEEE Transactions on Nuclear Science (IEEE, 2014), pp. 1126–1135.

Garrido, C. A.

Alfredo Valles Navarro, Andres Valles Navarro, and C. A. Garrido, “Anamorphic Lens,” US 9063321 B2 (2015).

Greenaway, A. H.

Hill, E.

J. Qiao, P. A. Jaanimagi, R. Boni, J. Bromage, and E. Hill, “Beam-homogenization and space-charge-broadening calibration for accurately measuring high-intensity laser pulses using a high-speed streak camera,” in Conference on Lasers and Electro-Optics (CLEO, 1–2 (2012).

Iemmi, J. C. Claudio

J. C. Claudio Iemmi, “Anamorphic zoom system based on liquid crystal displays,” J. Eur. Opt. Soc. 4, 09029 (2009).
[Crossref]

Jaanimagi, P. A.

J. Qiao, P. A. Jaanimagi, R. Boni, J. Bromage, and E. Hill, “Beam-homogenization and space-charge-broadening calibration for accurately measuring high-intensity laser pulses using a high-speed streak camera,” in Conference on Lasers and Electro-Optics (CLEO, 1–2 (2012).

Jonas, R. P.

J. W. Bowron and R. P. Jonas, “Optical System Including An Anamorphic Lens,” US 7289272 B2 (2007).

Kelly, S. L.

S. L. Kelly, “Anamorphic Optical System,” US 7995282 B2 (2011).

Kryszczynski, T.

T. Kryszczyński, “Development of the double-sided telecentric three-component zoom systems by means of matrix optics,” Proc. SPIE 7141, 71411Y (2008).
[Crossref]

Li, G.

G. Li, “Adaptive lens,” Prog. Opt. 55, 199–283 (2010).
[Crossref]

Miks, A.

Navarro, Alfredo Valles

Alfredo Valles Navarro, Andres Valles Navarro, and C. A. Garrido, “Anamorphic Lens,” US 9063321 B2 (2015).

Navarro, Andres Valles

Alfredo Valles Navarro, Andres Valles Navarro, and C. A. Garrido, “Anamorphic Lens,” US 9063321 B2 (2015).

Neil, A.

A. Neil, “Compound zoom lenses,” Proc. SPIE 5865, 586507 (2005).
[Crossref]

Neil, I. A.

I. A. Neil, “Anamorphic Imaging System,” US 7085066 B2 (2006).

I. A. Neil, “Anamorphic Objective Zoom Lens,” US 9239449 B2 (2016).

I. A. Neil, “Anamorphic Objective Lens,” US 9341827 B2 (2016).

Novak, J.

Plotkin, M.

D. K. Towner and M. Plotkin, “Anamorphic Prisms,” US 7751124 B2 (2010).

Qiao, J.

J. Qiao, P. A. Jaanimagi, R. Boni, J. Bromage, and E. Hill, “Beam-homogenization and space-charge-broadening calibration for accurately measuring high-intensity laser pulses using a high-speed streak camera,” in Conference on Lasers and Electro-Optics (CLEO, 1–2 (2012).

Runners, B. N.

B. N. Runners, Transform Optical Images (Art, 1965).

Sasian, J.

Schley-Seebold, H. M.

Sehr, H.

A. Dodoc, C. Bannert, V. Blahnik, and H. Sehr, “Anamorphic Objective,” US 8858099 B2 (2014).

Slyusarev, G.G.

G.G. Slyusarev, Aberration and Optical Design Theory, 2nd ed. (CRC Press, 1984).

Soskind, M.

M. Soskind and Y. G. Soskind, “Propagation invariant laser beams for optical metrology applications,” Proc. SPIE 9526, 95261O (2015).
[Crossref]

Soskind, Y. G.

M. Soskind and Y. G. Soskind, “Propagation invariant laser beams for optical metrology applications,” Proc. SPIE 9526, 95261O (2015).
[Crossref]

Towers, C. E.

Towers, D. P.

Towner, D. K.

D. K. Towner and M. Plotkin, “Anamorphic Prisms,” US 7751124 B2 (2010).

Wu, Z.

J. Zhang, X. Chen, J. Xi, and Z. Wu, “Paraxial analysis of double-sided telecentric zoom lenses with four components,” Opt. Eng. 53(11), 115103 (2014).
[Crossref]

Xi, J.

J. Zhang, X. Chen, J. Xi, and Z. Wu, “Paraxial analysis of double-sided telecentric zoom lenses with four components,” Opt. Eng. 53(11), 115103 (2014).
[Crossref]

Yuan, S.

Zhang, J.

J. Zhang, X. Chen, J. Xi, and Z. Wu, “Paraxial analysis of double-sided telecentric zoom lenses with four components,” Opt. Eng. 53(11), 115103 (2014).
[Crossref]

Zhang, S.

Appl. Opt. (5)

J. Eur. Opt. Soc. (1)

J. C. Claudio Iemmi, “Anamorphic zoom system based on liquid crystal displays,” J. Eur. Opt. Soc. 4, 09029 (2009).
[Crossref]

Opt. Eng. (1)

J. Zhang, X. Chen, J. Xi, and Z. Wu, “Paraxial analysis of double-sided telecentric zoom lenses with four components,” Opt. Eng. 53(11), 115103 (2014).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (4)

A. Neil, “Compound zoom lenses,” Proc. SPIE 5865, 586507 (2005).
[Crossref]

J. A. B. Francois Blais, “Calibration of an anamorphic laser based 3-D range sensor,” Proc. SPIE 3174, 113–122 (1997).
[Crossref]

M. Soskind and Y. G. Soskind, “Propagation invariant laser beams for optical metrology applications,” Proc. SPIE 9526, 95261O (2015).
[Crossref]

T. Kryszczyński, “Development of the double-sided telecentric three-component zoom systems by means of matrix optics,” Proc. SPIE 7141, 71411Y (2008).
[Crossref]

Prog. Opt. (1)

G. Li, “Adaptive lens,” Prog. Opt. 55, 199–283 (2010).
[Crossref]

Other (13)

B. N. Runners, Transform Optical Images (Art, 1965).

J. Qiao, P. A. Jaanimagi, R. Boni, J. Bromage, and E. Hill, “Beam-homogenization and space-charge-broadening calibration for accurately measuring high-intensity laser pulses using a high-speed streak camera,” in Conference on Lasers and Electro-Optics (CLEO, 1–2 (2012).

G.G. Slyusarev, Aberration and Optical Design Theory, 2nd ed. (CRC Press, 1984).

Alfredo Valles Navarro, Andres Valles Navarro, and C. A. Garrido, “Anamorphic Lens,” US 9063321 B2 (2015).

J. W. Bowron and R. P. Jonas, “Optical System Including An Anamorphic Lens,” US 7289272 B2 (2007).

I. A. Neil, “Anamorphic Imaging System,” US 7085066 B2 (2006).

A. Dodoc, C. Bannert, V. Blahnik, and H. Sehr, “Anamorphic Objective,” US 8858099 B2 (2014).

J. F. Forkner, “Anamorphic Prism for Beam Shaping,” US 4623225 (1986).

S. L. Kelly, “Anamorphic Optical System,” US 7995282 B2 (2011).

D. K. Towner and M. Plotkin, “Anamorphic Prisms,” US 7751124 B2 (2010).

I. A. Neil, “Anamorphic Objective Zoom Lens,” US 9239449 B2 (2016).

I. A. Neil, “Anamorphic Objective Lens,” US 9341827 B2 (2016).

H. L. Durko, H. H. Barrett, and L. R. Furenlid, “High-Resolution Anamorphic SPECT Imaging,” in Proceedings of IEEE Transactions on Nuclear Science (IEEE, 2014), pp. 1126–1135.

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Figures (19)

Fig. 1.
Fig. 1. Imaging of a paraxial lens with stop lying on the lens. The focal length is 30 mm, and the distance between the object and the lens is 60 mm in (a), 50 mm in (b), and 40 mm in (c).
Fig. 2.
Fig. 2. Imaging of a paraxial lens with stop lying on the left of the lens, and the distance is 10 mm. The focal length is 30 mm, and the distance between the object and the lens is 60 mm in (a), 50 mm in (b), and 40 mm in (c).
Fig. 3.
Fig. 3. Imaging of a paraxial lens with stop lying on the right of the lens, and the distance is 7.5 mm. The entrance pupil is on the right of the lens, and the distance is 10 mm. The focal length is 30 mm, and the distance between the object and the lens is 60 mm in (a), 50 mm in (b), and 40 mm in (c).
Fig. 4.
Fig. 4. The pinhole model of the imaging processes for Figs. 1, 2, and 3.
Fig. 5.
Fig. 5. Imaging of a paraxial anamorphic lens with the entrance pupils or OCIMVs coincident with each other along the optical axes. (a) refers to the imaging in X-O-Z plane, and (b) refers to the imaging in Y-O-Z plane.
Fig. 6.
Fig. 6. Imaging of a paraxial anamorphic lens with different entrance pupils or OCIMVs along the optical axes, and the distance between OCIMVx and OCIMVy is a. (a) refers to the imaging in X-O-Z plane, and (b) refers to the imaging in Y-O-Z plane.
Fig. 7.
Fig. 7. Configuration of a three-component lens system for calculation of matrix optics.
Fig. 8.
Fig. 8. Configuration of a rear anamorphic lens system in form of Y-Y type when the object is at infinity.
Fig. 9.
Fig. 9. Configuration of a rear anamorphic lens system in form of Y-Y type when the object is at a finite distance.
Fig. 10.
Fig. 10. Configuration of a rear anamorphic lens system in form of Y-Y-Y type when the object is at infinity.
Fig. 11.
Fig. 11. Configuration of a rear anamorphic lens system in form of Y-Y-Y type when the object is at a finite distance.
Fig. 12.
Fig. 12. Configuration of a rear anamorphic lens system in form of Y-X-Y type when the object is at infinity.
Fig. 13.
Fig. 13. Configuration of a rear anamorphic lens system in form of Y-X-Y type when the object is at a finite distance.
Fig. 14.
Fig. 14. Configuration of a rear anamorphic lens system in form of X-Y-Y type when the object is at infinity.
Fig. 15.
Fig. 15. Configuration of a rear anamorphic lens system in form of X-Y-Y type when the object is at a finite distance.
Fig. 16.
Fig. 16. Configuration of a rear anamorphic lens system in form of XY-XY type when the object is at infinity.
Fig. 17.
Fig. 17. Configuration of a rear anamorphic lens system in form of XY-XY type when the object is at a finite distance.
Fig. 18.
Fig. 18. Configuration of a rear anamorphic lens system in form of XY-Y type when the object is at infinity.
Fig. 19.
Fig. 19. Configuration of a rear anamorphic lens system in form of XY-Y type when the object is at a finite distance.

Equations (183)

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αI=hyhx=fyfx
αI=hyhx=fyfxlxlx+a
[1An1Bn1Cn1Dn][h1u1]=[hnun]
1An=[φ1,e1,,φn1,en1]
1Bn=[e1,,φn1,en1]
1Cn=[φ1,e1,,φn1,en1,φn]
1Dn=[e1,,φn1,en1,φn]
[1An1Bn1Cn1Dn][10]=[hnun]
F=11Cn
E=1An1Cn
[1An1Bn1Cn1Dn][e01]=[HnUn]
BFL=1Ane01Bn1Cne01Dn
m=11Dn1Cne0
αF=FyFx
BFLy=BFLx
e1=1φ1y+1φ2y=F1y+F2y
αF=φ2yφ1y
αI=IyIx
αI=φ2yφ1y=αF
Ly1=e1+es
(e1φ1yφ2yφ1yφ2y)es2+(e12φ1yφ2y2e1φ1y)ese12φ1y(e1φ1yφ2yφ1yφ2y)es+e12φ1yφ2ye1φ1y1=0
es=φ1y+φ2yφ2y(φ2yφ1y)
E3x=E3y
(φ2ye0φ1yφ2y)e12+(2e0φ2ye02φ1yφ2y)e1+e02(φ1y+φ2y)=0
e1=e0(φ1y+φ2y)1e0φ1yφ2yφ2y
(φ1y+φ2ye1φ1yφ2y)e02+(2e1φ2ye12φ1yφ2y)e0+e12φ2y=0
e0=e1φ2y1e1φ1yφ2yφ1yφ2y
αI=1e0φ1y+e0φ2y+e1φ2ye0e1φ1yφ2y1
Ly1=e1+es
es2es2+es1es+es0=0
es2=e1φ1yφ2yφ1yφ2y
es1=e12φ1yφ2y2e1φ1y
es0=e12φ1y
Ly1=e1+Lx2
es=12(1φ1y1φ2x)=e12
Fy=αFFx
BFLy=BFLx
1C4x=αF1C4y
1A4x=αF1A4y
Ly1=e1+e2+es
e1=αF(φ1y+φ2y)+φ3yαFφ1yφ2y
e2=αFφ1y+φ2y+φ3yφ2yφ3y
es=φ1yφ2yαF3+(φ1y2+φ2y2)αF2+(φ1yφ2y+2φ1yφ3y+2φ2yφ3y)αF+φ3y2(αF21)αFφ1yφ2yφ3y
e1=φ1y+φ2yφ3y(e2(φ1y+φ2y)1)φ3y(φ1ye2φ1yφ2y)+φ1yφ2y
l3=e0e1e2
e02e02+e01e0+e00=0
e02=φ1y+φ2y+φ3ye1φ1y(φ2y+φ3y)e2φ3y(φ1y+φ2y)+e1e2φ1yφ2yφ3y
e01=2(e1φ2y+e1φ3y+e2φ3y)e12φ1y(φ2y+φ3y)e2φ3y(e2+2e1)(φ1y+φ2y)+e1e2φ1yφ2yφ3y(e1+e2)
e00=e12(φ2y+φ3y)+e2φ3y(2e1+e2)e1e2φ2yφ3y(e1+e2)
E01e0+E00=0
E01=φ1y+φ2y+φ3ye1φ1y(φ2y+φ3y)e2φ3y(φ1y+φ2y)+e1e2φ1yφ2yφ3y
E00=e1φ2y+e1φ3y+e2φ3ye1e2φ2yφ3y1
aI=1A1e0+A0
A1=φ1y+φ2y+φ3ye1φ1y(φ2y+φ3y)e2φ3y(φ1y+φ2y)+e1e2φ1yφ2yφ3y
A0=e1φ2y+e1φ3y+e2φ3ye1e2φ2yφ3y1
Ly1=e1+e2+es
es2es2+es1es+es0=0
es2=φ1y+φ2y+φ3ye1φ1y(φ2y+φ3y)e2φ3y(φ1y+φ2y)+e1e2φ1yφ2yφ3y
es1=2(e1φ1y+e2φ1y+e2φ2y)e1φ1y(e1+2e2)(φ2y+φ3y)e22φ3y(φ1y+φ2y)+e1e2φ1yφ2yφ3y(e1+e2)
es0=φ1y(e1+e2)2e1e2φ1yφ2y(e1+e2)+e22φ2y
Es1es+Es0=0
Es1=φ1y+φ2y+φ3ye1φ1y(φ2y+φ3y)e2φ3y(φ1y+φ2y)+e1e2φ1yφ2yφ3y
Es0=e1φ1y+e2φ1y+e2φ2ye1e2φ1yφ2y1
Fy=αFFx
BFLy=BFLx
e1=φ1y2αF2+(2φ1yφ2x+φ2xφ3yφ1yφ3y)αFφ2x2αFφ1yφ2xφ3y
e2=αFφ1yφ2x+φ3yφ2xφ3y
Ly1=e1+Lx2
es=B3a3+B2a2+B1a1+B0A2a2+A1a1+A0
B3=φ1y3
B2=2φ1y2φ3y+3φ1y2φ2x+2φ1yφ2xφ3y+φ2xφ3y2
B1=2φ2x2φ3y3φ1yφ2x2+2φ1yφ2xφ3yφ1yφ3y2
B0=φ2x3
A2=φ1y2φ2xφ3y+φ2xφ3y3
A1=2φ2xφ3y(φ1yφ3yφ2xφ3y2φ1yφ2x)
A0=φ2x3φ3y
es=φ2xφ1yαFφ2xφ3y
lx2=ly3+e2
e02e02+e01e0+e00=0
e02=φ1yφ2x+φ3y+e1φ1yφ2xe2φ2xφ3ye1φ1yφ3ye2φ1yφ3y+e22φ1yφ2xφ3y+e1e2φ1yφ2xφ3y
e01=2(e1φ3ye1φ2x+e2φ3y)+e12φ1y(φ2xφ3y)e2φ3y(φ1y+φ2x)(e2+2e1)+e1e2φ1yφ2xφ3y(e1+e2)
e00=e12(φ3yφ2x)+e2φ3y(e2+2e1)e1e2φ2xφ3y(e2+e1)
E02e02+E01e0+E00=0
E02=φ1yφ2xφ3y(e1+e2)φ2x(φ1y+φ3y)
E01=(φ1y+φ2x+φ3y)e1φ1y(φ2x+φ3y)φ2xφ3y(e1+e2)φ3y(e1φ2x+e2φ1y)+e1φ1yφ2xφ3y(e1+e2)
E00=e1φ2xφ3y(e1+e2)+e1(φ2x+φ3y)+e1φ3y1
Ly1=Lx2+e1
es2es2+es1es+es0=0
es2=(φ1yφ2x+φ3y)+φ2x(e2φ3ye1φ1y)+(e1+e2)φ1yφ3y(e1φ2x1)
es1=2(e1φ1ye2φ2x+e2φ1y)(φ2x+φ3y)(e12φ1y+2e1e2φ1y)+e22φ3y(φ2xφ1y)+e1e2φ1yφ2xφ3y(e1+e2)
es0=e1φ1y(e1+2e2)+e22(φ1yφ2x)e1e2φ1yφ2x(e1+e2)
Es2es2+Es1es+Es0=0
Es2=φ1yφ2xφ3y(e1+e2)φ2x(φ1y+φ3y)
Es1=φ1y+φ2x+φ3ye2φ2x(φ1y+φ3y)+φ1y(e1+e2)(e2φ2xφ3yφ2xφ3y)
Es0=e2φ2x+φ1y(e1+e2)e2φ1yφ2x(e1+e2)1
αI=IyIx=φ2x(e0+e1)1e0(φ1y+φ3y)+φ3y(e1+e2)(1e0φ1y)1
Fy=αFFx
BFLy=BFLx
e1=φ2y2αF2+(φ2yφ3yφ1xφ3y2φ1xφ2y)αF+φ1x2φ1xφ2yφ3yαF
e2=(φ2y+φ3y)αFφ1xφ2yφ3yαF
e1=φ1xφ2yφ3ye22+(φ1xφ3y+φ2yφ3y)e2+(φ1xφ2yφ3y)φ1xφ2yφ3ye2φ1x(φ2y+φ3y)
Lx1=e1+Ly2
es=B3a3+B2a2+B1a1+B0A2a2+A1a1+A0
B3=φ2y3
B2=3φ1xφ2y2+2φ1xφ2yφ3y+φ1xφ3y22φ2y2φ3y
B1=2φ1xφ2yφ3y2φ1x2φ3y3φ1x2φ2yφ2yφ3y2
B0=φ1x3
A2=φ1xφ2y2φ3y+φ1xφ3y3
A1=2φ1xφ2yφ3y22φ1x2φ3y22φ1x2φ2yφ3y
A0=φ1x3φ3y
es=φ1xαFφ2yφ1xφ3y
lx1=ly3+e1+e2
e02e02+e01e0+e00=0
e02=φ1x+φ2y+φ3ye1φ1x(φ2y+φ3y)e2φ3y(φ1x+φ2y)+e2φ1xφ2yφ3y(e1+e2)
e01=2(e1φ2y+e1φ3y+e2φ3y)e12φ1x(φ2y+φ3y)e2φ3y(e2+2e1)(φ1x+φ2y)+e1e2φ1xφ2yφ3y(e1+e2)
e00=e12(φ2y+φ3y)+e2φ3y(e2+2e1)e1e2φ2yφ3y(e1+e2)
E02e02+E01e0+E00=0
E02=φ1x(φ2y+φ3y)+e2φ1xφ2yφ3y
E01=φ1x+φ2y+φ3ye1φ1x(φ2y+φ3y)e2φ3y(φ1x+φ2y)+e1e2φ1xφ2yφ3y
E00=e1e2φ2yφ3y+e1(φ2y+φ3y)+e2φ3y1
αI=IyIx=e0φ1x1(e0+e1)(φ2y+φ3ye2φ2yφ3y)+e2φ3y1
Lx1=e1+Ly2
es2es2+es1es+es0=0
es2=φ1xφ2yφ3ye1φ1x(φ2y+φ3y)e2φ3y(φ2yφ1x)+e1e2φ1xφ2yφ3y
es1=2(e1φ1x+e2φ1xe2φ2y)e1φ1x(e1+2e2)(φ2y+φ3y)e22φ3y(φ1xφ2y)+e1e2φ1xφ2yφ3y(e1+e2)
es0=φ1x(e12+e22)+e2(2e1φ1xe2φ2y)e1e2φ1xφ2y(e1+e2)
Es2es2+Es1es+Es0=0
Es2=e2φ1xφ2yφ3yφ1x(φ2y+φ3y)
Es1=φ1x+φ2y+φ3ye2φ2y(φ1x+φ3y)+φ1x(e1+e2)(e2φ2yφ3yφ2yφ3y)
Es0=φ1x(1e2φ2y)(e1+e2)+e2φ2y1
Fy=αFFx
BFLy=BFLx
φ1xαFφ1y=1αFe1
φ2xφ2y=αF1e1(1e1φ1x)
Lx1=Ly1
φ2xαFφ2y=(e1+es)(1αF)e1es
φ1x=αFφ1yαF1e1
φ2x=(e1+es)(1e1φ1y)+ese1es(1e1φ1y)
φ2y=αF(e1+es)(1e1φ1y)+esαFe1es(1e1φ1y)
φ1y2αF2+(φ1xφ2x2φ1xφ1yφ1xφ2yφ1yφ2x+φ1yφ2y)αF+φ1x2(φ1xφ1y)αF=0
φ1xφ1ye12(φ1x+φ1y)e1+1+φ1xφ1yφ2xφ2y=0
es=e1αFe1αF+e1φ2xαFe1φ2y1
lx2=ly2
e02e02+e01e0+e00=0
e02=φ1x+φ2xφ1yφ2y+e1(φ1x+φ1ye1φ1xφ1y)(φ2yφ2x)
e01=(φ2yφ2x)(e12φ1x+e12φ1y2e1)
e00=e12(φ2xφ2y)
E02e02+E01e0+E00=0
E02=(φ1x+φ2xe1φ1xφ2x)(φ1y+φ2ye1φ1yφ2y)
E01=(φ1x+φ1y+φ2x+φ2y)+e1(φ2x+φ2y)(φ1x+φ1y)+e1φ2xφ2y(2e1(φ1x+φ1y))
E00=e12φ2xφ2ye1(φ2x+φ2y)+1
Lx1=Ly1
es2es2+es1es+es0=0
es2=φ1x+φ1yφ2x+φ2y+e1(φ1xφ1y)(φ2x+φ2ye1φ2xφ2y)
es1=(φ1xφ1y)(e12(φ2x+φ2y)2e1)
es0=e12(φ1yφ1x)
Es2es2+Es1es+Es0=0
Es2=(φ1x+φ2x)(φ1y+φ2y)e1(φ1xφ2x(φ1y+φ2y)+φ1yφ2y(φ1x+φ2x)e1φ1xφ2xφ1yφ2y)
Es1=(φ1x+φ1y+φ2x+φ2y)+e1(φ1x(2φ1y+φ2x+φ2y)+φ1y(φ2x+φ2y)(1e1φ1x))
Es0=e12φ1xφ1ye1(φ1x+φ1y)+1
Fy=αFFx
BFLy=BFLx
φ1xφ1yφ2ye12φ2y(φ1x+φ1y)e1+φ1y+φ2yφ1xφ1ye11=0
αF=e1φ1x1e1φ1y1
Lx1=Ly1
es2es2+es1es+es0=0
es2=(φ1xφ1y)(e1φ2y1)+φ2y
es1=e1(φ1xφ1y)(e1φ2y2)
es0=e12(φ1yφ1x)
Es2es2+Es1es+Es0=0
Es2=e1φ1xφ1yφ2yφ1x(φ1y+φ2y)
Es1=φ1x+φ1y+φ2ye1φ1x(φ1y+φ2y)e1φ1y(φ1x+φ2y)+e12φ1xφ1yφ2y
Es0=e1(φ1x+φ1y)e12φ1xφ1y1
lx1=ly2+e1
e02e02+e01e0+e00=0
e02=φ1xφ1yφ2y+e1φ2y(φ1x+φ1y)e12φ1xφ1yφ2y
e01=e12φ2y(φ1x+φ1y)2e1φ2y
e00=e12φ2y
E02e02+E01e0+E00=0
E02=e1φ1xφ1yφ2yφ1x(φ1y+φ2y)
E01=(1e1φ2y)(φ1x+φ1y)+φ2y
E00=e1φ2y1
αI=IyIx=e0φ1x1e0(φ1y+φ2ye1φ1yφ2y)+e1φ2y1

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