Abstract

To the best of our knowledge, the generalized ray matrix, an augmented 5×5 ray matrix for a spherical mirror reflection with all the possible perturbation sources including three kinds of displacements and its detailed deducing process have been proposed in this paper for the first time. Square ring resonators and monolithic triaxial ring resonators have been chosen as examples to show its application, and some novel results of the optical-axis perturbation have been obtained. A novel method to eliminate the diaphragm mismatching error and the gain capillary mismatching error in monolithic triaxial ring resonators more effectively has also been proposed. Both those results and method have been confirmed by related experiments and the experimental results have been described with diagrammatic representation. This generalized ray matrix is valuable for ray analysis of various kinds of resonators. These results are important for the cavity design, cavity improvement and alignment of high accuracy and super high accuracy ring laser gyroscopes.

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  1. W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–86 (1985).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2010 (1)

2008 (2)

J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008).
[CrossRef] [PubMed]

J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun. 281(5), 1204–1210 (2008).
[CrossRef]

2007 (1)

2005 (1)

J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76(12), 125106 (2005).
[CrossRef]

2003 (1)

J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74(3), 1362–1365 (2003).
[CrossRef]

2002 (1)

2000 (1)

A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1389–1399 (2000).
[CrossRef]

1994 (1)

1990 (1)

1988 (1)

M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988).
[CrossRef]

1987 (2)

R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron. 23(4), 438–445 (1987).
[CrossRef]

H. R. Bilger and G. E. Stedman, “Stability of planar ring lasers with mirror misalignment,” Appl. Opt. 26(17), 3710–3716 (1987).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–86 (1985).
[CrossRef]

1984 (2)

A. L. Levkit and V. M. Ovchinnikov, “Stability of a ring resonator with a nonplane axial contour,” J. Appl. Spectrosc. (USSR) 40(6), 657–660 (1984).
[CrossRef]

D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. 23(17), 2944–2949 (1984).
[CrossRef] [PubMed]

1983 (1)

I. W. Smith, “Optical resonator axis stability and instability from first principles,” Proc. SPIE 412, 203–206 (1983).

1977 (1)

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. 7, 857–859 (1977).
[CrossRef]

1969 (1)

Altshuler, G. B.

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. 7, 857–859 (1977).
[CrossRef]

Anderson, D. Z.

Arnaud, J. A.

Bilger, H. R.

Chow, W. W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–86 (1985).
[CrossRef]

Currie, B. E.

Dunn, R. W.

Faucheux, M.

M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988).
[CrossRef]

Fayoux, D.

M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988).
[CrossRef]

Gea-Banacloche, J.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–86 (1985).
[CrossRef]

Isyanova, E. D.

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. 7, 857–859 (1977).
[CrossRef]

Karasev, V. B.

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. 7, 857–859 (1977).
[CrossRef]

Latham, W. P.

Levit, A. L.

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. 7, 857–859 (1977).
[CrossRef]

Levkit, A. L.

A. L. Levkit and V. M. Ovchinnikov, “Stability of a ring resonator with a nonplane axial contour,” J. Appl. Spectrosc. (USSR) 40(6), 657–660 (1984).
[CrossRef]

Liang, L. M.

Long, X. W.

X. W. Long and J. Yuan, “Method for eliminating mismatching error in monolithic triaxial ring resonators,” Chin. Opt. Lett. 8(12), 1135–1138 (2010).
[CrossRef]

J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun. 281(5), 1204–1210 (2008).
[CrossRef]

J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008).
[CrossRef] [PubMed]

J. Yuan, X. W. Long, B. Zhang, F. Wang, and H. C. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt. 46(25), 6314–6322 (2007).
[CrossRef] [PubMed]

J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76(12), 125106 (2005).
[CrossRef]

J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74(3), 1362–1365 (2003).
[CrossRef]

Ovchinnikov, V. M.

A. L. Levkit and V. M. Ovchinnikov, “Stability of a ring resonator with a nonplane axial contour,” J. Appl. Spectrosc. (USSR) 40(6), 657–660 (1984).
[CrossRef]

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. 7, 857–859 (1977).
[CrossRef]

Paxton, A. H.

Pedrotti, L. M.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–86 (1985).
[CrossRef]

Rodloff, R.

R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron. 23(4), 438–445 (1987).
[CrossRef]

Roland, J. J.

M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988).
[CrossRef]

Sampas, N. M.

Sanders, V. E.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–86 (1985).
[CrossRef]

Schleich, W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–86 (1985).
[CrossRef]

Scully, M. O.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–86 (1985).
[CrossRef]

Sharlai, S. F.

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. 7, 857–859 (1977).
[CrossRef]

Sheng, S.-C.

Siegman, A. E.

A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1389–1399 (2000).
[CrossRef]

Smith, I. W.

I. W. Smith, “Optical resonator axis stability and instability from first principles,” Proc. SPIE 412, 203–206 (1983).

Stedman, G. E.

Wang, F.

Yang, K. Y.

J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76(12), 125106 (2005).
[CrossRef]

Yuan, J.

X. W. Long and J. Yuan, “Method for eliminating mismatching error in monolithic triaxial ring resonators,” Chin. Opt. Lett. 8(12), 1135–1138 (2010).
[CrossRef]

J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun. 281(5), 1204–1210 (2008).
[CrossRef]

J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008).
[CrossRef] [PubMed]

J. Yuan, X. W. Long, B. Zhang, F. Wang, and H. C. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt. 46(25), 6314–6322 (2007).
[CrossRef] [PubMed]

J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76(12), 125106 (2005).
[CrossRef]

J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74(3), 1362–1365 (2003).
[CrossRef]

Zhang, B.

Zhao, H. C.

Appl. Opt. (8)

Chin. Opt. Lett. (1)

IEEE J. Quantum Electron. (1)

R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron. 23(4), 438–445 (1987).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1389–1399 (2000).
[CrossRef]

J. Appl. Spectrosc. (USSR) (1)

A. L. Levkit and V. M. Ovchinnikov, “Stability of a ring resonator with a nonplane axial contour,” J. Appl. Spectrosc. (USSR) 40(6), 657–660 (1984).
[CrossRef]

J. Opt. (1)

M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988).
[CrossRef]

Opt. Commun. (1)

J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun. 281(5), 1204–1210 (2008).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

I. W. Smith, “Optical resonator axis stability and instability from first principles,” Proc. SPIE 412, 203–206 (1983).

Rev. Mod. Phys. (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–86 (1985).
[CrossRef]

Rev. Sci. Instrum. (2)

J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74(3), 1362–1365 (2003).
[CrossRef]

J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76(12), 125106 (2005).
[CrossRef]

Sov. J. Quantum Electron. (1)

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. 7, 857–859 (1977).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Translational displacements of a spherical mirror Mi. (a) axial displacement δ i z of Mi and (b) radial displacement δ i x of Mi. Ri: the radius of spherical mirror Mi, Ai: the incident angle, Mi1: the blue solid arc which is the initial position of Mi, Mi2: the red solid arc in (a) which is the position of Mi after axial displacement δ i z , Mi3: the red solid arc in (b) which is the position of Mi after radial displacement δ i x , P1, P2, P3, P4 and P5: the reflecting points, O1 and O2: spherical centers of Mi1 and Mi3, T i x , T i y and T i z : three translational axes, R i x , R i y and R i z : three rotational axes, L1i and L2i: two parallel incident rays, L1o1, L1o2, L2o1 and L2o2: four reflection rays, x i and y i : the coordinate axes of the incident ray, x o and y o : the coordinate axes of the reflected ray, θ1: the angle between line O1P1 and line O1P3, θ2: the angle between line O2P4 and line O2P5, Δ x 1 : the distance between P1 and P3, Δ x 2 : the distance between P4 and P5, Δ y : the distance between P1 and P4.

Fig. 2
Fig. 2

Angular misalignments of a spherical mirror Mi. (a) definition of a mirror’s misalignments angle θ i x and (b) angular misalignment of the spherical mirror Mi. around rotational axis Rix. Ri: the radius of spherical mirror Mi, Ai: the incident angle, Mi4: the red solid line which is the position of Mi after angular misalignment θ i x > 0 , Mi5: the red dashed line which is the position of Mi after angular misalignment θ i x < 0 , Mi1: the blue solid arc which is the initial position of Mi, Mi6: the red solid arc which is the position of Mi after angular misalignment θ i x < 0 , P1: the reflecting points, O1 and O3: spherical centers of Mi1 and Mi6, R i x , R i y and R i z : three rotational axes, L1i: incident ray, L1o1 and L1o3: two reflection rays, x i and y i : the coordinate axes of the incident ray, x o and y o : the coordinate axes of the reflected ray. θ3: the angle between P1O1 and P1O3.

Fig. 3
Fig. 3

(a) Schematic diagram of square ring resonator and (b) schematic diagram of alignment experiment. Ma and Mb: spherical mirrors, Mc and Md: planar mirrors, the incident angle is 45°; a, b, c and d: terminal points of the resonator, e: the center of the diaphragm, g: the center of the gain capillary, f: the midpoint between b and c, h: the midpoint between a and d, x j , y j ( j = e , f , g , h ) :x and y coordinate axes at points e, f, g and h, δ i z ( i = a , b , c , d ) : axial displacement of Mi(i = a,b,c,d), δ i x , δ i y ( i = a , b ) : radial displacements of the spherical mirrors Ma and Mb, HNLP: He-Ne laser with path length control device, RAD1 and RAD2: reflectors with adjusting device, LB: light bulb, CFLAD: CCD area with focusing lens and adjusting device, PCIBS: personal computer with image grabber and image processing software, FI: facular image.

Fig. 4
Fig. 4

Schematic diagram of experimental results on optical-axis perturbations in square ring resonator. (a) optical-axis perturbation caused by spherical mirror’s axial displacements δ i z ( i = a , b ) , (b) optical-axis perturbation caused by planar mirror’s axial displacements δ i z ( i = c , d ) , (c) optical-axis perturbation caused by spherical mirror’s radial displacements δ i x ( i = a , b ) and (d) optical-axis perturbations caused by spherical mirror’s axial displacements δ i y ( i = a , b ) . The ideal optical-axes and the real optical axes after special perturbations are represented by blue solid lines and red dashed lines respectively, spherical mirror’s positions after axial displacements and radial displacements are illustrated with red solid arcs, and planar mirror’s positions after axial displacements are illustrated with red solid lines.

Fig. 5
Fig. 5

Schematic diagram of MTRR with all 3 spherical mirror’s radial displacements and all 6 mirror’s axial displacements. M1, M2 and M3: spherical mirrors with radius R, M4, M5 and M6: planar mirrors, Q1, Q2, Q3, Q4, Q5 and Q6: terminal points of the resonator, PA:, PB:, PC, PD, PE and PB: the midpoints of straight lines Q2Q3, Q1Q3, Q1Q2, Q4Q6, Q5Q6 and Q4Q5 respectively, δ 1 x , δ 1 y , δ 2 x , δ 2 y , δ 3 x , δ 3 y : radial displacements of spherical mirrors M1, M2 and M3, δ 1 z , δ 2 z , δ 3 z , δ 4 z , δ 5 z , δ 6 z : axial displacements of mirrors M1, M2, M3, M4, M5 and M6.

Fig. 6
Fig. 6

Schematic diagram of experimental results on the method for sharing and eliminating the mismatching errors C and C2 in SRR of MTRR. (a) ideal optical-axis of SRR, (b) optical-axis perturbation of SRR during mismatching error sharing process, (c) optical-axis perturbation of SRR during the mismatching error eliminating process by utilizing spherical mirror’s axial displacements δ i z ( i = a , b ) , and (d) optical-axis perturbation of SRR during the eliminating process by utilizing both spherical and planar mirror’s axial displacements δ i z ( i = a , b , c , d ) . The ideal optical-axis, the optical-axis after the mismatching error sharing process and the optical axis after the mismatching error eliminating process are represented by blue solid line, green solid line and red dashed line respectively, spherical mirror’s positions after axial displacements are illustrated with red solid arc, planar mirror’s positions after axial displacements are illustrated with red solid line.

Equations (14)

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( r o x r o x ' r o y r o y ' 1 ) = ( A x B x 0 0 E x C x D x 0 0 F x 0 0 A y B y E y 0 0 C y D y F y 0 0 0 0 1 ) ( r i x r i x ' r i y r i y ' 1 ) ,
M ( M i ) = [ 1 0 0 0 2 δ i z sin ( A i ) 2 R i × cos ( A i ) 1 0 0 2 δ i z tan ( A i ) / R i + 2 ( θ i x + δ i x / R i ) 0 0 1 0 0 0 0 2 × cos ( A i ) R i 1 2 ( θ i y + δ i y / R i ) 0 0 0 0 1 ]
Δ x e = 2 4 ( δ a x + δ b x + δ a z + δ b z ) , Δ y e = 2 2 ( δ a y + δ b y ) ; Δ x g = 2 4 ( δ a x + δ b x δ a z δ b z + 2 δ c z + 2 δ d z ) , Δ y g = 2 2 ( δ a y + δ b y ) .
Δ x A = 2 4 [ δ 2 y δ 3 x + δ 2 z + δ 3 z ] , Δ y A = 2 2 ( δ 2 x δ 3 y ) Δ x B = 2 4 [ δ 3 y δ 1 x + δ 1 z + δ 3 z ] , Δ y B = 2 2 ( δ 3 x δ 1 y ) , Δ x C = 2 4 [ δ 1 y δ 2 x + δ 1 z + δ 2 z ] , Δ y C = 2 2 ( δ 1 x δ 2 y )
Δ x D = 2 4 [ δ 2 y δ 3 x δ 2 z δ 3 z + 2 δ 4 z + 2 δ 6 z ] , Δ y D = 2 2 ( δ 2 x δ 3 y ) Δ x E = 2 4 [ δ 3 y δ 1 x δ 1 z δ 3 z + 2 δ 5 z + 2 δ 6 z ] , Δ y E = 2 2 ( δ 3 x δ 1 y ) Δ x F = 2 4 [ δ 1 y δ 2 x δ 1 z δ 2 z + 2 δ 4 z + 2 δ 5 z ] , Δ y F = 2 2 ( δ 1 x δ 2 y )
Δ x A + Δ y A / 2 + Δ x B + Δ y B / 2 + Δ x C + Δ y C / 2 = 2 2 ( δ 1 z + δ 2 z + δ 3 z ) Δ x C + Δ y D / 2 + Δ x E + Δ y E / 2 + Δ x F + Δ y F / 2 = 2 2 ( 2 δ 4 z + 2 δ 5 z + 2 δ 6 z δ 1 z δ 2 z δ 3 z )
Δ x j ( 0 t ) = x j ( t ) x j ( 0 ) Δ y j ( 0 t ) = y j ( t ) y j ( 0 ) , j = A , B , C , D , E , F
j = A , B , C Δ x j ( 0 t ) + Δ y j ( 0 t ) / 2 = 2 2 ( δ 1 z + δ 2 z + δ 3 z ) j = D , E , F Δ x j ( 0 t ) + Δ y j ( 0 t ) / 2 = 2 2 ( 2 δ 4 z + 2 δ 5 z + 2 δ 6 z δ 1 z δ 2 z δ 3 z )
C ( t ) = j = A , B , C x j ( t ) + y j ( t ) / 2 = j = A , B , C x j ( 0 ) + y j ( 0 ) / 2 + 2 2 ( δ 1 z + δ 2 z + δ 3 z ) = C ( 0 ) + 2 2 ( δ 1 z + δ 2 z + δ 3 z ) C 2 ( t ) = j = D , E , F x j ( t ) + y j ( t ) / 2 = j = D , E , F x j ( 0 ) + y j ( 0 ) / 2 + 2 2 ( 2 δ 4 z + 2 δ 5 z + 2 δ 6 z δ 1 z δ 2 z δ 3 z ) = C 2 ( 0 ) + 2 2 ( 2 δ 4 z + 2 δ 5 z + 2 δ 6 z δ 1 z δ 2 z δ 3 z )
D j ( t ) = x j ( t ) 2 + y j ( t ) 2 ( j = A , B , C , D , E , F )
x A ( 0 ) = x D ( 0 ) = x B ( 0 ) = x E ( 0 ) = x C ( 0 ) = x F ( 0 ) = C / 3 y A ( 0 ) = y D ( 0 ) = y B ( 0 ) = y E ( 0 ) = y C ( 0 ) = y F ( 0 ) = 0 D A ( 0 ) = D B ( 0 ) = D C ( 0 ) = D D ( 0 ) = D E ( 0 ) = D F ( 0 ) = | C | 3
δ 1 z + δ 2 z + δ 3 z = 2 × C ( 0 ) δ 4 z + δ 5 z + δ 6 z = 2 × C ( 0 )
δ 1 z = δ 2 z = δ 3 z = 2 × C ( 0 ) / 3 δ 4 z = δ 5 z = δ 6 z = 2 × C ( 0 ) / 3 ,
C ( t ) = 0 , C 2 ( t ) = 0 x A ( t ) = x B ( t ) = x C ( t ) = x D ( t ) = x E ( t ) = x F ( t ) = 0 y A ( t ) = y B ( t ) = y C ( t ) = y D ( t ) = y E ( t ) = y F ( t ) = 0 D A ( t ) = D B ( t ) = D C ( t ) = D D ( t ) = D E ( t ) = D F ( t ) = 0

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