Abstract

A spatiotemporal phase-unwrapping method is presented that combines the dynamic fringe-projection method and the phase-shifting technique and extends the phase-unwrapping method, which measures two phase maps at different sensitivities. The most important feature of the method is that it makes possible the automatic three-dimensional shape measurement of discontinuous objects with large dynamic range limits and high precision because the effective wavelength of the fringe-projection profilometry can be continuously varied over several orders of magnitude by rotation of the projection grating in its own plane. Only one grating and several steps of rotating the grating are required; therefore the method is inherently simple, fast, and robust. In the experiment, choosing the rotation angle was crucial for optimizing the measurement speed and the measurement accuracy. A criterion is presented for the choice of the minimum number of rotational steps for a given accuracy. The experimental results demonstrate the validity of the proposed method.

© 1999 Optical Society of America

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  1. R. E. Brooks, L. O. Heflinger, “Moiré gauging using optical interference patterns,” Appl. Opt. 8, 935–939 (1969).
    [CrossRef] [PubMed]
  2. P. Benoit, E. Mathieu, J. Hormièr, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
    [CrossRef]
  3. M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
    [CrossRef]
  4. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  5. V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105–3108 (1984).
    [CrossRef] [PubMed]
  6. D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
    [CrossRef]
  7. G. T. Reid, R. C. Rixon, H. Stewart, “Moiré topography with large contour intervals,” in International Conference on Photomechanics and Speckle Metrology, F.-P. Chiang, ed., Proc. SPIE814, 307–313 (1987).
    [CrossRef]
  8. H. Zhao, W. Chen, Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. 33, 4497–4500 (1994).
    [CrossRef] [PubMed]
  9. T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherent radar,” Appl. Opt. 31, 919–925 (1992).
    [CrossRef] [PubMed]
  10. J. M. Huntley, H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  11. H. O. Saldner, J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36, 2770–2775 (1997).
    [CrossRef] [PubMed]
  12. J. M. Huntley, H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
    [CrossRef]
  13. W. Nadeborn, P. Addrä, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
    [CrossRef]
  14. X. Xie, J. T. Atkinson, M. J. Lalor, D. R. Burton, “Three-map absolute moiré contouring,” Appl. Opt. 35, 6990–6995 (1996).
    [CrossRef] [PubMed]
  15. H. Zhang, M. J. Lalor, D. R. Burton, “A new error-compensating seven-sample phase-shifting algorithm and application in 3-D fringe projection profilometry,” in Laser Interferometry IX: Technique and Analysis, G. M. Brown, M. Kujawinska, M. Takeda, eds., Proc. SPIE3478, 121–132 (1998).
    [CrossRef]
  16. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  17. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithm,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  18. Y. Surrel, “Design of algorithms for phase measurement by the use phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  19. D. R. Burton, M. J. Lalor, “Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping,” Appl. Opt. 33, 2939–2948 (1994).
    [CrossRef] [PubMed]
  20. J. H. Bruning, J. E. Gallagher, D. P. Rosenfield, A. D. White, D. J. Brangaccio, D. R. Herriott, “Digital wave-front measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  21. H.-J. Su, J.-L. Li, X.-Y. Su, “Phase algorithm without the influence of carrier frequency,” Opt. Eng. 36, 1799–1805 (1997).
    [CrossRef]
  22. X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–151 (1993).
    [CrossRef]
  23. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  24. M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
    [CrossRef]

1997 (3)

H. O. Saldner, J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36, 2770–2775 (1997).
[CrossRef] [PubMed]

J. M. Huntley, H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
[CrossRef]

H.-J. Su, J.-L. Li, X.-Y. Su, “Phase algorithm without the influence of carrier frequency,” Opt. Eng. 36, 1799–1805 (1997).
[CrossRef]

1996 (4)

Y. Surrel, “Design of algorithms for phase measurement by the use phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

W. Nadeborn, P. Addrä, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

X. Xie, J. T. Atkinson, M. J. Lalor, D. R. Burton, “Three-map absolute moiré contouring,” Appl. Opt. 35, 6990–6995 (1996).
[CrossRef] [PubMed]

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

1995 (1)

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

1994 (2)

1993 (2)

X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–151 (1993).
[CrossRef]

J. M. Huntley, H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

1992 (2)

1991 (1)

1990 (1)

1989 (1)

M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

1984 (1)

1982 (1)

1975 (1)

P. Benoit, E. Mathieu, J. Hormièr, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

1974 (1)

1969 (1)

Abe, T.

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

Addrä, P.

W. Nadeborn, P. Addrä, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Atkinson, J. T.

X. Xie, J. T. Atkinson, M. J. Lalor, D. R. Burton, “Three-map absolute moiré contouring,” Appl. Opt. 35, 6990–6995 (1996).
[CrossRef] [PubMed]

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

Benoit, P.

P. Benoit, E. Mathieu, J. Hormièr, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

Bone, D. J.

Brangaccio, D. J.

Brooks, R. E.

Bruning, J. H.

Burton, D. R.

X. Xie, J. T. Atkinson, M. J. Lalor, D. R. Burton, “Three-map absolute moiré contouring,” Appl. Opt. 35, 6990–6995 (1996).
[CrossRef] [PubMed]

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

D. R. Burton, M. J. Lalor, “Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping,” Appl. Opt. 33, 2939–2948 (1994).
[CrossRef] [PubMed]

H. Zhang, M. J. Lalor, D. R. Burton, “A new error-compensating seven-sample phase-shifting algorithm and application in 3-D fringe projection profilometry,” in Laser Interferometry IX: Technique and Analysis, G. M. Brown, M. Kujawinska, M. Takeda, eds., Proc. SPIE3478, 121–132 (1998).
[CrossRef]

Chen, W.

Dresel, T.

Freischlad, K.

Gallagher, J. E.

Goodall, A. J.

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

Halioua, M.

M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105–3108 (1984).
[CrossRef] [PubMed]

Häusler, G.

Heflinger, L. O.

Herriott, D. R.

Hormièr, J.

P. Benoit, E. Mathieu, J. Hormièr, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

Huntley, J. M.

Ina, H.

Kobayashi, S.

Koliopoulos, C. L.

Lalor, M. J.

X. Xie, J. T. Atkinson, M. J. Lalor, D. R. Burton, “Three-map absolute moiré contouring,” Appl. Opt. 35, 6990–6995 (1996).
[CrossRef] [PubMed]

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

D. R. Burton, M. J. Lalor, “Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping,” Appl. Opt. 33, 2939–2948 (1994).
[CrossRef] [PubMed]

H. Zhang, M. J. Lalor, D. R. Burton, “A new error-compensating seven-sample phase-shifting algorithm and application in 3-D fringe projection profilometry,” in Laser Interferometry IX: Technique and Analysis, G. M. Brown, M. Kujawinska, M. Takeda, eds., Proc. SPIE3478, 121–132 (1998).
[CrossRef]

Larkin, G.

Li, J.-L.

H.-J. Su, J.-L. Li, X.-Y. Su, “Phase algorithm without the influence of carrier frequency,” Opt. Eng. 36, 1799–1805 (1997).
[CrossRef]

Liu, H. C.

Liu, H.-C.

M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

Mathieu, E.

P. Benoit, E. Mathieu, J. Hormièr, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

Nadeborn, W.

W. Nadeborn, P. Addrä, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Oreb, B. F.

Osten, W.

W. Nadeborn, P. Addrä, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Reid, G. T.

G. T. Reid, R. C. Rixon, H. Stewart, “Moiré topography with large contour intervals,” in International Conference on Photomechanics and Speckle Metrology, F.-P. Chiang, ed., Proc. SPIE814, 307–313 (1987).
[CrossRef]

Rixon, R. C.

G. T. Reid, R. C. Rixon, H. Stewart, “Moiré topography with large contour intervals,” in International Conference on Photomechanics and Speckle Metrology, F.-P. Chiang, ed., Proc. SPIE814, 307–313 (1987).
[CrossRef]

Rosenfield, D. P.

Saldner, H. O.

Srinivasan, V.

Stewart, H.

G. T. Reid, R. C. Rixon, H. Stewart, “Moiré topography with large contour intervals,” in International Conference on Photomechanics and Speckle Metrology, F.-P. Chiang, ed., Proc. SPIE814, 307–313 (1987).
[CrossRef]

Su, H.-J.

H.-J. Su, J.-L. Li, X.-Y. Su, “Phase algorithm without the influence of carrier frequency,” Opt. Eng. 36, 1799–1805 (1997).
[CrossRef]

Su, X.-Y.

H.-J. Su, J.-L. Li, X.-Y. Su, “Phase algorithm without the influence of carrier frequency,” Opt. Eng. 36, 1799–1805 (1997).
[CrossRef]

X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–151 (1993).
[CrossRef]

Surrel, Y.

Takeda, M.

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

Tan, Y.

Thomas, A.

P. Benoit, E. Mathieu, J. Hormièr, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

Venzke, H.

von Bally, G.

X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–151 (1993).
[CrossRef]

Vukicevic, D.

X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–151 (1993).
[CrossRef]

White, A. D.

Xie, X.

Zhang, H.

H. Zhang, M. J. Lalor, D. R. Burton, “A new error-compensating seven-sample phase-shifting algorithm and application in 3-D fringe projection profilometry,” in Laser Interferometry IX: Technique and Analysis, G. M. Brown, M. Kujawinska, M. Takeda, eds., Proc. SPIE3478, 121–132 (1998).
[CrossRef]

Zhao, H.

Appl. Opt. (11)

R. E. Brooks, L. O. Heflinger, “Moiré gauging using optical interference patterns,” Appl. Opt. 8, 935–939 (1969).
[CrossRef] [PubMed]

V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105–3108 (1984).
[CrossRef] [PubMed]

H. Zhao, W. Chen, Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. 33, 4497–4500 (1994).
[CrossRef] [PubMed]

T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherent radar,” Appl. Opt. 31, 919–925 (1992).
[CrossRef] [PubMed]

J. M. Huntley, H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

H. O. Saldner, J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36, 2770–2775 (1997).
[CrossRef] [PubMed]

X. Xie, J. T. Atkinson, M. J. Lalor, D. R. Burton, “Three-map absolute moiré contouring,” Appl. Opt. 35, 6990–6995 (1996).
[CrossRef] [PubMed]

Y. Surrel, “Design of algorithms for phase measurement by the use phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

D. R. Burton, M. J. Lalor, “Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping,” Appl. Opt. 33, 2939–2948 (1994).
[CrossRef] [PubMed]

J. H. Bruning, J. E. Gallagher, D. P. Rosenfield, A. D. White, D. J. Brangaccio, D. R. Herriott, “Digital wave-front measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (1)

J. M. Huntley, H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
[CrossRef]

Nouv. Rev. Opt. (1)

P. Benoit, E. Mathieu, J. Hormièr, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

Opt. Commun. (1)

X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–151 (1993).
[CrossRef]

Opt. Eng. (2)

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

H.-J. Su, J.-L. Li, X.-Y. Su, “Phase algorithm without the influence of carrier frequency,” Opt. Eng. 36, 1799–1805 (1997).
[CrossRef]

Opt. Lasers Eng. (3)

M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

W. Nadeborn, P. Addrä, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Other (2)

G. T. Reid, R. C. Rixon, H. Stewart, “Moiré topography with large contour intervals,” in International Conference on Photomechanics and Speckle Metrology, F.-P. Chiang, ed., Proc. SPIE814, 307–313 (1987).
[CrossRef]

H. Zhang, M. J. Lalor, D. R. Burton, “A new error-compensating seven-sample phase-shifting algorithm and application in 3-D fringe projection profilometry,” in Laser Interferometry IX: Technique and Analysis, G. M. Brown, M. Kujawinska, M. Takeda, eds., Proc. SPIE3478, 121–132 (1998).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Optical geometry of the dynamic fringe-projection and -recording system.

Fig. 2
Fig. 2

Variation of effective wavelength λ e with rotation angle α.

Fig. 3
Fig. 3

Experimental setup for dynamic fringe-projection phase-shifting profilometry.

Fig. 4
Fig. 4

(a) Fringe pattern projected onto the step object with rotation angle α = 0, (b) wrapped phase map with α = 0, showing discontinuities of the step object. The grayscale represents phases of -π and +π.

Fig. 5
Fig. 5

(a) Fringe pattern projected onto a human hand with rotation angle α = 0, (b) wrapped phase map with α = 0, showing discontinuities of the profile of the human hand compared with the reference plane. The grayscale represents phases of -π and +π.

Fig. 6
Fig. 6

Shading display showing the profile of the step object obtained by unwrapping Fig. 4(b) by the spatiotemporal phase-unwrapping method through five intermediate rotation angles.

Fig. 7
Fig. 7

Cross sections of the step object surface profile (row 450 of Fig. 6) obtained by the spatiotemporal phase-unwrapping method through (a) α = 82°; (b) α = 82°, 75°; (c) α = 82°, 75°, 65°; (d) α = 82°, 75°, 65°, 45°; (e) α = 82°, 75°, 65°, 45°, 0° intermediate rotation angles.

Fig. 8
Fig. 8

Shading display showing the profile of the human hand obtained by unwrapping Fig. 5(b) by the spatiotemporal phase-unwrapping method through four intermediate rotation angles.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

gix, y, ti=s0x, y+n=1 snx, ycosnϕx, y+nti,
ϕx, y=2π/p0x cos θ cos α-y sin α+zx, ysin θ cos α,
ϕx, y=2π/p0x cos θ+zx, ysin θ=2π/px+zx, ytan θ,
λ0=p/tan θ
λe=p0sin θ cos α=ptan θ cos α=λ0cos α
tan ϕ=i=1m aigii=1m bigi,
f1t=i=1m aiδt-ti, f2t=i=1m biδt-ti,
grix, y, ti=s0x, y+n=1 snx, ycosnϕrx, y+nti,
ϕrx, y=2π/p0x cos θ cos α-y sin α
ϕx, y=arctani=1m aigii=1m bigi=arctanSmCm,
ϕrx, y=arctani=1m aigrii=1m bigi=arctanSrCr,
ϕ0x, y=ϕx, y-ϕrx, y=arctanSmCr-CmSrCmCr+SmSr.
Φ0x, y=2nx, yπ+ϕ0x, y,
zx, y=Φ0x, yλ02π cos α.
Φ01x, y=2n1x, yπ+ϕ01x, y, z1x, y=Φ01x, yλ02π cos α1.
Φ0ix, y=Φ0i-1x,ycos αicos αi-1, nix, y=int12πΦ0ix, y-ϕ0ix, y,
Φ0ix, y=2nix, yπ+ϕ0ix, y, zix, y=Φ0ix, yλ02π cos αi.
tan ϕ1=i=110 aig1ii=110 big1i=S1C1,
tan ϕ2=i=110 aig2ii=110 big2i=S2C2.
Δϕ=ϕ2-ϕ1=arctanS2C1-C2S1C2C1+S2S1,
λ0=2πΔlΔϕ.
cos αicos αi-1=Ai  i=2,, k

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