Abstract

High-speed video stereo-microscopy relies on illumination from two distinct angles to create two views of a sample from different directions. The 3D trajectory of a microscopic object can then be reconstructed using parallax to combine 2D measurements of its position in each image. In this work, we evaluate the accuracy of 3D particle tracking using this technique, by extending the number of views from two to four directions. This allows us to record two independent sets of measurements of the 3D coordinates of tracked objects, and comparison of these enables measurement and minimisation of the tracking error in all dimensions. We demonstrate the method by tracking the motion of an optically trapped microsphere of 5 μm in diameter, and find an accuracy of 2–5 nm laterally, and 5–10 nm axially, representing a relative error of less than 2.5% of its range of motion in each dimension.

© 2014 Optical Society of America

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2014

2013

2012

M. Hasler, T. Haist, and W. Osten, “Stereo vision in spatial-light-modulator-based microscopy,” Opt. Lett. 37, 2238–2240 (2012).
[CrossRef] [PubMed]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20, 29679–29693 (2012).
[CrossRef]

S. N. Olof, J. A. Grieve, D. B. Phillips, H. Rosenkranz, M. L. Yallop, M. J. Miles, A. J. Patil, S. Mann, and D. M. Carberry, “Measuring nanoscale forces with living probes,” Nano Lett. 12, 6018–6023 (2012).
[CrossRef] [PubMed]

D. B. Phillips, S. H. Simpson, J. A. Grieve, R. Bowman, G. M. Gibson, M. J. Padgett, J. Rarity, S. Hanna, M. J. Miles, and D. Carberry, “Force sensing with a shaped dielectric micro-tool,” Europhys. Lett. 99, 58004 (2012).
[CrossRef]

2011

R. Bowman, D. Preece, G. Gibson, and M. Padgett, “Stereoscopic particle tracking for 3D touch, vision and closed-loop control in optical tweezers,” J. Opt. 13(4), 044003 (2011).
[CrossRef]

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

2010

2009

J. S. Dam, I. Perch-Nielsen, D. Palima, and J. Glückstad, “Multi-particle three-dimensional coordinate estimation in real-time optical manipulation,” J. Eur. Opt. Soc. Rapid Publ. 4, 09045 (2009).
[CrossRef]

2008

2007

P. Cicuta and A. M. Donald, “Microrheology: a review of the method and applications,” Soft Matter 3(12), 1449–1455 (2007).
[CrossRef]

2004

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

2003

2002

V. J. Anderson and H. N. W. Lekkerkerker, “Insights into phase transition kinetics from colloid science,” Nature 416, 811–815 (2002).
[CrossRef] [PubMed]

1999

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Alpmann, C.

Anderson, V. J.

V. J. Anderson and H. N. W. Lekkerkerker, “Insights into phase transition kinetics from colloid science,” Nature 416, 811–815 (2002).
[CrossRef] [PubMed]

Bernet, S.

M. P. Lee, G. M. Gibson, R. Bowman, S. Bernet, M. Ritsch-Marte, D. B. Phillips, and M. J. Padgett, “A multi-modal stereo microscope based on a spatial light modulator,” Opt. Express 21(14), 16541–16551 (2013).
[CrossRef] [PubMed]

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

Bianchi, S.

Bowman, R.

Carberry, D.

D. B. Phillips, S. H. Simpson, J. A. Grieve, R. Bowman, G. M. Gibson, M. J. Padgett, J. Rarity, S. Hanna, M. J. Miles, and D. Carberry, “Force sensing with a shaped dielectric micro-tool,” Europhys. Lett. 99, 58004 (2012).
[CrossRef]

Carberry, D. M.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photon. 8, 400–405 (2014).
[CrossRef]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20, 29679–29693 (2012).
[CrossRef]

S. N. Olof, J. A. Grieve, D. B. Phillips, H. Rosenkranz, M. L. Yallop, M. J. Miles, A. J. Patil, S. Mann, and D. M. Carberry, “Measuring nanoscale forces with living probes,” Nano Lett. 12, 6018–6023 (2012).
[CrossRef] [PubMed]

Cicuta, P.

P. Cicuta and A. M. Donald, “Microrheology: a review of the method and applications,” Soft Matter 3(12), 1449–1455 (2007).
[CrossRef]

Czerwinski, F.

Dam, J. S.

J. S. Dam, I. Perch-Nielsen, D. Palima, and J. Glückstad, “Multi-particle three-dimensional coordinate estimation in real-time optical manipulation,” J. Eur. Opt. Soc. Rapid Publ. 4, 09045 (2009).
[CrossRef]

J. S. Dam, I. R. Perch-Nielsen, D. Palima, and J. Glückstad, “Three-dimensional imaging in three-dimensional optical multi-beam micromanipulation,” Opt. Express 16(10), 7244–7250 (2008).
[CrossRef] [PubMed]

Denz, C.

Di Leonardo, R.

Donald, A. M.

P. Cicuta and A. M. Donald, “Microrheology: a review of the method and applications,” Soft Matter 3(12), 1449–1455 (2007).
[CrossRef]

Florin, E.-L.

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

Gibson, G.

R. Bowman, D. Preece, G. Gibson, and M. Padgett, “Stereoscopic particle tracking for 3D touch, vision and closed-loop control in optical tweezers,” J. Opt. 13(4), 044003 (2011).
[CrossRef]

R. Bowman, G. Gibson, and M. Padgett, “Particle tracking stereomicroscopy in optical tweezers: Control of trap shape,” Opt. Express 18(11), 11785–11790 (2010).
[CrossRef] [PubMed]

Gibson, G. M.

Glückstad, J.

J. S. Dam, I. Perch-Nielsen, D. Palima, and J. Glückstad, “Multi-particle three-dimensional coordinate estimation in real-time optical manipulation,” J. Eur. Opt. Soc. Rapid Publ. 4, 09045 (2009).
[CrossRef]

J. S. Dam, I. R. Perch-Nielsen, D. Palima, and J. Glückstad, “Three-dimensional imaging in three-dimensional optical multi-beam micromanipulation,” Opt. Express 16(10), 7244–7250 (2008).
[CrossRef] [PubMed]

Gornall, J. L.

Grieve, J. A.

D. B. Phillips, S. H. Simpson, J. A. Grieve, R. Bowman, G. M. Gibson, M. J. Padgett, J. Rarity, S. Hanna, M. J. Miles, and D. Carberry, “Force sensing with a shaped dielectric micro-tool,” Europhys. Lett. 99, 58004 (2012).
[CrossRef]

S. N. Olof, J. A. Grieve, D. B. Phillips, H. Rosenkranz, M. L. Yallop, M. J. Miles, A. J. Patil, S. Mann, and D. M. Carberry, “Measuring nanoscale forces with living probes,” Nano Lett. 12, 6018–6023 (2012).
[CrossRef] [PubMed]

Haist, T.

Hanna, S.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photon. 8, 400–405 (2014).
[CrossRef]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20, 29679–29693 (2012).
[CrossRef]

D. B. Phillips, S. H. Simpson, J. A. Grieve, R. Bowman, G. M. Gibson, M. J. Padgett, J. Rarity, S. Hanna, M. J. Miles, and D. Carberry, “Force sensing with a shaped dielectric micro-tool,” Europhys. Lett. 99, 58004 (2012).
[CrossRef]

Hasler, M.

Ho, Y.-L.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photon. 8, 400–405 (2014).
[CrossRef]

Jericho, M. H.

Jesacher, A.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

Keyser, U. F.

Kreuzer, H. J.

Lee, M. P.

Lekkerkerker, H. N. W.

V. J. Anderson and H. N. W. Lekkerkerker, “Insights into phase transition kinetics from colloid science,” Nature 416, 811–815 (2002).
[CrossRef] [PubMed]

Lepore, A.

Mann, S.

S. N. Olof, J. A. Grieve, D. B. Phillips, H. Rosenkranz, M. L. Yallop, M. J. Miles, A. J. Patil, S. Mann, and D. M. Carberry, “Measuring nanoscale forces with living probes,” Nano Lett. 12, 6018–6023 (2012).
[CrossRef] [PubMed]

Maurer, C.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

Mehta, A. D.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Meinertzhagen, I. A.

Menq, C.-H.

Miles, M. J.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photon. 8, 400–405 (2014).
[CrossRef]

S. N. Olof, J. A. Grieve, D. B. Phillips, H. Rosenkranz, M. L. Yallop, M. J. Miles, A. J. Patil, S. Mann, and D. M. Carberry, “Measuring nanoscale forces with living probes,” Nano Lett. 12, 6018–6023 (2012).
[CrossRef] [PubMed]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20, 29679–29693 (2012).
[CrossRef]

D. B. Phillips, S. H. Simpson, J. A. Grieve, R. Bowman, G. M. Gibson, M. J. Padgett, J. Rarity, S. Hanna, M. J. Miles, and D. Carberry, “Force sensing with a shaped dielectric micro-tool,” Europhys. Lett. 99, 58004 (2012).
[CrossRef]

Neumayer, D.

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

Oddershede, L. B.

Olof, S. N.

S. N. Olof, J. A. Grieve, D. B. Phillips, H. Rosenkranz, M. L. Yallop, M. J. Miles, A. J. Patil, S. Mann, and D. M. Carberry, “Measuring nanoscale forces with living probes,” Nano Lett. 12, 6018–6023 (2012).
[CrossRef] [PubMed]

Osten, W.

Otto, O.

Padgett, M.

Padgett, M. J.

Palima, D.

J. S. Dam, I. Perch-Nielsen, D. Palima, and J. Glückstad, “Multi-particle three-dimensional coordinate estimation in real-time optical manipulation,” J. Eur. Opt. Soc. Rapid Publ. 4, 09045 (2009).
[CrossRef]

J. S. Dam, I. R. Perch-Nielsen, D. Palima, and J. Glückstad, “Three-dimensional imaging in three-dimensional optical multi-beam micromanipulation,” Opt. Express 16(10), 7244–7250 (2008).
[CrossRef] [PubMed]

Patil, A. J.

S. N. Olof, J. A. Grieve, D. B. Phillips, H. Rosenkranz, M. L. Yallop, M. J. Miles, A. J. Patil, S. Mann, and D. M. Carberry, “Measuring nanoscale forces with living probes,” Nano Lett. 12, 6018–6023 (2012).
[CrossRef] [PubMed]

Perch-Nielsen, I.

J. S. Dam, I. Perch-Nielsen, D. Palima, and J. Glückstad, “Multi-particle three-dimensional coordinate estimation in real-time optical manipulation,” J. Eur. Opt. Soc. Rapid Publ. 4, 09045 (2009).
[CrossRef]

Perch-Nielsen, I. R.

Phillips, D.

Phillips, D. B.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photon. 8, 400–405 (2014).
[CrossRef]

M. P. Lee, G. M. Gibson, R. Bowman, S. Bernet, M. Ritsch-Marte, D. B. Phillips, and M. J. Padgett, “A multi-modal stereo microscope based on a spatial light modulator,” Opt. Express 21(14), 16541–16551 (2013).
[CrossRef] [PubMed]

D. B. Phillips, S. H. Simpson, J. A. Grieve, R. Bowman, G. M. Gibson, M. J. Padgett, J. Rarity, S. Hanna, M. J. Miles, and D. Carberry, “Force sensing with a shaped dielectric micro-tool,” Europhys. Lett. 99, 58004 (2012).
[CrossRef]

S. N. Olof, J. A. Grieve, D. B. Phillips, H. Rosenkranz, M. L. Yallop, M. J. Miles, A. J. Patil, S. Mann, and D. M. Carberry, “Measuring nanoscale forces with living probes,” Nano Lett. 12, 6018–6023 (2012).
[CrossRef] [PubMed]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20, 29679–29693 (2012).
[CrossRef]

Preece, D.

R. Bowman, D. Preece, G. Gibson, and M. Padgett, “Stereoscopic particle tracking for 3D touch, vision and closed-loop control in optical tweezers,” J. Opt. 13(4), 044003 (2011).
[CrossRef]

Rarity, J.

D. B. Phillips, S. H. Simpson, J. A. Grieve, R. Bowman, G. M. Gibson, M. J. Padgett, J. Rarity, S. Hanna, M. J. Miles, and D. Carberry, “Force sensing with a shaped dielectric micro-tool,” Europhys. Lett. 99, 58004 (2012).
[CrossRef]

Rief, M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Ritsch-Marte, M.

M. P. Lee, G. M. Gibson, R. Bowman, S. Bernet, M. Ritsch-Marte, D. B. Phillips, and M. J. Padgett, “A multi-modal stereo microscope based on a spatial light modulator,” Opt. Express 21(14), 16541–16551 (2013).
[CrossRef] [PubMed]

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

Rohrbach, A.

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

Rosenkranz, H.

S. N. Olof, J. A. Grieve, D. B. Phillips, H. Rosenkranz, M. L. Yallop, M. J. Miles, A. J. Patil, S. Mann, and D. M. Carberry, “Measuring nanoscale forces with living probes,” Nano Lett. 12, 6018–6023 (2012).
[CrossRef] [PubMed]

Saglimbeni, F.

Seidel, R.

Simmons, R. M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Simpson, S. H.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photon. 8, 400–405 (2014).
[CrossRef]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20, 29679–29693 (2012).
[CrossRef]

D. B. Phillips, S. H. Simpson, J. A. Grieve, R. Bowman, G. M. Gibson, M. J. Padgett, J. Rarity, S. Hanna, M. J. Miles, and D. Carberry, “Force sensing with a shaped dielectric micro-tool,” Europhys. Lett. 99, 58004 (2012).
[CrossRef]

Smith, D. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Spudich, J. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Stelzer, E. H. K.

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

Stober, G.

Tassieri, M.

Tischer, C.

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

Woerdemann, M.

Xu, W.

Yallop, M. L.

S. N. Olof, J. A. Grieve, D. B. Phillips, H. Rosenkranz, M. L. Yallop, M. J. Miles, A. J. Patil, S. Mann, and D. M. Carberry, “Measuring nanoscale forces with living probes,” Nano Lett. 12, 6018–6023 (2012).
[CrossRef] [PubMed]

Zhang, Z.

Appl. Opt.

Europhys. Lett.

D. B. Phillips, S. H. Simpson, J. A. Grieve, R. Bowman, G. M. Gibson, M. J. Padgett, J. Rarity, S. Hanna, M. J. Miles, and D. Carberry, “Force sensing with a shaped dielectric micro-tool,” Europhys. Lett. 99, 58004 (2012).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ.

J. S. Dam, I. Perch-Nielsen, D. Palima, and J. Glückstad, “Multi-particle three-dimensional coordinate estimation in real-time optical manipulation,” J. Eur. Opt. Soc. Rapid Publ. 4, 09045 (2009).
[CrossRef]

J. Opt.

R. Bowman, D. Preece, G. Gibson, and M. Padgett, “Stereoscopic particle tracking for 3D touch, vision and closed-loop control in optical tweezers,” J. Opt. 13(4), 044003 (2011).
[CrossRef]

Laser Photon. Rev.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

Nano Lett.

S. N. Olof, J. A. Grieve, D. B. Phillips, H. Rosenkranz, M. L. Yallop, M. J. Miles, A. J. Patil, S. Mann, and D. M. Carberry, “Measuring nanoscale forces with living probes,” Nano Lett. 12, 6018–6023 (2012).
[CrossRef] [PubMed]

Nat. Photon.

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Figures (3)

Fig. 1
Fig. 1

a) Schematic of the optical layout. It is based on a transmission light microscope with an SLM placed in the Fourier plane of the image. The sample is illuminated from four directions with red LEDs (Luxeon Rebel, peak emission at 636 nm) coupled into light guides. Light transmitted through the sample is collected by the objective lens (Zeiss Plan-Neofluar, 100×,1.3 NA). The polarising beamsplitter selects light of the correct polarisation for the SLM. An iris in a conjugate image plane controls the field-of-view to ensure all images fit adjacently onto the camera. A narrowband filter (635 nm/10 nm) minimises dispersion of the images when diffracted from the SLM. The polarising beamsplitter is also used to couple in a single beam optical tweezer (λ = 635 nm). The focal lengths of the lenses are: L1 = 120 mm, L2 = 150 mm and L3 = 200 mm. b) Illustration of the four illumination paths through the system. Here the SLM is shown as a transmissive element rather than reflective, for clarity. The SLM diffracts light corresponding to each illumination direction to a separate region of the camera sensor. (c) A representative phase pattern on the SLM. (d) The resulting image on the camera. Undiffracted zero-order light is sent to the central region. The scaling is 73.5 nm per pixel.

Fig. 2
Fig. 2

(a) The relationship between microsphere’s 3D position to its projected image on the focal plane. The measured motion of the image parallel to x in the focal plane is given by xm = A + δx, where A = δztanϕ cosθ. Likewise the measured motion of the image parallel to y is given by ym = B + δy, where B = δztanϕ sinθ. (b) shows a vertical crosssection through a z-stack of images (200 images recorded at 100 nm height separations) of a 5 μm diameter microsphere illuminated by a single off axis illuminator. The approximate position of the microsphere is marked with a dashed white circle. Optimum tracking is achieved by choosing the most symmetrical horizontal plane for centre of symmetry tracking. (c) shows a column of four images of a microsphere at different heights with a spatial filter diameter of 2.5 mm. (d) shows the same microsphere with a reduced diameter spatial filter. In (d) the images are now more symmetrical, facilitating accurate tracking.

Fig. 3
Fig. 3

Accuracy of 3D particle tracking using stereo-microscopy. (a), (b) and (c) show traces of the measured x, y and z positions respectively. In each case the two independent measurements (green and blue lines) are overlaid by subtracting their mean positions. The red trace indicates the difference between the measurements. (d) shows a zoom in on the highlighted region of (c) to more clearly observe the similarity of the measurements. (e) shows how the tracking accuracy in each dimension depends upon the level of spatial filtering. (f) shows how the accuracy depends upon the camera exposure time.

Equations (4)

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δ x = ( x m j tan ϕ i cos θ i x m i tan ϕ j cos θ j ) / ( tan ϕ i cos θ i tan ϕ j cos θ j )
δ y = ( y m j tan ϕ i sin θ i y m i tan ϕ j sin θ j ) / ( tan ϕ i sin θ i tan ϕ j sin θ j )
δ z from x = ( x m j x m i ) / ( tan ϕ j cos θ j tan ϕ i cos θ i )
δ z from y = ( y m j y m i ) / ( tan ϕ j sin θ j tan ϕ i sin θ i )

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