Abstract

We explore the potential use of the Fourier-transform profilometry technique in in vivo studies of muscular contractions through the variation of muscle-group cross sections. Thanks to a tensorial analysis of the technique, a general expression of its sensitivity vector is established. It allows derivation of the expression of the resolution and the limit condition imposed by the spatial sampling of the fringe pattern. Key parameters that maximize the sensitivity are then simulated. A measurement system is accordingly built up and characterized. It is then successfully applied to the evaluation of the deformation of the forearm muscles during grasping exertions.

© 2005 Optical Society of America

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References

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  1. B. Saltin, P. Gollnick, “Skeletal muscle adaptability: significance for metabolism and performance,” in Handbook of Physiology,L. D. Peachey, R. H. Adrian, S. R. Geiger, eds. (American Physiological Society, Bethesda, Maryland, 1983), pp. 555–631.
  2. A. F. Mannion, G. A. Dumas, J. M. Stevenson, R. G. Cooper, “The influence of muscle fiber size and type distribution on electromyographic measures of back muscle fatigability,” Spine 23, 576–584 (1996).
    [CrossRef]
  3. T. R. Dangaria, O. Naesh, “Changes in cross-sectional area of Psoas major muscle in unilateral Sciateca caused by disc herniation,” Spine 23, 928–931 (1992).
    [CrossRef]
  4. I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics XXII,E. Wolf, ed. (North Holland, Amsterdam, Netherlands, 1985), pp. 273–336.
  5. J. L. Doty, “Projection moiré for remote contour analysis,” J. Opt. Soc. Am. 73, 366–372 (1983).
    [CrossRef]
  6. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).
  7. D. Post, B. Han, P. Ifju, High Sensitivity Moiré. Experimental Analysis for Mechanics and Materials (Springer-Verlag, Berlin, Germany, 1994).
    [CrossRef]
  8. J. D. Trolinger, “Optical methods for wide scale displacement measurements,” in Fringe ’97: Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten (Akademie Verlag, Berlin, 1997), pp. 255–266.
  9. T. E. Denton, F. M. Randall, D. A. Deinlein, “The use of instant moiré photographs to reduce exposure from scoliosis radiographs,” Spine 17, 509–512 (1992).
    [CrossRef]
  10. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef]
  11. W. Schumann, J. P. Zürcher, D. Cuche, Holography and Deformation Analysis (Springer-Verlag, Berlin, 1986).
  12. W. Schumann, “Deformation measurement and analysis on several curved surfaces,” in Fringe ’97: Automatic Processing of Fringe Patterns,W. Jüptner, W. Osten (Akademie Verlag, Berlin, 1997), pp. 289–298.
  13. P. Tatasciore, “Récupération des franges d’interférence en interférométrie holographique appliquée aux grandes déformations des corps opaques,” Ph.D. Dissertation 8917 (Ecole Polytechnique Fédérale de Zurich, 1989).
  14. P. Tatasciore, E. K. Hack, “Projection moiré using tensor calculus for general geometries of optical setups,” Opt. Eng. 34, 1887–1899 (1995).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).
  16. K. G. Harding, S. L. Cartwright, “Phase grating use in moiré interferometry,” Appl. Opt. 38, 1517–1520 (1984).
    [CrossRef]
  17. X. Y. Su, W. S. Zhou, G. Von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
    [CrossRef]
  18. D. J. Bone, H. A. Bachor, R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
    [CrossRef] [PubMed]
  19. A. Hanafi, T. Gharbi, J. Y. Cornu, “In vivo measurement of the lower back deformations with Fourier-transform profilometry,” Appl. Opt. 44, 2266–2273 (2005).
    [CrossRef] [PubMed]
  20. 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]
  21. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  22. A. Hanafi, J. Y. Cornu, T. Gharbi, A. Courteville, “An approach to the evaluation of hand strength and the ability to control force given visual biofeedback,” J. Eng. Mech. 123, 1215–1218 (1997).
    [CrossRef]
  23. J. Y. Cornu, A. Hanafi, T. Gharbi, “A rapid evaluation of individual hand strength,” J. Eng. Mech. 125, 1056–1061 (1999).
    [CrossRef]

2005

1999

J. Y. Cornu, A. Hanafi, T. Gharbi, “A rapid evaluation of individual hand strength,” J. Eng. Mech. 125, 1056–1061 (1999).
[CrossRef]

1997

A. Hanafi, J. Y. Cornu, T. Gharbi, A. Courteville, “An approach to the evaluation of hand strength and the ability to control force given visual biofeedback,” J. Eng. Mech. 123, 1215–1218 (1997).
[CrossRef]

1996

A. F. Mannion, G. A. Dumas, J. M. Stevenson, R. G. Cooper, “The influence of muscle fiber size and type distribution on electromyographic measures of back muscle fatigability,” Spine 23, 576–584 (1996).
[CrossRef]

1995

P. Tatasciore, E. K. Hack, “Projection moiré using tensor calculus for general geometries of optical setups,” Opt. Eng. 34, 1887–1899 (1995).
[CrossRef]

1992

X. Y. Su, W. S. Zhou, G. Von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

T. R. Dangaria, O. Naesh, “Changes in cross-sectional area of Psoas major muscle in unilateral Sciateca caused by disc herniation,” Spine 23, 928–931 (1992).
[CrossRef]

T. E. Denton, F. M. Randall, D. A. Deinlein, “The use of instant moiré photographs to reduce exposure from scoliosis radiographs,” Spine 17, 509–512 (1992).
[CrossRef]

1986

1984

K. G. Harding, S. L. Cartwright, “Phase grating use in moiré interferometry,” Appl. Opt. 38, 1517–1520 (1984).
[CrossRef]

1983

1982

Bachor, H. A.

Bone, D. J.

Cartwright, S. L.

K. G. Harding, S. L. Cartwright, “Phase grating use in moiré interferometry,” Appl. Opt. 38, 1517–1520 (1984).
[CrossRef]

Cooper, R. G.

A. F. Mannion, G. A. Dumas, J. M. Stevenson, R. G. Cooper, “The influence of muscle fiber size and type distribution on electromyographic measures of back muscle fatigability,” Spine 23, 576–584 (1996).
[CrossRef]

Cornu, J. Y.

A. Hanafi, T. Gharbi, J. Y. Cornu, “In vivo measurement of the lower back deformations with Fourier-transform profilometry,” Appl. Opt. 44, 2266–2273 (2005).
[CrossRef] [PubMed]

J. Y. Cornu, A. Hanafi, T. Gharbi, “A rapid evaluation of individual hand strength,” J. Eng. Mech. 125, 1056–1061 (1999).
[CrossRef]

A. Hanafi, J. Y. Cornu, T. Gharbi, A. Courteville, “An approach to the evaluation of hand strength and the ability to control force given visual biofeedback,” J. Eng. Mech. 123, 1215–1218 (1997).
[CrossRef]

Courteville, A.

A. Hanafi, J. Y. Cornu, T. Gharbi, A. Courteville, “An approach to the evaluation of hand strength and the ability to control force given visual biofeedback,” J. Eng. Mech. 123, 1215–1218 (1997).
[CrossRef]

Cuche, D.

W. Schumann, J. P. Zürcher, D. Cuche, Holography and Deformation Analysis (Springer-Verlag, Berlin, 1986).

Dangaria, T. R.

T. R. Dangaria, O. Naesh, “Changes in cross-sectional area of Psoas major muscle in unilateral Sciateca caused by disc herniation,” Spine 23, 928–931 (1992).
[CrossRef]

Deinlein, D. A.

T. E. Denton, F. M. Randall, D. A. Deinlein, “The use of instant moiré photographs to reduce exposure from scoliosis radiographs,” Spine 17, 509–512 (1992).
[CrossRef]

Denton, T. E.

T. E. Denton, F. M. Randall, D. A. Deinlein, “The use of instant moiré photographs to reduce exposure from scoliosis radiographs,” Spine 17, 509–512 (1992).
[CrossRef]

Doty, J. L.

Dumas, G. A.

A. F. Mannion, G. A. Dumas, J. M. Stevenson, R. G. Cooper, “The influence of muscle fiber size and type distribution on electromyographic measures of back muscle fatigability,” Spine 23, 576–584 (1996).
[CrossRef]

Gharbi, T.

A. Hanafi, T. Gharbi, J. Y. Cornu, “In vivo measurement of the lower back deformations with Fourier-transform profilometry,” Appl. Opt. 44, 2266–2273 (2005).
[CrossRef] [PubMed]

J. Y. Cornu, A. Hanafi, T. Gharbi, “A rapid evaluation of individual hand strength,” J. Eng. Mech. 125, 1056–1061 (1999).
[CrossRef]

A. Hanafi, J. Y. Cornu, T. Gharbi, A. Courteville, “An approach to the evaluation of hand strength and the ability to control force given visual biofeedback,” J. Eng. Mech. 123, 1215–1218 (1997).
[CrossRef]

Gollnick, P.

B. Saltin, P. Gollnick, “Skeletal muscle adaptability: significance for metabolism and performance,” in Handbook of Physiology,L. D. Peachey, R. H. Adrian, S. R. Geiger, eds. (American Physiological Society, Bethesda, Maryland, 1983), pp. 555–631.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

Hack, E. K.

P. Tatasciore, E. K. Hack, “Projection moiré using tensor calculus for general geometries of optical setups,” Opt. Eng. 34, 1887–1899 (1995).
[CrossRef]

Han, B.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré. Experimental Analysis for Mechanics and Materials (Springer-Verlag, Berlin, Germany, 1994).
[CrossRef]

Hanafi, A.

A. Hanafi, T. Gharbi, J. Y. Cornu, “In vivo measurement of the lower back deformations with Fourier-transform profilometry,” Appl. Opt. 44, 2266–2273 (2005).
[CrossRef] [PubMed]

J. Y. Cornu, A. Hanafi, T. Gharbi, “A rapid evaluation of individual hand strength,” J. Eng. Mech. 125, 1056–1061 (1999).
[CrossRef]

A. Hanafi, J. Y. Cornu, T. Gharbi, A. Courteville, “An approach to the evaluation of hand strength and the ability to control force given visual biofeedback,” J. Eng. Mech. 123, 1215–1218 (1997).
[CrossRef]

Harding, K. G.

K. G. Harding, S. L. Cartwright, “Phase grating use in moiré interferometry,” Appl. Opt. 38, 1517–1520 (1984).
[CrossRef]

Ifju, P.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré. Experimental Analysis for Mechanics and Materials (Springer-Verlag, Berlin, Germany, 1994).
[CrossRef]

Ina, H.

Kobayashi, S.

Macy, W. W.

Mannion, A. F.

A. F. Mannion, G. A. Dumas, J. M. Stevenson, R. G. Cooper, “The influence of muscle fiber size and type distribution on electromyographic measures of back muscle fatigability,” Spine 23, 576–584 (1996).
[CrossRef]

Mutoh, K.

Naesh, O.

T. R. Dangaria, O. Naesh, “Changes in cross-sectional area of Psoas major muscle in unilateral Sciateca caused by disc herniation,” Spine 23, 928–931 (1992).
[CrossRef]

Patorski, K.

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).

Post, D.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré. Experimental Analysis for Mechanics and Materials (Springer-Verlag, Berlin, Germany, 1994).
[CrossRef]

Randall, F. M.

T. E. Denton, F. M. Randall, D. A. Deinlein, “The use of instant moiré photographs to reduce exposure from scoliosis radiographs,” Spine 17, 509–512 (1992).
[CrossRef]

Saltin, B.

B. Saltin, P. Gollnick, “Skeletal muscle adaptability: significance for metabolism and performance,” in Handbook of Physiology,L. D. Peachey, R. H. Adrian, S. R. Geiger, eds. (American Physiological Society, Bethesda, Maryland, 1983), pp. 555–631.

Sandeman, R. J.

Schumann, W.

W. Schumann, J. P. Zürcher, D. Cuche, Holography and Deformation Analysis (Springer-Verlag, Berlin, 1986).

W. Schumann, “Deformation measurement and analysis on several curved surfaces,” in Fringe ’97: Automatic Processing of Fringe Patterns,W. Jüptner, W. Osten (Akademie Verlag, Berlin, 1997), pp. 289–298.

Stevenson, J. M.

A. F. Mannion, G. A. Dumas, J. M. Stevenson, R. G. Cooper, “The influence of muscle fiber size and type distribution on electromyographic measures of back muscle fatigability,” Spine 23, 576–584 (1996).
[CrossRef]

Su, X. Y.

X. Y. Su, W. S. Zhou, G. Von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Takeda, M.

Tatasciore, P.

P. Tatasciore, E. K. Hack, “Projection moiré using tensor calculus for general geometries of optical setups,” Opt. Eng. 34, 1887–1899 (1995).
[CrossRef]

P. Tatasciore, “Récupération des franges d’interférence en interférométrie holographique appliquée aux grandes déformations des corps opaques,” Ph.D. Dissertation 8917 (Ecole Polytechnique Fédérale de Zurich, 1989).

Trolinger, J. D.

J. D. Trolinger, “Optical methods for wide scale displacement measurements,” in Fringe ’97: Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten (Akademie Verlag, Berlin, 1997), pp. 255–266.

Von Bally, G.

X. Y. Su, W. S. Zhou, G. Von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Vukicevic, D.

X. Y. Su, W. S. Zhou, G. Von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Yamaguchi, I.

I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics XXII,E. Wolf, ed. (North Holland, Amsterdam, Netherlands, 1985), pp. 273–336.

Zhou, W. S.

X. Y. Su, W. S. Zhou, G. Von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Zürcher, J. P.

W. Schumann, J. P. Zürcher, D. Cuche, Holography and Deformation Analysis (Springer-Verlag, Berlin, 1986).

Appl. Opt.

J. Eng. Mech.

A. Hanafi, J. Y. Cornu, T. Gharbi, A. Courteville, “An approach to the evaluation of hand strength and the ability to control force given visual biofeedback,” J. Eng. Mech. 123, 1215–1218 (1997).
[CrossRef]

J. Y. Cornu, A. Hanafi, T. Gharbi, “A rapid evaluation of individual hand strength,” J. Eng. Mech. 125, 1056–1061 (1999).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

X. Y. Su, W. S. Zhou, G. Von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Opt. Eng.

P. Tatasciore, E. K. Hack, “Projection moiré using tensor calculus for general geometries of optical setups,” Opt. Eng. 34, 1887–1899 (1995).
[CrossRef]

Spine

A. F. Mannion, G. A. Dumas, J. M. Stevenson, R. G. Cooper, “The influence of muscle fiber size and type distribution on electromyographic measures of back muscle fatigability,” Spine 23, 576–584 (1996).
[CrossRef]

T. R. Dangaria, O. Naesh, “Changes in cross-sectional area of Psoas major muscle in unilateral Sciateca caused by disc herniation,” Spine 23, 928–931 (1992).
[CrossRef]

T. E. Denton, F. M. Randall, D. A. Deinlein, “The use of instant moiré photographs to reduce exposure from scoliosis radiographs,” Spine 17, 509–512 (1992).
[CrossRef]

Other

B. Saltin, P. Gollnick, “Skeletal muscle adaptability: significance for metabolism and performance,” in Handbook of Physiology,L. D. Peachey, R. H. Adrian, S. R. Geiger, eds. (American Physiological Society, Bethesda, Maryland, 1983), pp. 555–631.

I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics XXII,E. Wolf, ed. (North Holland, Amsterdam, Netherlands, 1985), pp. 273–336.

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).

D. Post, B. Han, P. Ifju, High Sensitivity Moiré. Experimental Analysis for Mechanics and Materials (Springer-Verlag, Berlin, Germany, 1994).
[CrossRef]

J. D. Trolinger, “Optical methods for wide scale displacement measurements,” in Fringe ’97: Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten (Akademie Verlag, Berlin, 1997), pp. 255–266.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

W. Schumann, J. P. Zürcher, D. Cuche, Holography and Deformation Analysis (Springer-Verlag, Berlin, 1986).

W. Schumann, “Deformation measurement and analysis on several curved surfaces,” in Fringe ’97: Automatic Processing of Fringe Patterns,W. Jüptner, W. Osten (Akademie Verlag, Berlin, 1997), pp. 289–298.

P. Tatasciore, “Récupération des franges d’interférence en interférométrie holographique appliquée aux grandes déformations des corps opaques,” Ph.D. Dissertation 8917 (Ecole Polytechnique Fédérale de Zurich, 1989).

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Figures (10)

Fig. 1
Fig. 1

Spatial modulation of a fringe pattern in a triangular, nontelecentric system with cross-optical axes configuration.

Fig. 2
Fig. 2

Geometrical elements of the projection grating and the array of sensors.14 The fringe order in Pg is D(Pg) = (Δrg · gg)/p considering that D(P0g) = 0.

Fig. 3
Fig. 3

Geometrical configuration of a nontelecentric fringe projection system; S, R are, respectively, the projection centers of the projection and observation systems, which are located in the exit pupils.

Fig. 4
Fig. 4

(a) Determination of the grating pitches that satisfy the sampling condition p/m ≥ 2pc/mc for two projection lenses of 25 mm and 50 mm focal lengths, respectively; pc = 10 μm.

Fig. 5
Fig. 5

Computed sensitivity of the setup and its limits are plotted as function of the grating pitch for different variations of the height (10°, 20°, 30°, 40°): (a) 25 mm focal-length projection lens, (b) 50 mm focal-length projection lens.

Fig. 6
Fig. 6

Evaluation of the resolution of the system.

Fig. 7
Fig. 7

Three-dimensional reconstruction of a human hand.

Fig. 8
Fig. 8

Reconstruction of the lower back profile with dark skin.

Fig. 9
Fig. 9

Variation of the cross-section of forearm muscle group during grasping exertions: (a) intensity distribution during a grasping exertion at 30% of the maximum voluntary contraction (MVC); (b) intensity distribution during a grasping exertion at 40% of the MVC. (c) Profiles reconstruction of the forearm muscles before and after the contraction.

Fig. 10
Fig. 10

Distribution of the out-of-plane displacements generated by the contraction of the forearm muscles.

Tables (1)

Tables Icon

Table 1 Characteristics of Measurement System

Equations (34)

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

T g ( P g ) = 1 2 { 1 - cos [ 2 π D p ( P g ) ] } ,
I ( P c ) = I ( P ) = I s ( P ) T g ( P ) R ( P ) = A ( P ) 2 { 1 - cos [ 2 π D o ( P ) ] } ,
D p ( P ) = D p ( P g ) = 1 p g g · O g P g + constant ,
OH = N g O P 2 ,
H P 3 = - ( O P 2 · h m ) l g + L g O P 3 ,
N g O P 2 = [ 1 + ( O P 2 · h m ) l g + L g ] O P 3 .
D p ( P ) = D p ( P 2 ) = m / p [ 1 + ( O P 2 · h m ) / ( l g + L g ) ] g g N g · O P 2 + constant ,
δ D ( P ) = D o ( P ) - D p ( P ) = D p ( P 1 ) - D p ( P 2 ) .
δ D = m / p [ 1 + ( O P 2 · h m ) / ( l g + L g ) ] g g N g · P 2 P 1 m p g g N g · P 2 P 1 ,
P 2 P 1 = e ( Δ r · k m ) l c + L c + ( Δ r · k m ) q ,
δ D m p e ( Δ r · k m ) l c + L c + ( Δ r · k m ) g g N g · q .
δ D S · Δ r , S = m p e l c + L c ( g g · q ) k m ,
δ D ( P ) m p e l c + L c ( g g · q ) z ( P ) = S z ( P ) .
I ( P ) = A ( P ) 2 { 1 - cos [ 2 π D p ( P ) + 2 π δ D ( P ) ] } .
d ( δ D ) = ( δ D ) · d Δ r = S · d Δ r = S d z .
d I = π A sin { 2 π [ D p + δ D ] } d ( δ D ) .
min Δ ( δ D ) = 1 π max ( A ) .
D p = m p 1 m c g g N g · r c + constant ,
ν = α = m p 1 m c g g + 1 m c S ¯ z .
max ν · g c 1 2 p c ,
S max ( ¯ z · g c ) ( m c 2 p c - m p ) ,
gM · r = g ( I - k n k · n ) · r = ( gI - k · g k · n n ) · r = ( g · r ) - k · g k · n ( n · r ) .
g · Mr = g · ( I - k n k · n ) r = g · ( Ir - n · r k · n k ) = g · r - n · r k · n ( g · k ) .
D p ( r ) = a g g N g · u ( r )             with             N g = I - h m h m .
d D p ( r ) = D p ( r ) · d r .
D p ( r ) = a { [ ( g g N g ) ] u ( r ) + [ u ( r ) ] g g N g } = a g g N g [ u ( r ) ]
d D p ( r ) = a g g N g [ u ( r ) ] · d r .
u ( r ) = 1 b ( r ) ( r ) - 1 b 2 ( r ) { [ b ( r ) ] r } = 1 b ( r ) I - 1 b 2 ( r ) { [ b ( r ) ] r } .
b ( r ) = c { ( r ) h m + ( h m ) r } = c ( r ) h m = c I h m = c h m .
u ( r ) = 1 b ( r ) I - c b 2 ( r ) ( h m r ) .
d D p ( r ) = a g g N g { 1 b ( r ) I - c b 2 ( r ) ( h m r ) } · d r .
d D p ( r ) = a g g N g · d r b ( r ) .
N g q = q - ( h m h m ) q = q - ( q · h m ) h m .
g g N g · q = g g · q - ( q · h m ) ( g g · h m ) = g g · q ,

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