Abstract

The measurement of astigmatic lenses, optical surfaces or wavefronts are a highly studied problem and many different instruments have been commercially fabricated to perform this task. Many of them use a Hartmann arrangement to obtain the result. In this paper, we analyze with detail the algorithms that can be used to make the necessary calculations and propose several alternatives with different advantages and disadvantages. Different mathematical algorithms that are involved in the calculation process have been given whereas any description of the instrument itself is not proposed, but only the different mathematical algorithms that are involved in the calculation process.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. Ikezawa, E. Yanagi, K. Takeyuki, and Y. Eiichi, “Lens meter,” European patent EP 0 977 026 A1 (24November2010).
  2. K. Yoshimi, “Keratometer,” U.S. patent5,325,134 (28June1994).
  3. T. Iwane, “Automatic lens meter,” U.S. patent4,779,979 (25October1988).
  4. H. Akihiro, “Lens measurement apparatus providing measurements of multiple lens characteristics,” U.S. patent5,432,596 (11July1995).
  5. Y. Ueno, “Apparatus for measuring refractive power and radius of curvature of a lens,” U.S. patent5,742,381 (21April1998).
  6. S. Imaizumi, “Lens meter,” European patent EP 1 679 499 A2 (12July2006).
  7. S. E. Feldon, B. G. Broome, and D. A. Wallace, “Keratometer,” U.S. patent4,597,648 (1July1986).
  8. R. Snook, “Surgical keratometer system for measuring surface topography of a cornea during surgery,” U.S. patent5,735,283 (7April1998).
  9. Y. Fukuma, K. Matsumoto, Y. Ikezawa, E. Yanagi, and K. Takeyuki, “Layout judgment apparatus and layout judgment system,” U.S. patent5,973,772 (26October1999).
  10. M. Abitbol, E. Tederry, and E. Meimoun, “Methods and apparatus for measuring and mapping ophthalmic elements,” U.S. patent5,855,074 (5January1999).
  11. M. Abitbol, E. Tederry, and E. Meimoun, “Apparatus for mapping optical elements,” U.S. patent5,825,476 (20October1998).
  12. C. Castellini, F. Francini, and B. Tiribilli, “Hartmann test modification for measuring ophthalmic progressive lenses,” Appl. Opt. 33, 4120–4124 (1994).
    [CrossRef]
  13. B. C. Platt and R. V. Shack, “Lenticular Hartmann screen,” Optical Sciences Center Newsletter 5, 15–16 (1971).
  14. R. Cubalchini, “Modal wavefront estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  15. I. Ghozeil and J. E. Simmons, “Screen test for large mirrors,” Appl. Opt. 13, 1773–1777 (1974).
    [CrossRef]
  16. S. Rios, E. Acosta, and S. Bara, “Hartmann sensing with Albrecht grids,” Opt. Commun. 133, 443–453 (1997).
    [CrossRef]
  17. S. K. Mishra, A. K. Gupta, and A. Sharma, “An ocular wavefront sensor based on binary phase element: design and analysis,” J. Mod. Opt. 59, 1034–1048 (2012).
    [CrossRef]
  18. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. A 70, 998–1005 (1980).
    [CrossRef]
  19. D. Malacara-Doblado and I. Ghozeil, Optical Shop Testing, 3rd. ed. (Wiley, 2007), p. 361.
  20. S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 945–962 (1997).
    [CrossRef]
  21. S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. 74, 931–944 (1997).
    [CrossRef]
  22. Y. Mejía-Barbosa and D. Malacara-Hernández, “Object surface for applying a modified Hartmann test to measure corneal topography,” Appl. Opt. 40, 5778–5786 (2001).
    [CrossRef]

2012

S. K. Mishra, A. K. Gupta, and A. Sharma, “An ocular wavefront sensor based on binary phase element: design and analysis,” J. Mod. Opt. 59, 1034–1048 (2012).
[CrossRef]

2001

1997

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 945–962 (1997).
[CrossRef]

S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. 74, 931–944 (1997).
[CrossRef]

S. Rios, E. Acosta, and S. Bara, “Hartmann sensing with Albrecht grids,” Opt. Commun. 133, 443–453 (1997).
[CrossRef]

1994

1980

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. A 70, 998–1005 (1980).
[CrossRef]

1979

1974

1971

B. C. Platt and R. V. Shack, “Lenticular Hartmann screen,” Optical Sciences Center Newsletter 5, 15–16 (1971).

Abitbol, M.

M. Abitbol, E. Tederry, and E. Meimoun, “Methods and apparatus for measuring and mapping ophthalmic elements,” U.S. patent5,855,074 (5January1999).

M. Abitbol, E. Tederry, and E. Meimoun, “Apparatus for mapping optical elements,” U.S. patent5,825,476 (20October1998).

Acosta, E.

S. Rios, E. Acosta, and S. Bara, “Hartmann sensing with Albrecht grids,” Opt. Commun. 133, 443–453 (1997).
[CrossRef]

Akihiro, H.

H. Akihiro, “Lens measurement apparatus providing measurements of multiple lens characteristics,” U.S. patent5,432,596 (11July1995).

Bara, S.

S. Rios, E. Acosta, and S. Bara, “Hartmann sensing with Albrecht grids,” Opt. Commun. 133, 443–453 (1997).
[CrossRef]

Broome, B. G.

S. E. Feldon, B. G. Broome, and D. A. Wallace, “Keratometer,” U.S. patent4,597,648 (1July1986).

Castellini, C.

Cubalchini, R.

Eiichi, Y.

Y. Ikezawa, E. Yanagi, K. Takeyuki, and Y. Eiichi, “Lens meter,” European patent EP 0 977 026 A1 (24November2010).

Feldon, S. E.

S. E. Feldon, B. G. Broome, and D. A. Wallace, “Keratometer,” U.S. patent4,597,648 (1July1986).

Francini, F.

Fukuma, Y.

Y. Fukuma, K. Matsumoto, Y. Ikezawa, E. Yanagi, and K. Takeyuki, “Layout judgment apparatus and layout judgment system,” U.S. patent5,973,772 (26October1999).

Ghozeil, I.

I. Ghozeil and J. E. Simmons, “Screen test for large mirrors,” Appl. Opt. 13, 1773–1777 (1974).
[CrossRef]

D. Malacara-Doblado and I. Ghozeil, Optical Shop Testing, 3rd. ed. (Wiley, 2007), p. 361.

Gupta, A. K.

S. K. Mishra, A. K. Gupta, and A. Sharma, “An ocular wavefront sensor based on binary phase element: design and analysis,” J. Mod. Opt. 59, 1034–1048 (2012).
[CrossRef]

Ikezawa, Y.

Y. Ikezawa, E. Yanagi, K. Takeyuki, and Y. Eiichi, “Lens meter,” European patent EP 0 977 026 A1 (24November2010).

Y. Fukuma, K. Matsumoto, Y. Ikezawa, E. Yanagi, and K. Takeyuki, “Layout judgment apparatus and layout judgment system,” U.S. patent5,973,772 (26October1999).

Imaizumi, S.

S. Imaizumi, “Lens meter,” European patent EP 1 679 499 A2 (12July2006).

Iwane, T.

T. Iwane, “Automatic lens meter,” U.S. patent4,779,979 (25October1988).

Klein, S. A.

S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. 74, 931–944 (1997).
[CrossRef]

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 945–962 (1997).
[CrossRef]

Malacara-Doblado, D.

D. Malacara-Doblado and I. Ghozeil, Optical Shop Testing, 3rd. ed. (Wiley, 2007), p. 361.

Malacara-Hernández, D.

Matsumoto, K.

Y. Fukuma, K. Matsumoto, Y. Ikezawa, E. Yanagi, and K. Takeyuki, “Layout judgment apparatus and layout judgment system,” U.S. patent5,973,772 (26October1999).

Meimoun, E.

M. Abitbol, E. Tederry, and E. Meimoun, “Apparatus for mapping optical elements,” U.S. patent5,825,476 (20October1998).

M. Abitbol, E. Tederry, and E. Meimoun, “Methods and apparatus for measuring and mapping ophthalmic elements,” U.S. patent5,855,074 (5January1999).

Mejía-Barbosa, Y.

Mishra, S. K.

S. K. Mishra, A. K. Gupta, and A. Sharma, “An ocular wavefront sensor based on binary phase element: design and analysis,” J. Mod. Opt. 59, 1034–1048 (2012).
[CrossRef]

Platt, B. C.

B. C. Platt and R. V. Shack, “Lenticular Hartmann screen,” Optical Sciences Center Newsletter 5, 15–16 (1971).

Rios, S.

S. Rios, E. Acosta, and S. Bara, “Hartmann sensing with Albrecht grids,” Opt. Commun. 133, 443–453 (1997).
[CrossRef]

Shack, R. V.

B. C. Platt and R. V. Shack, “Lenticular Hartmann screen,” Optical Sciences Center Newsletter 5, 15–16 (1971).

Sharma, A.

S. K. Mishra, A. K. Gupta, and A. Sharma, “An ocular wavefront sensor based on binary phase element: design and analysis,” J. Mod. Opt. 59, 1034–1048 (2012).
[CrossRef]

Simmons, J. E.

Snook, R.

R. Snook, “Surgical keratometer system for measuring surface topography of a cornea during surgery,” U.S. patent5,735,283 (7April1998).

Southwell, W. H.

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. A 70, 998–1005 (1980).
[CrossRef]

Takeyuki, K.

Y. Fukuma, K. Matsumoto, Y. Ikezawa, E. Yanagi, and K. Takeyuki, “Layout judgment apparatus and layout judgment system,” U.S. patent5,973,772 (26October1999).

Y. Ikezawa, E. Yanagi, K. Takeyuki, and Y. Eiichi, “Lens meter,” European patent EP 0 977 026 A1 (24November2010).

Tederry, E.

M. Abitbol, E. Tederry, and E. Meimoun, “Apparatus for mapping optical elements,” U.S. patent5,825,476 (20October1998).

M. Abitbol, E. Tederry, and E. Meimoun, “Methods and apparatus for measuring and mapping ophthalmic elements,” U.S. patent5,855,074 (5January1999).

Tiribilli, B.

Ueno, Y.

Y. Ueno, “Apparatus for measuring refractive power and radius of curvature of a lens,” U.S. patent5,742,381 (21April1998).

Wallace, D. A.

S. E. Feldon, B. G. Broome, and D. A. Wallace, “Keratometer,” U.S. patent4,597,648 (1July1986).

Yanagi, E.

Y. Ikezawa, E. Yanagi, K. Takeyuki, and Y. Eiichi, “Lens meter,” European patent EP 0 977 026 A1 (24November2010).

Y. Fukuma, K. Matsumoto, Y. Ikezawa, E. Yanagi, and K. Takeyuki, “Layout judgment apparatus and layout judgment system,” U.S. patent5,973,772 (26October1999).

Yoshimi, K.

K. Yoshimi, “Keratometer,” U.S. patent5,325,134 (28June1994).

Appl. Opt.

J. Mod. Opt.

S. K. Mishra, A. K. Gupta, and A. Sharma, “An ocular wavefront sensor based on binary phase element: design and analysis,” J. Mod. Opt. 59, 1034–1048 (2012).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. A 70, 998–1005 (1980).
[CrossRef]

Opt. Commun.

S. Rios, E. Acosta, and S. Bara, “Hartmann sensing with Albrecht grids,” Opt. Commun. 133, 443–453 (1997).
[CrossRef]

Optical Sciences Center Newsletter

B. C. Platt and R. V. Shack, “Lenticular Hartmann screen,” Optical Sciences Center Newsletter 5, 15–16 (1971).

Optom. Vis. Sci.

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 945–962 (1997).
[CrossRef]

S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. 74, 931–944 (1997).
[CrossRef]

Other

D. Malacara-Doblado and I. Ghozeil, Optical Shop Testing, 3rd. ed. (Wiley, 2007), p. 361.

Y. Ikezawa, E. Yanagi, K. Takeyuki, and Y. Eiichi, “Lens meter,” European patent EP 0 977 026 A1 (24November2010).

K. Yoshimi, “Keratometer,” U.S. patent5,325,134 (28June1994).

T. Iwane, “Automatic lens meter,” U.S. patent4,779,979 (25October1988).

H. Akihiro, “Lens measurement apparatus providing measurements of multiple lens characteristics,” U.S. patent5,432,596 (11July1995).

Y. Ueno, “Apparatus for measuring refractive power and radius of curvature of a lens,” U.S. patent5,742,381 (21April1998).

S. Imaizumi, “Lens meter,” European patent EP 1 679 499 A2 (12July2006).

S. E. Feldon, B. G. Broome, and D. A. Wallace, “Keratometer,” U.S. patent4,597,648 (1July1986).

R. Snook, “Surgical keratometer system for measuring surface topography of a cornea during surgery,” U.S. patent5,735,283 (7April1998).

Y. Fukuma, K. Matsumoto, Y. Ikezawa, E. Yanagi, and K. Takeyuki, “Layout judgment apparatus and layout judgment system,” U.S. patent5,973,772 (26October1999).

M. Abitbol, E. Tederry, and E. Meimoun, “Methods and apparatus for measuring and mapping ophthalmic elements,” U.S. patent5,855,074 (5January1999).

M. Abitbol, E. Tederry, and E. Meimoun, “Apparatus for mapping optical elements,” U.S. patent5,825,476 (20October1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Sampling points distribution used at the entrance pupil of a Hartmann or Shack–Hartmann test when only the paraxial parameters are needed. (a) Three sampling points, (b) four sampling points, and (c) five sampling points.

Fig. 2.
Fig. 2.

Sampling points distribution used at the entrance pupil of a Hartmann test when the local curvatures over the whole pupil for a wavefront with high-order wavefront deformations are to be measured. (a) A square array and (b) a hexagonal array. In the case of the rectangular cells the algorithm for four points in Eq. (26) can be used. In the case of the hexagonal array, the algorithm for three points in Eq. (18) can be used. The sampling points are at the vertices of the triangles and the evaluated points are at the centers of the triangles, which are the vertices of a hexagonal cells array.

Fig. 3.
Fig. 3.

Three configurations for measuring instruments. (a) The first arrangement: classical Hartmann test, (b) second arrangement: array of light sources, and (c) the third arrangement: with a dot array in a screen.

Tables (1)

Tables Icon

Table 1. Calculation of Astigmatism and the Curvature with Three, Four, and Five Measuring Spots and with Two Different Values of ρ0

Equations (45)

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

W(x,y)=a0+a1x+a2y+a3x2+a4xy+a5y2.
W(ρ,θ)=b0+b1ρcosθ+b2ρsinθ+b3ρ2+b4ρ2sin2θ+b5ρ2cos2θ,
b1=a1,b2=a2,b3=a3+a52,b4=a42,b5=a3a52.
W(ρ,θ)=b0+b1ρcosθ+b2ρsinθ+b3ρ2+(b42+b52)1/2ρ2cos(2θ+ϕ)=b0+b1ρcosθ+b2ρsinθ+csρ22+ccρ22(1+cos(2θ+ϕ)),
cs=2[b3(b42+b52)1/2]cc=2(b42+b52)1/2ϕ=b5b4,
TAxr=W(x,y)x=a1+2a3x+a4y
TAyr=W(x,y)y=a2+a4x+2a5y.
TAρr=W(ρ,θ)ρ=b1cosθ+b2sinθ+2b3ρ+2b4ρsin2θ+2b5ρcos2θ
TAθr=1ρW(ρ,θ)θ=b1sinθ+b2cosθ2b4ρcos2θ+2b5ρsin2θ.
TAxar=a1+2a3xa+a4ya,TAxbr=a1+2a3xb+a4yb,TAxcr=a1+2a3xc+a4yc,TAyar=a2+a4xa+2a5ya,TAybr=a2+a4xb+2a5yb,TAycr=a2+a4xc+2a5yc.
a1=1r[(TAxaybTAxbya)(xbycxcyb)(TAxbycTAxcyb)(xaybxbya)(ybya)(xbycxcyb)(ycyb)(xaybxbya)],
a3=12r[(TAxaybTAxbya)(ycyb)(TAxbycTAxcyb)(ybya)(xaybxbya)(ycyb)(xbycxcyb)(ybya)],
a4=1r[(TAxaTAxb)(xbxc)(TAxbTAxc)(xaxb)(xbxc)(yayb)(xaxb)(ybyc)],
a2=1r[(TAyaxbTAybxa)(ybxcycxb)(TAybxcTAycxb)(yaxbybxa)(xbxa)(ybxcycxb)(xcxb)(yaxbybxa)],
a4=1r[(TAyaTAyb)(ybyc)(TAybTAyc)(yayb)(xaxb)(ybyc)(xbxc)(yayb)],
a5=12r[(TAyaxbTAybxa)(xcxb)(TAybxcTAycxb)(xbxa)(yaxbybxa)(xcxb)(ybxcycxb)(xbxa)],
xa=0;ya=ρ0xb=32ρ0;yb=12ρ0xc=32ρ0;yc=12ρ0,
a1=13r[TAxa+TAxb+TAxc],a2=13r[TAya+TAyb+TAyc],a3=123rρ0[TAxcTAxb],a4=12rρ0[2TAxaTAxbTAxc3TAybTAyc3],a5=16rρ0[2TAyaTAybTAyc].
a1=1r[(TAxbycTAxcyb)(xcydxdyc)(TAxcydTAxdyc)(xbycxcyb)(ycyb)(xcydxdyc)(ydyc)(xbycxcyb)],
a3=12r[(TAxbycTAxcyb)(ydyc)(TAxcydTAxdyc)(ycyb)(xbycxcyb)(ydyc)(xcydxdyc)(ycyb)],
a4=1r[(TAxbTAxc)(xcxd)(TAxcTAxd)(xbxc)(xcxd)(ybyc)(xbxc)(ycyd)],
a2=1r[(TAybxcTAycxb)(ycxdydxc)(TAycxdTAydxc)(ybxcycxb)(xcxb)(ycxdydxc)(xdxc)(ybxcycxb)],
a4=1r[(TAybTAyc)(ycyd)(TAycTAyd)(ybyc)(xbxc)(ycyd)(xcxd)(ybyc)],
a5=12r[(TAybxcTAycxb)(xdxc)(TAycxdTAydxc)(xcxb)(ybxcycxb)(xdxc)(ycxdydxc)(xcxb)],
a1=1r[(TAxcydTAxdyc)(xdyaxayd)(TAxdyaTAxayd)(xcydxdyc)(ydyc)(xdyaxayd)(yayd)(xcydxdyc)],
a3=12r[(TAxcydTAxdyc)(yayd)(TAxdyaTAxayd)(ydyc)(xcydxdyc)(yayd)(xdyaxayd)(ydya)],
a4=1r[(TAxcTAxd)(xdxa)(TAxdTAxa)(xcxd)(xdxa)(ycyd)(xcxd)(ydya)],
a2=1r[(TAycxdTAydxc)(ydxayaxd)(TAydxaTAyaxd)(ycxdydxc)(xdxc)(ydxayaxd)(xaxd)(ycxdydxc)],
a4=1r[(TAycTAyd)(ydya)(TAydTAya)(ycyd)(xcxd)(ydya)(xdxa)(ycyd)],
a5=12r[(TAycxdTAydxc)(xaxd)(TAydxaTAyaxd)(xdxc)(ycxdydxc)(xaxd)(ydxayaxd)(xdxc)].
a1=1r[(TAxdyaTAxayd)(xaybxbya)(TAxaybTAxbya)(xdyaxayd)(yayd)(xaybxbya)(ybya)(xdyaxayd)],
a3=12r[(TAxdyaTAxayd)(ybya)(TAxaybTAxbya)(yayd)(xdyaxayd)(ybya)(xaybxbya)(yayb)],
a4=1r[(TAxdTAxa)(xaxb)(TAxaTAxb)(xdxa)(xaxb)(ydya)(xdxa)(yayb)],
a2=1r[(TAydxaTAyaxd)(yaxbybxa)(TAyaxbTAybxa)(ydxayaxd)(xaxd)(yaxbybxa)(xbxa)(ydxayaxd)],
a4=1r[(TAydTAya)(yayb)(TAyaTAyb)(ydya)(xdxa)(yayb)(xaxb)(ydya)],
a5=12r[(TAydxaTAyaxd)(xbxa)(TAyaxbTAybxa)(xaxd)(ydxayaxd)(xbxa)(yaxbybxa)(xaxd)].
xa=ρ02;ya=ρ02xb=ρ02;yb=ρ02xc=ρ02;yc=ρ02xd=ρ02;yd=ρ02.
a1=12r[TAxa+TAxb+TAxc+TAxd],a2=12r[TAya+TAyb+TAyc+TAyd],a3=142rρ0[TAxa+TAxbTAxc+TAxd],a4=142rρ0[TAxaTAxc+TAxbTAxdTAya+TAybTAyc+TAyd],a5=142rρ0[TAyaTAyc+TAybTAyd].
TAρar=b1cosθa+b2sinθa+2b3ρa+2b4ρasin2θa+2b5ρacos2θa,TAρbr=b1cosθb+b2sinθb+2b3ρb+2b4ρbsin2θb+2b5ρbcos2θb,TAρcr=b1cosθc+b2sinθc+2b3ρc+2b4ρcsin2θc+2b5ρccos2θc,TAρdr=b1cosθd+b2sinθd+2b3ρc+2b4ρ0sin2θd+2b5ρccos2θd,TAρer=b1cosθe+b2sinθe+2b3ρd+2b4ρdsin2θe+2b5ρdcos2θe.
[b1b2b3b4b5]=1r[cos0°sin0°2ρ02ρ0sin0°2ρ0cos0°cos72°sin72°2ρ02ρ0sin144°2ρ0cos144°cos144°sin144°2ρ02ρ0sin288°2ρ0cos288°cos216°sin216°2ρ02ρ0sin432°2ρ0cos432°cos288°sin288°2ρ02ρ0sin576°2ρ0cos576°]1[TAρaTAρbTAρcTAρdTAρe].
[b1b2b3b4b5]=1r[102ρ002ρ0sin18°cos18°2ρ02ρ0sin36°2ρ0cos36°cos36°sin36°2ρ02ρ0cos18°2ρ0sin18°cos36°sin36°2ρ02ρ0cos18°2ρ0sin18°sin18°cos18°2ρ02ρ0sin36°2ρ0cos36°]1[TAρaTAρbTAρcTAρdTAρe].
b1=120r[8TAρa(1+5)((3+5)TAρb+2(TAρc+TAρd)+(3+5)TAρe)],b2=55(TAρcTAρd)+5+5(TAρbTAρe)52r,b3=TAρa+TAρb+TAρc+TAρd+TAρe10rρ0,b4=(1+5)TAρb2TAρc+2TAρd+(15)TAρe250105rρ0,b5=120rρ0[4TAρa(1+5)TAρb+(1+5)TAρc+(1+5)TAρd(1+5)TAρe].
TAθar=b1sinθa+b2cosθa2b4ρacos2θa+2b5ρasin2θa,TAθbr=b1sinθb+b2cosθb2b4ρbcos2θb+2b5ρbsin2θb,TAθcr=b1sinθc+b2cosθc2b4ρccos2θc+2b5ρcsin2θc,TAθdr=b1sinθd+b2cosθd2b4ρdcos2θd+2b5ρdsin2θd,
[b1b2b4b5]=1r[012ρ00cos18°sin18°2ρ0cos36°2ρ0sin36°sin36°cos36°2ρ0sin18°2ρ0cos18°sin36°cos36°2ρ0sin18°2ρ0cos18°]1[TAθaTAθbTAθdTAθe],
b1=TAθa+TAθb2r,b2=TAθb+TAθc2r,b4=(1+2)TAθa+22TAθb+(1+2)TAθc+22TAθd4rρ0b5=TAθaTAθc4rρ0.

Metrics