Abstract

An algorithm to compute the sagittal and meridional radii of curvature for a surface of revolution is presented. The sagittal radius is obtained from the surface normal, and the meridional radius is calculated from a function fitted to the derivative of the sagittal curvature by using the surface-normals raw data. A calibration spherical surface is tested by using the null-screen testing method. Experimental results of the spherical surface show that the sagittal and meridional radii of curvature differ by 2.600% and 2.604%, respectively, with respect to the actual radius of the calibration spherical surface.

© 2013 Optical Society of America

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References

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  1. R. Díaz-Uribe and M. Campos-García, “Null-screen testing of fast convex aspheric surface,” Appl. Opt. 39, 2670–2677 (2000).
    [CrossRef]
  2. M. Campos-García, R. Díaz-Uribe, and F. Granados-Agustín, “Testing fast aspheric convex surfaces with a linear array of sources,” Appl. Opt. 43, 6255–6264 (2004).
    [CrossRef]
  3. L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screen used for testing convex optical surfaces,” Rev. Mex. Fís. 53, 421–430 (2007).
  4. M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47, 849–859 (2008).
    [CrossRef]
  5. I. Funes-Maderey, “Flat field videokeratometry,” B. Sc. dissertation (College of Sciences of The Autonomous National University of Mexico, 1998).
  6. R. Colín, “New developments on flat field videokeratometry,” B. Sc. dissertation (College of Sciences of The Autonomous National University of Mexico, 2007).
  7. A. Estrada-Molina, “Design and construction of a portable videokeratometer for neonates,” M. Sc. dissertation (Graduate School of The Autonomous National University of Mexico, 2010).
  8. M. Campos-García, A. Estrada-Molina, and R. Díaz-Uribe, “New null screen design for corneal topography,” Proc. SPIE 8011, 801124 (2011).
    [CrossRef]
  9. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley-VCH, 2006).
  10. D. J. Struik, Lectures on Classical Differential Geometry(Addison-Wesley, 1961).
  11. M. P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice Hall, 1976).
  12. A. Estrada-Molina and R. Díaz-Uribe, “Tangential and sagittal curvature from the normals computed by the null screen method in corneal topography,” Proc. SPIE 8011, 80119J (2011).
    [CrossRef]
  13. Y. Mejía-Barbosa and D. Malacara-Hernández, “A review of methods for measuring corneal topography,” Optom. Vis. Sci. 78, 240–253 (2001).
    [CrossRef]
  14. O. Cardona-Nuñez, J. Pedraza-Contreras, A. Cornejo-Rodríguez, and A. Cordero-Dávila, “Significado de las superficies causticas en óptica,” Rev. Mex. Fís. 29, 245–258 (1983).
  15. S. A. Klein and R. B. Mandell, “Axial and instantaneous power conversion in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2155–2159 (1995).
  16. R. Díaz-Uribe, “Medium-precision null-screen testing of off axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. 39, 2790–2804 (2000).
    [CrossRef]
  17. C. Menchaca and D. Malacara, “Directional curvature in a conic mirror,” Appl. Opt. 23, 3258–3260 (1984).
    [CrossRef]
  18. K. S. Choi, E. Y. Lam, and K. K. Y. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 14, 11551–11565(2006).
    [CrossRef]
  19. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1992).
  20. J. R. Taylor, An Introduction to Error Analysis (University Science, 1997).
  21. 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]
  22. B. A. Barsky, S. A. Klein, and D. D. Garcia, “Gaussian power with cylinder vector field representation for corneal topography maps,” Optom. Vis. Sci. 74, 917–925 (1997).
    [CrossRef]
  23. S. A. Klein, “A corneal topography algorithm that produces continuous curvature,” Optom. Vis. Sci. 69, 829–834 (1992).
    [CrossRef]
  24. T. O. Salmon and D. G. Horner, “Comparison of elevation, curvature, and power descriptors for corneal topographic mapping,” Optom. Vis. Sci. 72, 800–808 (1995).
    [CrossRef]
  25. Y. Mejía and J. C. Galeano, “Corneal topographer based on the Hartmann test,” Optom. Vis. Sci. 86, 370–381 (2009).
    [CrossRef]
  26. C. Roberts, “The accuracy of ‘power’ maps to display curvature data in corneal topography data,” Invest. Ophthalmol. Visual Sci. 35, 3525–3532 (1994).
  27. J. Schwiegerling and J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vis. Sci. 74, 906–916 (1997).
    [CrossRef]
  28. S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. 74, 931–944(1997).
    [CrossRef]
  29. D. Malacara, “Mathematical representation of an optical surface and its characteristics,” in Optical Shop Testing, D. Malacara, ed., 3rd ed. (Wiley, 2007), pp. 832–851.

2011 (2)

M. Campos-García, A. Estrada-Molina, and R. Díaz-Uribe, “New null screen design for corneal topography,” Proc. SPIE 8011, 801124 (2011).
[CrossRef]

A. Estrada-Molina and R. Díaz-Uribe, “Tangential and sagittal curvature from the normals computed by the null screen method in corneal topography,” Proc. SPIE 8011, 80119J (2011).
[CrossRef]

2009 (1)

Y. Mejía and J. C. Galeano, “Corneal topographer based on the Hartmann test,” Optom. Vis. Sci. 86, 370–381 (2009).
[CrossRef]

2008 (1)

M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47, 849–859 (2008).
[CrossRef]

2007 (1)

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screen used for testing convex optical surfaces,” Rev. Mex. Fís. 53, 421–430 (2007).

2006 (1)

K. S. Choi, E. Y. Lam, and K. K. Y. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 14, 11551–11565(2006).
[CrossRef]

2004 (1)

M. Campos-García, R. Díaz-Uribe, and F. Granados-Agustín, “Testing fast aspheric convex surfaces with a linear array of sources,” Appl. Opt. 43, 6255–6264 (2004).
[CrossRef]

2001 (2)

Y. Mejía-Barbosa and D. Malacara-Hernández, “A review of methods for measuring corneal topography,” Optom. Vis. Sci. 78, 240–253 (2001).
[CrossRef]

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]

2000 (2)

R. Díaz-Uribe, “Medium-precision null-screen testing of off axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. 39, 2790–2804 (2000).
[CrossRef]

R. Díaz-Uribe and M. Campos-García, “Null-screen testing of fast convex aspheric surface,” Appl. Opt. 39, 2670–2677 (2000).
[CrossRef]

1997 (3)

B. A. Barsky, S. A. Klein, and D. D. Garcia, “Gaussian power with cylinder vector field representation for corneal topography maps,” Optom. Vis. Sci. 74, 917–925 (1997).
[CrossRef]

J. Schwiegerling and J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vis. Sci. 74, 906–916 (1997).
[CrossRef]

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

1995 (2)

S. A. Klein and R. B. Mandell, “Axial and instantaneous power conversion in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2155–2159 (1995).

T. O. Salmon and D. G. Horner, “Comparison of elevation, curvature, and power descriptors for corneal topographic mapping,” Optom. Vis. Sci. 72, 800–808 (1995).
[CrossRef]

1994 (1)

C. Roberts, “The accuracy of ‘power’ maps to display curvature data in corneal topography data,” Invest. Ophthalmol. Visual Sci. 35, 3525–3532 (1994).

1992 (1)

S. A. Klein, “A corneal topography algorithm that produces continuous curvature,” Optom. Vis. Sci. 69, 829–834 (1992).
[CrossRef]

1984 (1)

C. Menchaca and D. Malacara, “Directional curvature in a conic mirror,” Appl. Opt. 23, 3258–3260 (1984).
[CrossRef]

1983 (1)

O. Cardona-Nuñez, J. Pedraza-Contreras, A. Cornejo-Rodríguez, and A. Cordero-Dávila, “Significado de las superficies causticas en óptica,” Rev. Mex. Fís. 29, 245–258 (1983).

Barsky, B. A.

B. A. Barsky, S. A. Klein, and D. D. Garcia, “Gaussian power with cylinder vector field representation for corneal topography maps,” Optom. Vis. Sci. 74, 917–925 (1997).
[CrossRef]

Bevington, P. R.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1992).

Bolado-Gómez, R.

M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47, 849–859 (2008).
[CrossRef]

Campos-García, M.

M. Campos-García, A. Estrada-Molina, and R. Díaz-Uribe, “New null screen design for corneal topography,” Proc. SPIE 8011, 801124 (2011).
[CrossRef]

M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47, 849–859 (2008).
[CrossRef]

M. Campos-García, R. Díaz-Uribe, and F. Granados-Agustín, “Testing fast aspheric convex surfaces with a linear array of sources,” Appl. Opt. 43, 6255–6264 (2004).
[CrossRef]

R. Díaz-Uribe and M. Campos-García, “Null-screen testing of fast convex aspheric surface,” Appl. Opt. 39, 2670–2677 (2000).
[CrossRef]

Cardona-Nuñez, O.

O. Cardona-Nuñez, J. Pedraza-Contreras, A. Cornejo-Rodríguez, and A. Cordero-Dávila, “Significado de las superficies causticas en óptica,” Rev. Mex. Fís. 29, 245–258 (1983).

Carmona-Paredes, L.

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screen used for testing convex optical surfaces,” Rev. Mex. Fís. 53, 421–430 (2007).

Choi, K. S.

K. S. Choi, E. Y. Lam, and K. K. Y. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 14, 11551–11565(2006).
[CrossRef]

Colín, R.

R. Colín, “New developments on flat field videokeratometry,” B. Sc. dissertation (College of Sciences of The Autonomous National University of Mexico, 2007).

Cordero-Dávila, A.

O. Cardona-Nuñez, J. Pedraza-Contreras, A. Cornejo-Rodríguez, and A. Cordero-Dávila, “Significado de las superficies causticas en óptica,” Rev. Mex. Fís. 29, 245–258 (1983).

Cornejo-Rodríguez, A.

O. Cardona-Nuñez, J. Pedraza-Contreras, A. Cornejo-Rodríguez, and A. Cordero-Dávila, “Significado de las superficies causticas en óptica,” Rev. Mex. Fís. 29, 245–258 (1983).

Díaz-Uribe, R.

A. Estrada-Molina and R. Díaz-Uribe, “Tangential and sagittal curvature from the normals computed by the null screen method in corneal topography,” Proc. SPIE 8011, 80119J (2011).
[CrossRef]

M. Campos-García, A. Estrada-Molina, and R. Díaz-Uribe, “New null screen design for corneal topography,” Proc. SPIE 8011, 801124 (2011).
[CrossRef]

M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47, 849–859 (2008).
[CrossRef]

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screen used for testing convex optical surfaces,” Rev. Mex. Fís. 53, 421–430 (2007).

M. Campos-García, R. Díaz-Uribe, and F. Granados-Agustín, “Testing fast aspheric convex surfaces with a linear array of sources,” Appl. Opt. 43, 6255–6264 (2004).
[CrossRef]

R. Díaz-Uribe and M. Campos-García, “Null-screen testing of fast convex aspheric surface,” Appl. Opt. 39, 2670–2677 (2000).
[CrossRef]

R. Díaz-Uribe, “Medium-precision null-screen testing of off axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. 39, 2790–2804 (2000).
[CrossRef]

do Carmo, M. P.

M. P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice Hall, 1976).

Estrada-Molina, A.

A. Estrada-Molina and R. Díaz-Uribe, “Tangential and sagittal curvature from the normals computed by the null screen method in corneal topography,” Proc. SPIE 8011, 80119J (2011).
[CrossRef]

M. Campos-García, A. Estrada-Molina, and R. Díaz-Uribe, “New null screen design for corneal topography,” Proc. SPIE 8011, 801124 (2011).
[CrossRef]

A. Estrada-Molina, “Design and construction of a portable videokeratometer for neonates,” M. Sc. dissertation (Graduate School of The Autonomous National University of Mexico, 2010).

Funes-Maderey, I.

I. Funes-Maderey, “Flat field videokeratometry,” B. Sc. dissertation (College of Sciences of The Autonomous National University of Mexico, 1998).

Galeano, J. C.

Y. Mejía and J. C. Galeano, “Corneal topographer based on the Hartmann test,” Optom. Vis. Sci. 86, 370–381 (2009).
[CrossRef]

Garcia, D. D.

B. A. Barsky, S. A. Klein, and D. D. Garcia, “Gaussian power with cylinder vector field representation for corneal topography maps,” Optom. Vis. Sci. 74, 917–925 (1997).
[CrossRef]

Granados-Agustín, F.

M. Campos-García, R. Díaz-Uribe, and F. Granados-Agustín, “Testing fast aspheric convex surfaces with a linear array of sources,” Appl. Opt. 43, 6255–6264 (2004).
[CrossRef]

Greivenkamp, J. E.

J. Schwiegerling and J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vis. Sci. 74, 906–916 (1997).
[CrossRef]

Horner, D. G.

T. O. Salmon and D. G. Horner, “Comparison of elevation, curvature, and power descriptors for corneal topographic mapping,” Optom. Vis. Sci. 72, 800–808 (1995).
[CrossRef]

Klein, S. A.

B. A. Barsky, S. A. Klein, and D. D. Garcia, “Gaussian power with cylinder vector field representation for corneal topography maps,” Optom. Vis. Sci. 74, 917–925 (1997).
[CrossRef]

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

S. A. Klein and R. B. Mandell, “Axial and instantaneous power conversion in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2155–2159 (1995).

S. A. Klein, “A corneal topography algorithm that produces continuous curvature,” Optom. Vis. Sci. 69, 829–834 (1992).
[CrossRef]

Lam, E. Y.

K. S. Choi, E. Y. Lam, and K. K. Y. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 14, 11551–11565(2006).
[CrossRef]

Malacara, D.

C. Menchaca and D. Malacara, “Directional curvature in a conic mirror,” Appl. Opt. 23, 3258–3260 (1984).
[CrossRef]

D. Malacara, “Mathematical representation of an optical surface and its characteristics,” in Optical Shop Testing, D. Malacara, ed., 3rd ed. (Wiley, 2007), pp. 832–851.

Malacara-Hernández, D.

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]

Y. Mejía-Barbosa and D. Malacara-Hernández, “A review of methods for measuring corneal topography,” Optom. Vis. Sci. 78, 240–253 (2001).
[CrossRef]

Mandell, R. B.

S. A. Klein and R. B. Mandell, “Axial and instantaneous power conversion in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2155–2159 (1995).

Mejía, Y.

Y. Mejía and J. C. Galeano, “Corneal topographer based on the Hartmann test,” Optom. Vis. Sci. 86, 370–381 (2009).
[CrossRef]

Mejía-Barbosa, Y.

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]

Y. Mejía-Barbosa and D. Malacara-Hernández, “A review of methods for measuring corneal topography,” Optom. Vis. Sci. 78, 240–253 (2001).
[CrossRef]

Menchaca, C.

C. Menchaca and D. Malacara, “Directional curvature in a conic mirror,” Appl. Opt. 23, 3258–3260 (1984).
[CrossRef]

Pedraza-Contreras, J.

O. Cardona-Nuñez, J. Pedraza-Contreras, A. Cornejo-Rodríguez, and A. Cordero-Dávila, “Significado de las superficies causticas en óptica,” Rev. Mex. Fís. 29, 245–258 (1983).

Roberts, C.

C. Roberts, “The accuracy of ‘power’ maps to display curvature data in corneal topography data,” Invest. Ophthalmol. Visual Sci. 35, 3525–3532 (1994).

Robinson, D. K.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1992).

Salmon, T. O.

T. O. Salmon and D. G. Horner, “Comparison of elevation, curvature, and power descriptors for corneal topographic mapping,” Optom. Vis. Sci. 72, 800–808 (1995).
[CrossRef]

Schwiegerling, J.

J. Schwiegerling and J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vis. Sci. 74, 906–916 (1997).
[CrossRef]

Stavroudis, O. N.

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley-VCH, 2006).

Struik, D. J.

D. J. Struik, Lectures on Classical Differential Geometry(Addison-Wesley, 1961).

Taylor, J. R.

J. R. Taylor, An Introduction to Error Analysis (University Science, 1997).

Wong, K. K. Y.

K. S. Choi, E. Y. Lam, and K. K. Y. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 14, 11551–11565(2006).
[CrossRef]

Appl. Opt. (6)

M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47, 849–859 (2008).
[CrossRef]

R. Díaz-Uribe and M. Campos-García, “Null-screen testing of fast convex aspheric surface,” Appl. Opt. 39, 2670–2677 (2000).
[CrossRef]

M. Campos-García, R. Díaz-Uribe, and F. Granados-Agustín, “Testing fast aspheric convex surfaces with a linear array of sources,” Appl. Opt. 43, 6255–6264 (2004).
[CrossRef]

R. Díaz-Uribe, “Medium-precision null-screen testing of off axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. 39, 2790–2804 (2000).
[CrossRef]

C. Menchaca and D. Malacara, “Directional curvature in a conic mirror,” Appl. Opt. 23, 3258–3260 (1984).
[CrossRef]

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]

Invest. Ophthalmol. Visual Sci. (2)

C. Roberts, “The accuracy of ‘power’ maps to display curvature data in corneal topography data,” Invest. Ophthalmol. Visual Sci. 35, 3525–3532 (1994).

S. A. Klein and R. B. Mandell, “Axial and instantaneous power conversion in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2155–2159 (1995).

Opt. Express (1)

K. S. Choi, E. Y. Lam, and K. K. Y. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 14, 11551–11565(2006).
[CrossRef]

Optom. Vis. Sci. (7)

Y. Mejía-Barbosa and D. Malacara-Hernández, “A review of methods for measuring corneal topography,” Optom. Vis. Sci. 78, 240–253 (2001).
[CrossRef]

J. Schwiegerling and J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vis. Sci. 74, 906–916 (1997).
[CrossRef]

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

B. A. Barsky, S. A. Klein, and D. D. Garcia, “Gaussian power with cylinder vector field representation for corneal topography maps,” Optom. Vis. Sci. 74, 917–925 (1997).
[CrossRef]

S. A. Klein, “A corneal topography algorithm that produces continuous curvature,” Optom. Vis. Sci. 69, 829–834 (1992).
[CrossRef]

T. O. Salmon and D. G. Horner, “Comparison of elevation, curvature, and power descriptors for corneal topographic mapping,” Optom. Vis. Sci. 72, 800–808 (1995).
[CrossRef]

Y. Mejía and J. C. Galeano, “Corneal topographer based on the Hartmann test,” Optom. Vis. Sci. 86, 370–381 (2009).
[CrossRef]

Proc. SPIE (2)

A. Estrada-Molina and R. Díaz-Uribe, “Tangential and sagittal curvature from the normals computed by the null screen method in corneal topography,” Proc. SPIE 8011, 80119J (2011).
[CrossRef]

M. Campos-García, A. Estrada-Molina, and R. Díaz-Uribe, “New null screen design for corneal topography,” Proc. SPIE 8011, 801124 (2011).
[CrossRef]

Rev. Mex. Fís. (2)

O. Cardona-Nuñez, J. Pedraza-Contreras, A. Cornejo-Rodríguez, and A. Cordero-Dávila, “Significado de las superficies causticas en óptica,” Rev. Mex. Fís. 29, 245–258 (1983).

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screen used for testing convex optical surfaces,” Rev. Mex. Fís. 53, 421–430 (2007).

Other (9)

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1992).

J. R. Taylor, An Introduction to Error Analysis (University Science, 1997).

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley-VCH, 2006).

D. J. Struik, Lectures on Classical Differential Geometry(Addison-Wesley, 1961).

M. P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice Hall, 1976).

I. Funes-Maderey, “Flat field videokeratometry,” B. Sc. dissertation (College of Sciences of The Autonomous National University of Mexico, 1998).

R. Colín, “New developments on flat field videokeratometry,” B. Sc. dissertation (College of Sciences of The Autonomous National University of Mexico, 2007).

A. Estrada-Molina, “Design and construction of a portable videokeratometer for neonates,” M. Sc. dissertation (Graduate School of The Autonomous National University of Mexico, 2010).

D. Malacara, “Mathematical representation of an optical surface and its characteristics,” in Optical Shop Testing, D. Malacara, ed., 3rd ed. (Wiley, 2007), pp. 832–851.

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Figures (9)

Fig. 1.
Fig. 1.

Surface normal at an evaluation point.

Fig. 2.
Fig. 2.

Layout for the surface-normal evaluation in the meridional plane (rr, ri, I do not necessarily lie in the meridional plane).

Fig. 3.
Fig. 3.

Sagittal and meridional planes for a surface of revolution.

Fig. 4.
Fig. 4.

Profiles of the conic surfaces.

Fig. 5.
Fig. 5.

(a) Flat-printed null screen. (b) Image captured of the steel ball. (c) Picture of the experimental setup.

Fig. 6.
Fig. 6.

Plot of the coordinates of the experimental centroids (open circles) and ideal centroids (solid circles) on the image plane.

Fig. 7.
Fig. 7.

Sagittal radius of curvature versus the radial distance.

Fig. 8.
Fig. 8.

Meridional radius of curvature versus the radial distance.

Fig. 9.
Fig. 9.

(a) Plot of the sagittal radius obtained from the IFD procedure (solid squares) and the surface-normals data (open squares) versus the radial distance. (b) Plot of the meridional radius obtained from the IFD procedure (solid circles) and from surface-normals data (open circles) versus the radial distance.

Tables (3)

Tables Icon

Table 1. Design Parameters for the Test of the Spherical Surface

Tables Icon

Table 2. Deviations Obtained for the Steel Ball with Respect to Its Actual Radius r=6.375mm

Tables Icon

Table 3. Mean and Standard Deviations of the Measurements

Equations (37)

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

N=r^rr^i.
r^r=(x1,y1,a)(x12+y12+a2)1/2.
ψ(xs,ys,zs)=(k+1)(zsz0)22r(zsz0)+(xsx0)2+(ysy0)2=0,
I^=(xsx3,ysy3,zsz3)[(xsx3)2+(ysy3)2+(zsz3)2]1/2,
xs=tx1,
ys=ty1,
zs=(at+b),
t=λ+(λ24αβ)1/22α,
α=x12+y12+(k+1)a2,
λ=2[(k+1)(b+z0)a+ra(x1x0+y1y0)],
β=(k+1)(b+z0)2+2r(b+z0)+x02+y02.
Na=(Nax,Nay,Naz)=r^rI^,
Nax=x1(x12+y12+a2)1/2+(x3xs)[(xsx3)2+(ysy3)2+(zsz3)2]1/2,
Nay=y1(x12+y12+a2)1/2+(y3ys)[(xsx3)2+(ysy3)2+(zsz3)2]1/2,
Naz=a(x12+y12+a2)1/2+(z3zs)[(xsx3)2+(ysy3)2+(zsz3)2]1/2.
x=ρcosϕ,
y=ρsinϕ,
z=f(ρ),
κsag=dz/dρρ[1+(dz/dρ)2]1/2,
κmer=d2z/dρ2[1+(dz/dρ)2]3/2.
κmer=κsag+ρdκsagdρ.
dz=(NxNzdx+NyNzdy).
dz=zxdx+zydy,
zx=NxNz,
zy=NyNz.
dzdρ=zxdxdρ+zydydρ.
dzdρ=1(x2+y2)1/2[xNxNz+yNyNz],
κsag=xNxNz+yNyNz(x2+y2)1/2[x2+y2+(xNxNz+yNyNz)2]1/2.
κmer=(x2+y2)1/2{x2x(NxNz)+y2y(NyNz)+xy[y(NxNz)+x(NyNz)][x2+y2+(xNxNz+yNyNz)2]3/2}.
N^=(x,y,(k+1)zr){x2+y2+[(k+1)zr]2}1/2.
NxNz=x[r2(k+1)(x2+y2)]1/2,
NyNz=y[r2(k+1)(x2+y2)]1/2.
κsag=1[r2k(x2+y2)]1/2,
κmer=r2[r2k(x2+y2)]3/2.
η(ρ)(NxNz)2+(NyNz)2=x2+y2[r2(k+1)(x2+y2)]=ρ2[r2(k+1)ρ2].
κmer=κsag+kfit(x2+y2)[rfit2kfit(x2+y2)]3/2,
z=1kasp+1[rasprasp2(kasp+1)ρ2]+A1ρ4+A2ρ6+A3ρ8+A4ρ10+z0,

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