Abstract

The cylindrical coordinate machining method (CCM) is systematically studied in generating optical freeform surfaces, in which the feature points are fitted to typical Non-Uniform Rational B-Splines (NURBS). The given points have the mapping coordinates in the variable space using the point inversion technique, while the other points have their NURBS coordinates due to the interpolation technique. The derivation and mathematical features are obtained using the fitting formula. The compensation and optimized values for tool geometry are studied using a proposed sectional curve method for fabricating designed surfaces. Typical freeform surfaces fabricated by the CCM method are presented.

© 2008 Optical Society of America

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References

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  1. X. Jiang, P. Scott, and D. Whitehouse, "Freeform surface characterisation - A fresh strategy," Ann. CIRP 56, 553-556 (2007).
    [CrossRef]
  2. H. N. Hansen, K. Carneiro, H. Haitjema, and L. De Chiffre, "Dimensional micro and nano metrology," Ann. CIRP 55, 721-743 (2006).
    [CrossRef]
  3. L. De Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, "Surfaces in precision engineering, microengineering and nanotechnology," Ann. CIRP 52, 561-578 (2003).
    [CrossRef]
  4. Y. Takeuchia, S. Maedaa, T. Kawaib, K. Sawadab, "Manufacture of multiple-focus micro Fresnel lenses by means of nonrotational diamond grooving," Ann. CIRP 51, 343-346 (2002).
    [CrossRef]
  5. C. C. Chen, C. M. Chen, and J. R. Chen, "Toolpath generation of diamond shaping of aspheric lens array," J. Mater. Proc. Technol. 192-193, 194-199 (2007).
    [CrossRef]
  6. H. B. Wu, Z. Q. Wang, R. L. Fu, and J. Liu, "Design of a hybrid diffractive/refractive achromatized telecentric f-θ lens," Optik 117, 271-276 (2006).
    [CrossRef]
  7. L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, "Designing reflectors to generate a line-shaped directivity diagram," J. Mod. Opt. 52, 1529-1536 (2005).
    [CrossRef]
  8. C. F. Cheung, L. B. Kong, and W. B. Lee, "Modelling and simulation of freeform surface generation in ultra-precision raster milling," J. Eng. Manuf. 220, 1787-1801 (2006).
    [CrossRef]
  9. L. Piegl and W. Tiller, "The NURBS Book," (New York: Springer-Verlag, 1997).
  10. Q8. W. Ma, J. P. Kruth, "NURBS curve and surface fitting for reverse engineering," J. Adv. Manuf. Technol. 14, 918-927 (2005).
    [CrossRef]
  11. Y. L. Ma, W. T. Hewitt, "Point inversion and projection for NURBS curve and surface: control polygon approach," Comput. Aided Geom. Des. 20, 79-99 (2003).
    [CrossRef]
  12. K. H. Jeong, J. Kim, L. P. Lee, "Biological inspired artificial compound eyes," Science 312, 557-561 (2006).
    [CrossRef] [PubMed]

2007 (2)

X. Jiang, P. Scott, and D. Whitehouse, "Freeform surface characterisation - A fresh strategy," Ann. CIRP 56, 553-556 (2007).
[CrossRef]

C. C. Chen, C. M. Chen, and J. R. Chen, "Toolpath generation of diamond shaping of aspheric lens array," J. Mater. Proc. Technol. 192-193, 194-199 (2007).
[CrossRef]

2006 (4)

H. B. Wu, Z. Q. Wang, R. L. Fu, and J. Liu, "Design of a hybrid diffractive/refractive achromatized telecentric f-θ lens," Optik 117, 271-276 (2006).
[CrossRef]

C. F. Cheung, L. B. Kong, and W. B. Lee, "Modelling and simulation of freeform surface generation in ultra-precision raster milling," J. Eng. Manuf. 220, 1787-1801 (2006).
[CrossRef]

H. N. Hansen, K. Carneiro, H. Haitjema, and L. De Chiffre, "Dimensional micro and nano metrology," Ann. CIRP 55, 721-743 (2006).
[CrossRef]

K. H. Jeong, J. Kim, L. P. Lee, "Biological inspired artificial compound eyes," Science 312, 557-561 (2006).
[CrossRef] [PubMed]

2005 (2)

Q8. W. Ma, J. P. Kruth, "NURBS curve and surface fitting for reverse engineering," J. Adv. Manuf. Technol. 14, 918-927 (2005).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, "Designing reflectors to generate a line-shaped directivity diagram," J. Mod. Opt. 52, 1529-1536 (2005).
[CrossRef]

2003 (2)

Y. L. Ma, W. T. Hewitt, "Point inversion and projection for NURBS curve and surface: control polygon approach," Comput. Aided Geom. Des. 20, 79-99 (2003).
[CrossRef]

L. De Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, "Surfaces in precision engineering, microengineering and nanotechnology," Ann. CIRP 52, 561-578 (2003).
[CrossRef]

2002 (1)

Y. Takeuchia, S. Maedaa, T. Kawaib, K. Sawadab, "Manufacture of multiple-focus micro Fresnel lenses by means of nonrotational diamond grooving," Ann. CIRP 51, 343-346 (2002).
[CrossRef]

Bernard, S.

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, "Designing reflectors to generate a line-shaped directivity diagram," J. Mod. Opt. 52, 1529-1536 (2005).
[CrossRef]

Carneiro, K.

H. N. Hansen, K. Carneiro, H. Haitjema, and L. De Chiffre, "Dimensional micro and nano metrology," Ann. CIRP 55, 721-743 (2006).
[CrossRef]

Chen, C. C.

C. C. Chen, C. M. Chen, and J. R. Chen, "Toolpath generation of diamond shaping of aspheric lens array," J. Mater. Proc. Technol. 192-193, 194-199 (2007).
[CrossRef]

Chen, C. M.

C. C. Chen, C. M. Chen, and J. R. Chen, "Toolpath generation of diamond shaping of aspheric lens array," J. Mater. Proc. Technol. 192-193, 194-199 (2007).
[CrossRef]

Chen, J. R.

C. C. Chen, C. M. Chen, and J. R. Chen, "Toolpath generation of diamond shaping of aspheric lens array," J. Mater. Proc. Technol. 192-193, 194-199 (2007).
[CrossRef]

Cheung, C. F.

C. F. Cheung, L. B. Kong, and W. B. Lee, "Modelling and simulation of freeform surface generation in ultra-precision raster milling," J. Eng. Manuf. 220, 1787-1801 (2006).
[CrossRef]

De Chiffre, L.

H. N. Hansen, K. Carneiro, H. Haitjema, and L. De Chiffre, "Dimensional micro and nano metrology," Ann. CIRP 55, 721-743 (2006).
[CrossRef]

L. De Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, "Surfaces in precision engineering, microengineering and nanotechnology," Ann. CIRP 52, 561-578 (2003).
[CrossRef]

Doskolovich, L. L.

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, "Designing reflectors to generate a line-shaped directivity diagram," J. Mod. Opt. 52, 1529-1536 (2005).
[CrossRef]

Fu, R. L.

H. B. Wu, Z. Q. Wang, R. L. Fu, and J. Liu, "Design of a hybrid diffractive/refractive achromatized telecentric f-θ lens," Optik 117, 271-276 (2006).
[CrossRef]

Haitjema, H.

H. N. Hansen, K. Carneiro, H. Haitjema, and L. De Chiffre, "Dimensional micro and nano metrology," Ann. CIRP 55, 721-743 (2006).
[CrossRef]

Hansen, H. N.

H. N. Hansen, K. Carneiro, H. Haitjema, and L. De Chiffre, "Dimensional micro and nano metrology," Ann. CIRP 55, 721-743 (2006).
[CrossRef]

Hewitt, W. T.

Y. L. Ma, W. T. Hewitt, "Point inversion and projection for NURBS curve and surface: control polygon approach," Comput. Aided Geom. Des. 20, 79-99 (2003).
[CrossRef]

Jeong, K. H.

K. H. Jeong, J. Kim, L. P. Lee, "Biological inspired artificial compound eyes," Science 312, 557-561 (2006).
[CrossRef] [PubMed]

Jiang, X.

X. Jiang, P. Scott, and D. Whitehouse, "Freeform surface characterisation - A fresh strategy," Ann. CIRP 56, 553-556 (2007).
[CrossRef]

Kawaib, T.

Y. Takeuchia, S. Maedaa, T. Kawaib, K. Sawadab, "Manufacture of multiple-focus micro Fresnel lenses by means of nonrotational diamond grooving," Ann. CIRP 51, 343-346 (2002).
[CrossRef]

Kazanskiy, N. L.

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, "Designing reflectors to generate a line-shaped directivity diagram," J. Mod. Opt. 52, 1529-1536 (2005).
[CrossRef]

Kharitonov, S. I.

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, "Designing reflectors to generate a line-shaped directivity diagram," J. Mod. Opt. 52, 1529-1536 (2005).
[CrossRef]

Kim, J.

K. H. Jeong, J. Kim, L. P. Lee, "Biological inspired artificial compound eyes," Science 312, 557-561 (2006).
[CrossRef] [PubMed]

Kong, L. B.

C. F. Cheung, L. B. Kong, and W. B. Lee, "Modelling and simulation of freeform surface generation in ultra-precision raster milling," J. Eng. Manuf. 220, 1787-1801 (2006).
[CrossRef]

Kruth, J. P.

Q8. W. Ma, J. P. Kruth, "NURBS curve and surface fitting for reverse engineering," J. Adv. Manuf. Technol. 14, 918-927 (2005).
[CrossRef]

Kunzmann, H.

L. De Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, "Surfaces in precision engineering, microengineering and nanotechnology," Ann. CIRP 52, 561-578 (2003).
[CrossRef]

Lee, L. P.

K. H. Jeong, J. Kim, L. P. Lee, "Biological inspired artificial compound eyes," Science 312, 557-561 (2006).
[CrossRef] [PubMed]

Lee, W. B.

C. F. Cheung, L. B. Kong, and W. B. Lee, "Modelling and simulation of freeform surface generation in ultra-precision raster milling," J. Eng. Manuf. 220, 1787-1801 (2006).
[CrossRef]

Liu, J.

H. B. Wu, Z. Q. Wang, R. L. Fu, and J. Liu, "Design of a hybrid diffractive/refractive achromatized telecentric f-θ lens," Optik 117, 271-276 (2006).
[CrossRef]

Lucca, D. A.

L. De Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, "Surfaces in precision engineering, microengineering and nanotechnology," Ann. CIRP 52, 561-578 (2003).
[CrossRef]

Ma, W.

Q8. W. Ma, J. P. Kruth, "NURBS curve and surface fitting for reverse engineering," J. Adv. Manuf. Technol. 14, 918-927 (2005).
[CrossRef]

Ma, Y. L.

Y. L. Ma, W. T. Hewitt, "Point inversion and projection for NURBS curve and surface: control polygon approach," Comput. Aided Geom. Des. 20, 79-99 (2003).
[CrossRef]

Maedaa, S.

Y. Takeuchia, S. Maedaa, T. Kawaib, K. Sawadab, "Manufacture of multiple-focus micro Fresnel lenses by means of nonrotational diamond grooving," Ann. CIRP 51, 343-346 (2002).
[CrossRef]

Peggs, G. N.

L. De Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, "Surfaces in precision engineering, microengineering and nanotechnology," Ann. CIRP 52, 561-578 (2003).
[CrossRef]

Perlo, P.

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, "Designing reflectors to generate a line-shaped directivity diagram," J. Mod. Opt. 52, 1529-1536 (2005).
[CrossRef]

Sawadab, K.

Y. Takeuchia, S. Maedaa, T. Kawaib, K. Sawadab, "Manufacture of multiple-focus micro Fresnel lenses by means of nonrotational diamond grooving," Ann. CIRP 51, 343-346 (2002).
[CrossRef]

Scott, P.

X. Jiang, P. Scott, and D. Whitehouse, "Freeform surface characterisation - A fresh strategy," Ann. CIRP 56, 553-556 (2007).
[CrossRef]

Takeuchia, Y.

Y. Takeuchia, S. Maedaa, T. Kawaib, K. Sawadab, "Manufacture of multiple-focus micro Fresnel lenses by means of nonrotational diamond grooving," Ann. CIRP 51, 343-346 (2002).
[CrossRef]

Wang, Z. Q.

H. B. Wu, Z. Q. Wang, R. L. Fu, and J. Liu, "Design of a hybrid diffractive/refractive achromatized telecentric f-θ lens," Optik 117, 271-276 (2006).
[CrossRef]

Whitehouse, D.

X. Jiang, P. Scott, and D. Whitehouse, "Freeform surface characterisation - A fresh strategy," Ann. CIRP 56, 553-556 (2007).
[CrossRef]

Wu, H. B.

H. B. Wu, Z. Q. Wang, R. L. Fu, and J. Liu, "Design of a hybrid diffractive/refractive achromatized telecentric f-θ lens," Optik 117, 271-276 (2006).
[CrossRef]

Ann. CIRP (4)

X. Jiang, P. Scott, and D. Whitehouse, "Freeform surface characterisation - A fresh strategy," Ann. CIRP 56, 553-556 (2007).
[CrossRef]

H. N. Hansen, K. Carneiro, H. Haitjema, and L. De Chiffre, "Dimensional micro and nano metrology," Ann. CIRP 55, 721-743 (2006).
[CrossRef]

L. De Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, "Surfaces in precision engineering, microengineering and nanotechnology," Ann. CIRP 52, 561-578 (2003).
[CrossRef]

Y. Takeuchia, S. Maedaa, T. Kawaib, K. Sawadab, "Manufacture of multiple-focus micro Fresnel lenses by means of nonrotational diamond grooving," Ann. CIRP 51, 343-346 (2002).
[CrossRef]

Comput. Aided Geom. Des. (1)

Y. L. Ma, W. T. Hewitt, "Point inversion and projection for NURBS curve and surface: control polygon approach," Comput. Aided Geom. Des. 20, 79-99 (2003).
[CrossRef]

J. Adv. Manuf. Technol. (1)

Q8. W. Ma, J. P. Kruth, "NURBS curve and surface fitting for reverse engineering," J. Adv. Manuf. Technol. 14, 918-927 (2005).
[CrossRef]

J. Eng. Manuf. (1)

C. F. Cheung, L. B. Kong, and W. B. Lee, "Modelling and simulation of freeform surface generation in ultra-precision raster milling," J. Eng. Manuf. 220, 1787-1801 (2006).
[CrossRef]

J. Mater. Proc. Technol. (1)

C. C. Chen, C. M. Chen, and J. R. Chen, "Toolpath generation of diamond shaping of aspheric lens array," J. Mater. Proc. Technol. 192-193, 194-199 (2007).
[CrossRef]

J. Mod. Opt. (1)

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, "Designing reflectors to generate a line-shaped directivity diagram," J. Mod. Opt. 52, 1529-1536 (2005).
[CrossRef]

Optik (1)

H. B. Wu, Z. Q. Wang, R. L. Fu, and J. Liu, "Design of a hybrid diffractive/refractive achromatized telecentric f-θ lens," Optik 117, 271-276 (2006).
[CrossRef]

Science (1)

K. H. Jeong, J. Kim, L. P. Lee, "Biological inspired artificial compound eyes," Science 312, 557-561 (2006).
[CrossRef] [PubMed]

Other (1)

L. Piegl and W. Tiller, "The NURBS Book," (New York: Springer-Verlag, 1997).

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Figures (9)

Fig. 1.
Fig. 1.

Cylindrical coordinate machining.

Fig. 2.
Fig. 2.

Model of tool nose radius compensation.

Fig. 3.
Fig. 3.

Model of clearance angle compensation.

Fig. 4.
Fig. 4.

Sectional curve method.

Fig. 5.
Fig. 5.

Definition of tool nose radius and included angle.

Fig. 6.
Fig. 6.

Sketch of searching concave of sectional curve.

Fig. 7.
Fig. 7.

Compound eye lens by CCM.

Fig. 8.
Fig. 8.

Compound eye lens example.

Fig. 9.
Fig. 9.

Typical freeform parts machined by CCM.

Equations (13)

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{ x s = x cos φ y s = x sin φ { x = ± x s 2 + y s 2 φ = d [ tan 1 ( y s x s ) ]
( x t , ϕ t , z t ) = ( x + Δ x , φ + Δ φ , z + Δ z )
n = ( f x , f y , 1 ) = ( cos α , cos β , cos γ )
p 1 = p 0 + d · n
n t 0 = ( sin φ 0 , cos φ 0 , 0 )
n p = n ( n · n t 0 ) n t 0
n t 1 = ( sin φ 0 cos α 0 , cos φ 0 cos α 0 , sin α 0 )
o t ( x st , y st , z st ) = ( p 1 o ( p 1 o · B ) · B ) cos α 0
( B × p 1 o ) + sin α 0 + ( p 1 o · B ) · B + p 1
P ( x s , y s , z s ) = S ( u , v ) = i = 0 n j = 0 m N i , p ( u ) N j , q ( v ) Q i , j ( x s , y s , z s )
n = S u × S v S u × S v
{ u = u 0 + r 0 cos φ 0 v = v 0 + r 0 sin φ 0
A 2 θ max

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