Abstract

A light cone efficiently transports, distributes, and concentrates the incoming radiation. I derive a formula for the spatial irradiance (or illuminance) distribution at the exit aperture of a tapered light pipe. The theory is demonstrated by Monte Carlo ray-tracing for lightpipes with light-emitting diodes at the input face. The analysis is based on the addition of the radiation patterns of the multiple virtual sources that, as in a three-dimensional kaleidoscope, are seen through a tapered light tube. Given its explicit dependence on the structural and optical parameters, this analysis may be a useful tool in the development and application of light cones.

© 2010 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  20. I. Moreno and C. C. Sun, “Three-dimensional measurement of light-emitting diode radiation pattern: a rapid estimation,” Meas. Sci. Technol. 20, 075306 (2009).
    [CrossRef]
  21. I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46, 033001 (2007).
    [CrossRef]
  22. I. Moreno, C. C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48, 1190–1197 (2009).
    [CrossRef]
  23. I. Moreno, “Creating a desired lighting pattern with an LED array,” Proc. SPIE 7058, 705811 (2008).
    [CrossRef]

2009 (4)

M. Kocifaj, “Analytical solution for daylight transmission via hollow light pipes with a transparent glazing,” Sol. Energy 83, 186–192 (2009).
[CrossRef]

A. J. W. Whang, P. C. Li, Y. Y. Chen, and S. L. Hsieh, “Guiding light from LED array via tapered light pipe for illumination systems design,” J. Disp. Technol. 5, 104–108 (2009).
[CrossRef]

I. Moreno and C. C. Sun, “Three-dimensional measurement of light-emitting diode radiation pattern: a rapid estimation,” Meas. Sci. Technol. 20, 075306 (2009).
[CrossRef]

I. Moreno, C. C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48, 1190–1197 (2009).
[CrossRef]

2008 (7)

2007 (3)

Y. Qu, J. R. Howell, and O. A. Ezekoye, “Monte Carlo modeling of a light-pipe radiation thermometer,” IEEE Trans. Semicond. Manuf. 20, 39–50 (2007).
[CrossRef]

H. Murat, A. Gielen, and H. De Smet, “Gradually tapered light pipes for illumination of LED projectors,” J. Soc. Inf. Disp. 15, 519–526 (2007).
[CrossRef]

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46, 033001 (2007).
[CrossRef]

2006 (1)

2005 (1)

2004 (1)

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

2001 (1)

1982 (1)

M. Iona, “Virtual mirrors,” Phys. Teach. 20, 278 (1982).
[CrossRef]

1963 (1)

Benítez, P.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Berkowitz-Mattuck, J. B.

Cassarly, W.

W. Cassarly, “Nonimaging optics: concentration and illumination,” in OSA Handbook of Optics, 2nd ed., Vol. III (McGraw-Hill, 2001).

Cassarly, W. J.

Chaves, J.

J. Chaves, Introduction to Nonimaging Optics (CRC Press, 2008).
[CrossRef]

Chen, M. M.

Chen, Y. Y.

A. J. W. Whang, P. C. Li, Y. Y. Chen, and S. L. Hsieh, “Guiding light from LED array via tapered light pipe for illumination systems design,” J. Disp. Technol. 5, 104–108 (2009).
[CrossRef]

Cheng, C. M.

Cheng, Y.-K.

Chern, J. L.

Chern, J.-L..

Chu, S. C.

De Smet, H.

H. Murat, A. Gielen, and H. De Smet, “Gradually tapered light pipes for illumination of LED projectors,” J. Soc. Inf. Disp. 15, 519–526 (2007).
[CrossRef]

Ezekoye, O. A.

Y. Qu, J. R. Howell, and O. A. Ezekoye, “Monte Carlo modeling of a light-pipe radiation thermometer,” IEEE Trans. Semicond. Manuf. 20, 39–50 (2007).
[CrossRef]

Feuermann, D.

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Fournier, F.

Gentle, A.

P. D. Swift, R. Lawlor, G. B. Smith, and A. Gentle, “Rectangular-section mirror light pipes,” Sol. Energy Mater. Sol. Cells 92, 969–975 (2008).
[CrossRef]

Gielen, A.

H. Murat, A. Gielen, and H. De Smet, “Gradually tapered light pipes for illumination of LED projectors,” J. Soc. Inf. Disp. 15, 519–526 (2007).
[CrossRef]

Glaser, P. E.

Gordon, J. M.

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Gupta, A.

Howell, J. R.

Y. Qu, J. R. Howell, and O. A. Ezekoye, “Monte Carlo modeling of a light-pipe radiation thermometer,” IEEE Trans. Semicond. Manuf. 20, 39–50 (2007).
[CrossRef]

Hsieh, S. L.

A. J. W. Whang, P. C. Li, Y. Y. Chen, and S. L. Hsieh, “Guiding light from LED array via tapered light pipe for illumination systems design,” J. Disp. Technol. 5, 104–108 (2009).
[CrossRef]

Hua, H.

R. Zhang and H. Hua, “8.3: Design of a Compact Light Engine for FLCOS Microdisplays in a p-HMPD System,” SID Symposium Digest 39, 85 –87 (2008).
[CrossRef]

Huleihil, M.

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Iona, M.

M. Iona, “Virtual mirrors,” Phys. Teach. 20, 278 (1982).
[CrossRef]

Ivanov, R.

I. Moreno, C. C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48, 1190–1197 (2009).
[CrossRef]

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46, 033001 (2007).
[CrossRef]

Katz, E. A.

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Kocifaj, M.

M. Kocifaj, “Analytical solution for daylight transmission via hollow light pipes with a transparent glazing,” Sol. Energy 83, 186–192 (2009).
[CrossRef]

Koshel, R. J.

Lawlor, R.

P. D. Swift, R. Lawlor, G. B. Smith, and A. Gentle, “Rectangular-section mirror light pipes,” Sol. Energy Mater. Sol. Cells 92, 969–975 (2008).
[CrossRef]

Lee, J.

Li, P. C.

A. J. W. Whang, P. C. Li, Y. Y. Chen, and S. L. Hsieh, “Guiding light from LED array via tapered light pipe for illumination systems design,” J. Disp. Technol. 5, 104–108 (2009).
[CrossRef]

Miñano, J. C.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Moreno, I.

I. Moreno, C. C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48, 1190–1197 (2009).
[CrossRef]

I. Moreno and C. C. Sun, “Three-dimensional measurement of light-emitting diode radiation pattern: a rapid estimation,” Meas. Sci. Technol. 20, 075306 (2009).
[CrossRef]

I. Moreno, “Creating a desired lighting pattern with an LED array,” Proc. SPIE 7058, 705811 (2008).
[CrossRef]

I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16, 1808–1819 (2008).
[CrossRef] [PubMed]

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46, 033001 (2007).
[CrossRef]

Muñoz, J.

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46, 033001 (2007).
[CrossRef]

Murat, H.

H. Murat, A. Gielen, and H. De Smet, “Gradually tapered light pipes for illumination of LED projectors,” J. Soc. Inf. Disp. 15, 519–526 (2007).
[CrossRef]

Qu, Y.

Y. Qu, J. R. Howell, and O. A. Ezekoye, “Monte Carlo modeling of a light-pipe radiation thermometer,” IEEE Trans. Semicond. Manuf. 20, 39–50 (2007).
[CrossRef]

Rolland, J. P.

Smith, G. B.

P. D. Swift, R. Lawlor, G. B. Smith, and A. Gentle, “Rectangular-section mirror light pipes,” Sol. Energy Mater. Sol. Cells 92, 969–975 (2008).
[CrossRef]

Sun, C. C.

Swift, P. D.

P. D. Swift, R. Lawlor, G. B. Smith, and A. Gentle, “Rectangular-section mirror light pipes,” Sol. Energy Mater. Sol. Cells 92, 969–975 (2008).
[CrossRef]

Whang, A. J. W.

A. J. W. Whang, P. C. Li, Y. Y. Chen, and S. L. Hsieh, “Guiding light from LED array via tapered light pipe for illumination systems design,” J. Disp. Technol. 5, 104–108 (2009).
[CrossRef]

Winston, R.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Zhang, R.

R. Zhang and H. Hua, “8.3: Design of a Compact Light Engine for FLCOS Microdisplays in a p-HMPD System,” SID Symposium Digest 39, 85 –87 (2008).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

IEEE Trans. Semicond. Manuf. (1)

Y. Qu, J. R. Howell, and O. A. Ezekoye, “Monte Carlo modeling of a light-pipe radiation thermometer,” IEEE Trans. Semicond. Manuf. 20, 39–50 (2007).
[CrossRef]

J. Disp. Technol. (1)

A. J. W. Whang, P. C. Li, Y. Y. Chen, and S. L. Hsieh, “Guiding light from LED array via tapered light pipe for illumination systems design,” J. Disp. Technol. 5, 104–108 (2009).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Soc. Inf. Disp. (1)

H. Murat, A. Gielen, and H. De Smet, “Gradually tapered light pipes for illumination of LED projectors,” J. Soc. Inf. Disp. 15, 519–526 (2007).
[CrossRef]

Meas. Sci. Technol. (1)

I. Moreno and C. C. Sun, “Three-dimensional measurement of light-emitting diode radiation pattern: a rapid estimation,” Meas. Sci. Technol. 20, 075306 (2009).
[CrossRef]

Opt. Eng. (1)

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical light-emitting diode array,” Opt. Eng. 46, 033001 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Teach. (1)

M. Iona, “Virtual mirrors,” Phys. Teach. 20, 278 (1982).
[CrossRef]

Proc. SPIE (1)

I. Moreno, “Creating a desired lighting pattern with an LED array,” Proc. SPIE 7058, 705811 (2008).
[CrossRef]

SID Symposium Digest (1)

R. Zhang and H. Hua, “8.3: Design of a Compact Light Engine for FLCOS Microdisplays in a p-HMPD System,” SID Symposium Digest 39, 85 –87 (2008).
[CrossRef]

Sol. Energy (1)

M. Kocifaj, “Analytical solution for daylight transmission via hollow light pipes with a transparent glazing,” Sol. Energy 83, 186–192 (2009).
[CrossRef]

Sol. Energy Mater. Sol. Cells (1)

P. D. Swift, R. Lawlor, G. B. Smith, and A. Gentle, “Rectangular-section mirror light pipes,” Sol. Energy Mater. Sol. Cells 92, 969–975 (2008).
[CrossRef]

Other (3)

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

W. Cassarly, “Nonimaging optics: concentration and illumination,” in OSA Handbook of Optics, 2nd ed., Vol. III (McGraw-Hill, 2001).

J. Chaves, Introduction to Nonimaging Optics (CRC Press, 2008).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Diagram of a tapered lightpipe with a light source in the entrance. This figure also shows the irradiance (or illuminance) spatial distribution that would be measured or projected in a screen at the exit aperture of the lightpipe.

Fig. 2
Fig. 2

(a) Schematic that depicts what is viewed from the output of the lightpipe, an array of virtual images of the light source. (b), (c) 3D locus of the virtual sources of a lightpipe with square and rectangular cross section.

Fig. 3
Fig. 3

Geometry and coordinate system for calculation of irradiance distribution. (a) Angular shift along x-direction. (b) Cartesian coordinates of the s th source at the entrance plane. (c) Some lightpipe parameters.

Fig. 4
Fig. 4

Irradiance distribution at the output of a hollow tapered rod with two parallel walls ( d x = 3 , d y = 3 , D x = 6 , D y = d y , H = 4 , ρ = 0.9 ). One Lambertian LED is placed at the center of the input face. (a) Irradiance pattern obtained with Eq. (7a, 7b). (b) Irradiance pattern generated with ray-tracing using 10 7 rays. (c) Detailed comparison between (a) and (b) along x direction and y = 0 . (d) Also compares the irradiance distributions but with only 10 6 rays.

Fig. 5
Fig. 5

Output irradiance distribution of a hollow tapered rod with two parallel walls ( d x = 3 , d y = 3 , D x = 6 , D y = d y , H = 4 , ρ = 0.9 ). One Lambertian LED is placed at one corner of the input face ( x s = 1.5 , y s = 1.5 ) . (a) Irradiance pattern obtained with Eq. (7a, 7b). (b) Irradiance pattern generated with ray-tracing, using 107 rays. (c) Detailed comparison between (a) and (b) along x direction and y = 0 .

Fig. 6
Fig. 6

Virtual windows effect: As an observer views the sphere of images through the lightpipe, each image can be seen only through a specific wall or set of virtual walls. Dark (red online) lines show the four primary windows, light (blue online) lines the secondary virtual windows.

Fig. 7
Fig. 7

Output irradiance distribution of a square cross section tapered rod ( d = 3 , D = 6 , H = 4 , ρ = 0.9 ). One Lambertian LED is placed at different places of the input face. (a) LED is located at the center of the entrance ( x s = 0 , y s = 0 ) . (b) LED is placed at ( x s = 1.5 , y s = 0 ) . (c) LED is placed at ( x s = 0.75 , y s = 0.75 ) . (d) LED is placed at one corner of the input face ( x s = 1.5 , y s = 1.5 ) . The ray-tracing uses 10 7 rays.

Fig. 8
Fig. 8

Output irradiance distribution of a square cross section tapered rod ( d = 3 , D = 6 , H = 18 , ρ = 0.9 ). One Lambertian LED is placed at the center of the input face. (a) Irradiance pattern obtained with Eq. (10). (b) Irradiance pattern generated with ray-tracing using 2 × 10 7 rays. (c) Irradiance pattern by using 107 rays. (d) Detailed comparison between (a) and (b) along x direction and y = 0 .

Fig. 9
Fig. 9

Irradiance distribution at the exit face of a rectangular cross section tapered rod, where the parameters are d x = 3 , d y = 4 , D x = 6 , D y = 7 , and ρ = 0.9 . One Lambertian LED is placed at the center of the input face. Figure shows the irradiance pattern obtained with both Eq. (15) and ray-tracing for (a) one tube with length H = 5 , (b) a tube with length H = 10 . The ray-tracing uses 10 7 rays.

Fig. 10
Fig. 10

Geometry used to derive the maximum number of visible images N ± x along x-direction.

Equations (44)

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x s i = i a + ( 1 ) i x s , y s j = j b + ( 1 ) j y s ,
α s i = i Δ x + ( 1 ) i arctan ( x s R x ) ,
α s j = j Δ y + ( 1 ) j arctan ( y s R y ) ,
E L = E sources + E images .
E s ( X , Y , Z ) = Z ( X 2 + Y 2 + Z 2 ) 3 2 I s [ θ s ( X , Y , Z ) ] .
θ s i j ( x , y , z ) = arccos [ n i j ( r r s i j ) | n i j | | r r s i j | ] .
θ s i j ( x , y , z ) = arccos [ x 0 i j ( x x s i j ) + y 0 i j ( y y s i j ) + ( z 0 i j + R ) ( z z s i j ) R ( x x s i j ) 2 + ( y y s i j ) 2 + ( z z s i j ) 2 ] ,
θ s i j ( x , y , z ) = arccos [ x 0 i ( x x s i ) + ( z 0 i + R ) ( z z s i ) R ( x x s i ) 2 + ( y y s j ) 2 + ( z z s i ) 2 ] ,
θ s i j ( x , y , z ) = arccos [ y 0 j ( y y s j ) + ( z 0 j + R ) ( z z s j ) R ( x x s i ) 2 + ( y y s j ) 2 + ( z z s j ) 2 ] .
E L ( x , y ) = s = 1 K i = N x N x j = N N ρ | i | + | j | E s ( x x s i , y y s j , H z s i ) ,
E L ( x , y ) = s = 1 K i = N N j = N y N + y ρ | i | + | j | E s ( x x s i , y y s j , H z s j ) ,
x s i = i d x + ( 1 ) i x s , y s j = R s y sin α s j , z s j = R s y cos α s j R ,
x s i = R s x sin α s i , y s j = j d y + ( 1 ) j y s , z s i = R s x cos α s i R .
E L ( x , y ) = s = 1 K i = N x N + x j = N y N + y ρ | i | + | j | h i j [ W 1 s i j E s ( x x 1 s i j , y y 1 s j , H z 1 s i j ) + W 2 s i j E s ( x x 2 s i , y y 2 s i j , H z 2 s i j ) ] ,
x 1 s i j = R s ( x s , A j y s ) cos ( A j α s j ) sin α s i ,
y 1 s j = R s ( A j x s , y s ) sin α s j ,
z 1 s i j = R s ( x s , A j y s ) cos ( A j α s j ) cos α s i R ,
x 2 s i = R s ( x s , A i y s ) sin α s i ,
y 2 s i j = R s ( A i x s , y s ) cos ( A i α s i ) sin α s j ,
z 2 s i j = R s ( A i x s , y s ) cos ( A i α s i ) cos α s j R ,
W 1 s i j = { 1 if V 1 s i j 0 0 if V 1 s i j < 0 } , and
W 2 s i j = { 1 if V 2 s i j 0 0 if V 2 s i j < 0 }
V 1 s i j = Sign ( i ) Sign ( j ) { x 1 s i j [ ( z + R ) C j y H ] + y 1 s j [ x H ( z + R ) C i ] + ( z 1 s i j + R ) [ y C i x C j ] } ,
V 2 s i j = Sign ( i ) Sign ( j ) { x 2 s i [ ( z + R ) C j y H ] + y 2 s i j [ x H ( z + R ) C i ] + ( z 2 s i j + R ) [ y C i x C j ] } .
E L ( x , y ) = s = 1 K i = N x N + x j = N y N + y ρ | i | + | j | h i j [ W 1 s i j E s ( x x 1 s i j , y y 1 s j , H z 1 s i j ) + W 2 s i j E s ( x x 2 s i , y y 2 s i j , H z 2 s i j ) ] .
θ s i j ( x , y , z ) = arccos [ X 1 0 i j ( x x 1 s i j ) + Y 1 0 j ( y y 1 s j ) + Z 1 0 i j ( z z 1 s i j ) X 1 0 i j 2 + Y 1 0 j 2 + Z 1 0 i j 2 ( x x 1 s i j ) 2 + ( y y 1 s j ) 2 + ( z z 1 s i j ) 2 ] ,
θ s i j ( x , y , z ) = arccos [ X 2 0 i ( x x 2 s i ) + Y 2 0 i j ( y y 2 s i j ) + Z 2 0 i j ( z z 2 s i j ) X 2 0 i 2 + Y 2 0 i j 2 + Z 2 0 i j 2 ( x x 2 s i ) 2 + ( y y 2 s i j ) 2 + ( z z 2 s i j ) 2 ] ,
x 1 s i j = ( R s i R s j ) sin α s i + R s j cos ( A j α s j ) sin α s i ,
y 1 s j = R s j sin α s j ,
z 1 s i j = ( R s i R s j ) cos α s i + R s j cos ( A j α s j ) cos α s i R x ,
x 2 s i = R s i sin α s i , y 2 s i j = ( R s j R s i ) sin α s j + R s i cos ( A i α s i ) sin α s j ,
z 2 s i j = ( R s j R s i ) cos α s j + R s i cos ( A i α s i ) cos α s j R y .
W n s i j = { 1 if V n s i j 0 0 if V n s i j < 0 } , n = 1 , 2
V n s i j = ( 1 ) n + 1 Sign ( i ) Sign ( j ) [ X n s i j ( z C j Y j H ) + Y n s j ( X i H z C i ) + z n s i j ( Y j C i X i C j ) ] , n = 1 , 2
α s N + x = N + x Δ x + ( 1 ) N + x arctan ( x s R x ) .
N + x < α x ( 1 ) N + x arctan ( x s R x ) Δ x .
N + x > α x + ( 1 ) N + x arctan ( x s R x ) Δ x 1 .
arctan ( x s R x ) Δ x < 1 2 .
N ± x = Integer [ α x ± ( 1 ) Round [ α x Δ x ] + 1 arctan ( x s R x ) Δ x ] ,
N ± y = Integer [ α y ± ( 1 ) Round [ α y Δ y ] + 1 arctan ( y s R y ) Δ y ] ,
α x = arccos { [ x s 2 + R x 2 ( H + R x ) 2 + D x 2 4 ] 0.5 } + Δ x 2 ,
α y = arccos { [ y s 2 + R y 2 ( H + R y ) 2 + D y 2 4 ] 0.5 } + Δ y 2
N = Round [ log ρ 0 log ρ ] | i | ,
N = Round [ log ρ 0 log ρ ] | j | .

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