Abstract

The problem of designing a reflector to distribute the illumination of a nonisotropic point source on a plane aperture according to a pre-assigned pattern is analyzed. An integral equation and equivalent partial differential equation are derived. The form of the latter reveals this reflector-design problem to be a singular elliptic Monge–Ampère boundary-value problem.

© 1972 Optical Society of America

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References

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  1. H. G. Williams, J. Illum. Engr. Soc. 1, 143 (1972).
  2. B. O’Neill, Elementary Differential Geometry (Academic, New York and London, 1969), p. 281.
  3. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Wiley–Interscience, New York, London, Sydney, 1966), p. 324.
  4. L. Nirenberg, Commun. Pure Appl. Math. 6, 103 (1953).
    [CrossRef]

1972 (1)

H. G. Williams, J. Illum. Engr. Soc. 1, 143 (1972).

1953 (1)

L. Nirenberg, Commun. Pure Appl. Math. 6, 103 (1953).
[CrossRef]

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Wiley–Interscience, New York, London, Sydney, 1966), p. 324.

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Wiley–Interscience, New York, London, Sydney, 1966), p. 324.

Nirenberg, L.

L. Nirenberg, Commun. Pure Appl. Math. 6, 103 (1953).
[CrossRef]

O’Neill, B.

B. O’Neill, Elementary Differential Geometry (Academic, New York and London, 1969), p. 281.

Williams, H. G.

H. G. Williams, J. Illum. Engr. Soc. 1, 143 (1972).

Commun. Pure Appl. Math. (1)

L. Nirenberg, Commun. Pure Appl. Math. 6, 103 (1953).
[CrossRef]

J. Illum. Engr. Soc. (1)

H. G. Williams, J. Illum. Engr. Soc. 1, 143 (1972).

Other (2)

B. O’Neill, Elementary Differential Geometry (Academic, New York and London, 1969), p. 281.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Wiley–Interscience, New York, London, Sydney, 1966), p. 324.

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Figures (3)

F. 1
F. 1

Point source of light above a ceiling plane.

F. 2
F. 2

Stereographic coordinates.

F. 3
F. 3

Reflection geometry.

Equations (37)

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ν ( Ω ) L ( x , y ) d x d y = Ω I ( u , υ ) d Ω ,
x ( u , υ ) = ( 1 + 1 4 w 2 ) 1 ( u , υ , 1 1 4 w 2 ) ,
w 2 = u 2 + υ 2 .
x / u = y / υ = ( z + 1 ) / 2
x 2 + y 2 + z 2 = 1 .
d Ω = | x u × x υ | d u d υ ,
x u ( u , υ ) = ( 1 + 1 4 w 2 ) 2 ( 1 + 1 4 υ 2 1 4 u 2 , 1 2 u υ , u ) ,
x υ ( u , υ ) = ( 1 + 1 4 w 2 ) 2 ( 1 2 u υ , 1 + 1 4 u 2 1 4 υ 2 , υ ) .
d Ω = ( 1 + 1 4 w 2 ) 2 d u d υ .
ω ( S ) L ( x , y ) d x d y = S I ( u , υ ) ( 1 + 1 4 w 2 ) 2 d u d υ ,
R L ( x , y ) d x d y = D I ( u , υ ) ( 1 + 1 4 w 2 ) 2 d u d υ .
N = ( p x + ρ x u ) × ( q x + ρ x ν ) ,
N = ρ 2 ( 1 + 1 4 w 2 ) 2 x p ρ x u q ρ x υ .
| N | 2 = N N = ρ 2 ( 1 + 1 4 w 2 ) 2 [ ρ 2 ( 1 + 1 4 w 2 ) 2 + p 2 + q 2 ]
x N = ρ 2 ( 1 + 1 4 w 2 ) 2 .
A | A | + A X | X A | = 2 N ( A N ) | A | | N | 2 .
X = A + | X A | [ x 2 N ( x N ) / | N | 2 ] ,
A / | A | = x .
F | X A | ( 1 + 1 4 w 2 ) 1 [ ρ 2 ( 1 + 1 4 w 2 ) 2 + p 2 + q 2 ] 1
G ρ ( 1 + 1 4 w 2 ) 1 + F [ ρ 2 ( 1 + 1 4 w 2 ) 2 + p 2 + q 2 ρ ( p u + q υ ) ( 1 + 1 4 w 2 ) 1 ] ,
x = u G + 2 ρ p F ,
y = υ G + 2 ρ q F ,
F = 1 + ρ ( 1 1 4 w 2 ) ( 1 + 1 4 w 2 ) 1 ( 1 1 4 w 2 ) [ ρ 2 ( 1 + 1 4 w 2 ) 2 p 2 q 2 ] + 2 ρ ( p u + q υ ) ( 1 + 1 4 w 2 ) 1 .
x = x ( u , υ , ρ , p , q ) ,
y = y ( u , υ , ρ , p , q ) .
D = | x u + p x ρ + r x p + s x q x υ + q x ρ + s x p + t x q y u + p y ρ + r y p + s y q y υ + q y ρ + s y p + t y q | ,
L [ x ( u , υ , ρ , p , q ) , y ( u , υ , ρ , p , q ) ] D ( u , υ , ρ , p , q , r , s , t ) = I ( u , υ ) .
D = J p q ( r t s 2 ) + ( J p υ + q J p p ) r + ( J q υ + J u p + p J ρ p + q J q ρ ) s + ( J u q + p J ρ q ) t + J u υ + p J ρ υ + q J u ρ ,
J α β = x α y β x β y α for α , β { u , υ , ρ , p , q } .
J q υ = J u p , p J ρ p = q J q ρ .
Q 4 D τ D t D s 2
Q = 4 [ J p q D J p q ( J u υ + p J ρ υ + q J u ρ ) + ( J p υ + q J p ρ ) ( J u q + p J ρ q ) ( J u p + p J ρ p ) ( J q υ + q J q ρ ) ] .
J α β J γ δ = J α δ J γ β J γ α J β δ ,
Q = 4 J p q D .
D = I ( u , υ ) / L ( x , y )
J p q = 4 ρ 2 ( 1 + 1 4 w 2 ) [ ρ 2 ( 1 + 1 4 w 2 ) 2 + p 2 + q 2 ] [ 1 + ρ ( 1 1 4 w 2 ) ( 1 + 1 4 w 2 ) 1 ] 2 { ( 1 1 4 w 2 ) [ ρ 2 ( 1 + 1 4 w 2 ) 2 p 2 q 2 ] + 2 ρ ( p u + q υ ) ( 1 + 1 4 w 2 ) 1 } 3 .
1 + ρ ( 1 1 4 w 2 ) ( 1 + 1 4 w 2 ) 1 = 0 .