Abstract

The solution to a problem in asymmetric reflector design for illumination is described. Working backward from a required rectangular and nonuniform light distribution pattern, the reflector geometry is arrived at by a combination of calculus and geometric construction. The central curve is computed from the general reflector equation lnR = tan (θd/2) , the integration being performed on an approximate matching function or graphically. The off-center contours are then obtained by means of an original drafting-board construction which is described, and the complete reflector shape is thus defined.

© 1966 Optical Society of America

Full Article  |  PDF Article

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 (9)

Fig. 1
Fig. 1

Diagram of desired illumination coverage from street lighting unit.

Fig. 2
Fig. 2

Ideal vertical candlepower distribution curve for street lighting unit.

Fig. 3
Fig. 3

Reflector diagram.

Fig. 4
Fig. 4

Candlepower in percent of luminaire maximum. A, bare lamp candlepower; B, luminaire candlepower; C, luminaire candlepower less bare lamp candlepower (BA); D, reflector candlepower (luminaire candlepower with glassware removed less bare lamp candlepower).

Fig. 5
Fig. 5

Ray diagram of central reflector contour.

Fig. 6
Fig. 6

Bare reflector candlepower vs vertical angle or reflected light.

Fig. 7
Fig. 7

Reflector radius vs reflector angle in central contour plotted in (left) rectangular coordinates and (right) polar coordinates.

Fig. 8
Fig. 8

Diagram of incident and reflected light rays striking rotating mirror.

Fig. 9
Fig. 9

Geometric construction for determining off-center reflector contours.

Equations (35)

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

R s = O X O L .
C : n k C 0 = d θ : d α
C = n k C 0 d θ d α .
tan β = d R R d θ
θ = α + 2 β
β = θ - α 2 .
tan θ - α 2 = d R R d θ .
k = reflected flux incident flux = 0 α 1 C d α 0 θ 1 n C 0 d θ
n k 0 θ 1 C 0 d θ = 0 α 1 C d θ .
n k C 0 θ = 0 α 1 C d α .
C = A α 2 + B - C ( D - α ) - 1 / 2 .
C = 102.8 α 2 + 105.3 - 73.0 ( 1.4212 - α ) - 1 / 2 .
C d α = n k C 0 d θ .
C d α = n k C 0 d θ = n k C 0 θ .
n K C 0 θ n K C 0 θ 1 = C d α 0 α 1 C d α = θ θ 1
θ = θ 1 0 α 1 C d α C d α ,
θ = T C d α ,
T = θ 1 0 α 1 C d α .
d R R = tan T c d α - α 2 d θ ,
d R R = tan ( T 2 c d α - α 2 ) d θ ,
d θ = d ( T c d α ) = 2 d [ T 2 c d α - α 2 + α 2 ] ,
d θ = 2 d [ T 2 c d α - α 2 ] + d α ;
d R R = 2 tan ( T 2 C d α - α 2 ) d ( T 2 c d α - α 2 ) + tan ( T 2 c d α - α 2 ) d α .
ln R + K 1 = - 2 ln cos ( T 2 c d α - α 2 ) + tan ( T 2 c d α - α 2 ) d α .
ln R R 0 = - ln cos 2 ( T 2 c d α - α 2 ) + tan ( T 2 c d α - α 2 ) d α
R R 0 = ln - 1 tan ( T 2 c d α - α 2 ) d α cos 2 ( T 2 c d α - α 2 ) .
θ = T [ A α 2 + B - C ( D - α ) - 1 / 2 ] d α ,
θ = T [ A α 3 3 + B α - 2 C D 1 / 2 + 2 C ( D - α ) 1 / 2 .
R R 0 = ln - 1 tan T 2 [ A α 3 3 + ( B - 1 T ) α - 2 C D 1 / 2 + 2 C ( D - α ) 1 / 2 ] d α cos 2 T 2 [ A α 3 3 + ( B - 1 T ) α - 2 C D 1 / 2 + 2 C ( D - α ) 1 / 2 ]
C = A α 2 + B - C ( D - α ) - 1 / 2 .
44 = B - C D - 1 / 2 ,
101 = 1.13 2 A + B - C ( D - 1.13 ) - 1 / 2 ,
0 = 2.26 A - c 2 ( D - 1.13 ) - 3 / 2 ,
0 = 1.36 2 A + B - C ( D - 1.36 ) - 1 / 2 .
C = 102.8 α 2 + 105.25 - 73.0 ( 1.4212 - α ) - ½ .

Metrics