Abstract

Employing geometrical optics and assuming Lambert reflection, a general method is developed for calculating the diffusely reflected light from a class of surfaces known as quadrics. The analysis is applied to several well-known nondegenerate as well as degenerate quadric surfaces for cases which are considered to be convex or concave with respect to the directions of incident light and observer vectors. For several examples from the convex category, the results are exact and agree with those previously obtained with other methods of development. The effects of shadowing are included for a few concave-surface geometries, but to the extent that diffraction around edges and multiple reflections are omitted, the results are approximate.

© 1967 Optical Society of America

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References

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  1. M. Sussman, J. Opt. Soc. Am. 48, 275 (1958).
    [CrossRef]
  2. W. Rambauske and R. G. Gruenzel, J. Opt. Soc. Am. 55, 315 (1965).
    [CrossRef]
  3. D. C. Look, J. Opt. Soc. Am. 55, 462 (1965).
  4. H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).
  5. In Eq. (3) and the ensuing derivation, the usual vector notation (→) is dropped and the quantity κ is set equal to one.
  6. The symbol T, in vector-calculus notation, denotes the transpose of a matrix or vector.
  7. K. W. Brand and F. A. Spagnolo, Lambert Diffuse Reflection from General Quadric Surfaces, , January, 1967.
  8. D. J. Struik, Differential Geometry (Addison-Wesley Company, Reading, Massachusetts, 1950).

1965 (2)

1958 (1)

Brand, K. W.

K. W. Brand and F. A. Spagnolo, Lambert Diffuse Reflection from General Quadric Surfaces, , January, 1967.

Gruenzel, R. G.

Look, D. C.

Rambauske, W.

Spagnolo, F. A.

K. W. Brand and F. A. Spagnolo, Lambert Diffuse Reflection from General Quadric Surfaces, , January, 1967.

Struik, D. J.

D. J. Struik, Differential Geometry (Addison-Wesley Company, Reading, Massachusetts, 1950).

Sussman, M.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

J. Opt. Soc. Am. (3)

Other (5)

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

In Eq. (3) and the ensuing derivation, the usual vector notation (→) is dropped and the quantity κ is set equal to one.

The symbol T, in vector-calculus notation, denotes the transpose of a matrix or vector.

K. W. Brand and F. A. Spagnolo, Lambert Diffuse Reflection from General Quadric Surfaces, , January, 1967.

D. J. Struik, Differential Geometry (Addison-Wesley Company, Reading, Massachusetts, 1950).

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Figures (6)

Fig. 1
Fig. 1

Diffuse reflection from general surface.

Fig. 2
Fig. 2

General quadric surface q(x,y,z) = 0, in cartesian representation.

Fig. 3
Fig. 3

Ellipsoid of revolution (spheroid with a = b, c>a, b).

Fig. 4
Fig. 4

Cylinder (circular, a = b, height h, nonreflecting bases).

Fig. 5
Fig. 5

Hemisphere (radius a, no shadowing).

Fig. 6
Fig. 6

Hemisphere (radius a, shadowing).

Tables (1)

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Table I Quadric surfaces discussed in text.

Equations (53)

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d I α 0 = ( κ E / π ) cos γ s cos α 0 d S ,
I = κ E π [ n · r s ] [ n · r 0 ] d S ,
n = q / q .
I = E π ( [ q · r s ] [ q · r 0 ] q 2 ) d S .
q ( x , y , z ) = q 1 x 2 + q 2 y 2 + q 3 z 2 + q 4 y z + q 5 x z + q 6 x y + q 7 x + q 8 y + q 9 z - 1 = 0 ,
q = { q / x q / y q / z } = { 2 q 1 x + q 0 y + q 5 z + q 7 q 6 x + 2 q 2 y + q 4 z + q 8 q 5 x + q 4 y + 2 q 3 z + q 9 } ,
q = ( 2 q 1 q 6 q 5 q 6 2 q 2 q 4 q 5 q 4 2 q 3 ) { x y z } + { q 7 q 8 q 9 } .
( 2 q 1 q 6 q 5 q 6 2 q 2 q 4 q 5 q 4 2 q 3 )
{ x y z } and { q 7 q 8 q 9 } ,
q = Q r + c .
r s = { sin φ s cos θ s sin φ s sin θ s cos φ s } , r 0 = { sin φ 0 cos θ 0 sin φ 0 sin θ 0 cos φ 0 } .
[ q · r s ] = [ ( Q r + c ) · r s ] = [ r s T ( Q r + c ) ] [ q · r 0 ] = [ ( Q r + c ) · r 0 ] = [ r 0 T ( Q r + c ) ] ,
r i T = ( sin φ i cos θ i , sin φ i sin θ i , cos φ i ) .
[ q · r s ] [ q · r 0 ] = [ r s T ( Q r + c ) ] [ r 0 T ( Q r + c ) ] = r 0 T ( Q r + c ) ( r T Q T + c T ) r s ,
[ q · r s ] [ q · r 0 ] = r 0 T ( Q r + c ) ( r T Q + c T ) r s .
q 2 q · q = ( q ) T q = ( Q r + c ) T ( Q r + c ) .
q 2 = r T Q 2 r + c T Q r + r T Q c + c T c .
I = E π ( [ q · r s ] [ q · r 0 ] q 2 ) d S = E π ( r 0 T Q r r T Q r s + r 0 T Q r c T r s + r 0 T c r T Q r s + r 0 T c c T r s r T Q 2 r + c T Q r + r T Q c + c T c ) d S ,
q ( x , y , z ) = ( x 2 + y 2 ) / a 2 + z 2 / c 2 - 1 = 0.
r s = { cos θ s sin θ s 0 } ,             r 0 = { cos θ 0 sin θ 0 0 } ,             r 0 T = ( cos θ 0 , sin θ 0 , 0 ) , Q = ( 2 / a 2 0 0 0 2 / a 2 0 0 0 2 / c 2 ) ,             and             c = { 0 0 0 } .
Q r s = { 2 / a 2 cos θ s 2 / a 2 sin θ s 0 } ,             r T Q r s = 2 x a 2 cos θ s + 2 y a 2 sin θ s , r r T Q r s = 2 a 2 { x 2 cos θ s + x y sin θ s x y cos θ s + y 2 sin θ s x z cos θ s + y z sin θ s } ,             Q r r T Q r s = 4 a 2 { x 2 cos θ s / a 2 + x y sin θ s / a 2 x y cos θ s / a 2 + y 2 sin θ s / a 2 x z cos θ s / c 2 + y z sin θ s / c 2 } ,
r 0 T Q r r T Q r s = ( 4 / a 4 ) [ x 2 cos θ s cos θ 0 + x y ( sin θ s cos θ 0 + cos θ s sin θ 0 ) + y 2 sin θ s sin θ 0 ] .
Q 2 = 4 ( 1 / a 4 0 0 0 1 / a 4 0 0 0 1 / c 4 ) ,             Q 2 r = 4 { x / a 4 y / a 4 z / c 4 } .
r T Q 2 r = 4 { [ ( x 2 + y 2 ) / a 4 ] + z 2 / c 4 } ,
( 4 / a 2 c 2 ) ( c 2 - z 2 + a 2 z 2 / c 2 ) ,
c = { 0 0 0 } ,
I = E π c 2 [ x 2 cos θ s cos θ 0 + x y ( sin θ s cos θ 0 + cos θ s sin θ 0 ) + y 2 sin θ s sin θ 0 ] a 2 [ c 2 - z 2 + a 2 z 2 / c 2 ] d S .
x = a sin u cos v ,             y = a sin u sin v ,             z = c cos u .
E π c 2 sin 2 u [ cos v cos ( v - θ s ) ] ( a 2 cos 2 u + c 2 sin 2 u ) d S ,
d S = a sin u ( a 2 cos 2 u + c 2 sin 2 u ) 1 2 d u d v .
I = E π ( β - 1 2 π ) 1 2 π 0 π a c 2 sin 3 φ cos θ cos ( θ - θ s ) ( a 2 cos 2 φ + c 2 sin 2 φ ) 1 2 d φ d θ ,
I = E a 2 π ( β - 1 2 π ) 1 2 π 0 π sin 3 φ cos θ cos ( θ - β ) d φ d θ .
I = 2 3 ( E a 2 / π ) [ ( π - β ) cos β + sin β ] ,
q ( x , y , z ) = ( x 2 + y 2 ) / a 2 - 1 = 0.
r s = { cos θ s sin θ s 0 } ,             r 0 = { cos θ 0 sin θ 0 0 } ,             r 0 T = ( cos θ 0 , sin θ 0 , 0 ) Q = ( 2 / a 2 0 0 0 2 / a 2 0 0 0 0 ) ,             and             c = { 0 0 0 } . Q r s = { 2 / a 2 cos θ s 2 / a 2 sin θ s 0 } ,             r T Q r s = ( 2 x / a 2 ) cos θ s + ( 2 y / a 2 ) sin θ s , r r T Q r s = 2 / a 2 { x 2 cos θ s + y x sin θ s x y cos θ s + y 2 sin θ s x z cos θ z + y z sin θ s } , Q r r T Q r s = 4 / a 4 { x 2 cos θ s + x y sin θ s x y cos θ s + y 2 sin θ s 0 } ,
r 0 T Q r r T Q r s = ( 4 / a 4 ) ( x 2 cos θ s cos θ 0 + x y [ sin θ s cos θ 0 + cos θ s sin θ 0 ] + y 2 sin θ s sin θ 0 ) .
Q 2 r = 4 { x / a 4 y / b 4 0 }             and             r T Q 2 r = 4 / a 4 ( x 2 + y 2 ) = 4 / a 2 ,
I = E a π - 1 2 h 1 2 h ( β - 1 2 π ) 1 2 π cos θ cos ( θ - β ) d θ d z .
I = ( E a h / 2 π ) [ ( π - β ) cos β + sin β ] ,
q ( x , y , z ) = ( x 2 + y 2 + z 2 ) / a 2 - 1 = 0.
r s = { 0 - 1 0 } ,             r 0 = { 0 - 1 0 } ,             r 0 T = ( 0 , - 1 , 0 ) , Q = ( 2 / a 2 0 0 0 2 / a 2 0 0 0 2 / a 2 ) ,             and             c = { 0 0 0 } .
Q r s = { 0 - 2 / a 2 0 } ,             r T Q r s = - 2 y a 2 ,             r r T Q r s = - 2 a 2 { x y y 2 z y } , Q r r T Q r s = - 4 a 4 { x y y 2 z y } ,             and             r 0 T Q r r T Q r s = 4 a 4 y 2 .
Q 2 = 4 ( 1 / a 4 0 0 0 1 / a 4 0 0 0 1 / a 4 ) ,             Q 2 r = 4 a 4 { x y z } ,
r T Q 2 r = 4 a 4 ( x 2 + y 2 + z 2 ) = 4 a 2 ,
I E a 2 π 0 π 0 π sin 3 φ sin 2 θ d φ d θ ,
I 2 3 E a 2 .
r s = { 0 - 1 0 } ,             r 0 = { - 1 2 - 1 2 0 } ,             r 0 T = ( - 1 2 ,             - 1 2 ,             0 ) , Q = ( 2 / a 2 0 0 2 / a 2 0 0 0 2 / a 2 ) ,             and             c = { 0 0 0 } .
Q r r T Q r s = - 4 a 4 { x y y 2 z y } .
r 0 T Q r r T Q r s = ( 4 / a 4 2 ) ( x y + y 2 ) .
I E a 2 π 2 0 1 2 π 0 π ( sin 3 φ cos θ sin θ + sin 3 φ sin 2 θ ) d φ d θ ,
0 π sin 3 φ d φ = 4 3 ,             0 1 2 π sin 2 θ d θ = 1 4 π ,
0 1 2 π cos θ sin θ d θ = 1 2 .
I E a 2 3 2 π ( π + 2 ) .