Abstract

In this paper, the results of a theoretical study of the radiation characteristics of semicircular, circular, and rectangular surface sources have been presented when these surface sources radiate uniformly over their surfaces and obey Lambert’s cosine law. Equations for all three types of sources have been derived giving the total flux falling on an elementary receiving area when this elementary receiving area has arbitrary coordinates and its surface normal arbitrary direction cosines. The classical method of surface integration has been used in each case. These equations are very general in form so that a large number of the equations already published for these three sources become special cases of these equations for which either the coordinates or the direction cosines of the surface normal of the elementary receiving area have particular values. These equations are equally applicable to problems for which the previously designated elementary receiving area becomes the source and the semicircular, circular, and rectangular areas the receiving areas. A simple translation and rotation of coordinates also makes it possible to consider the equally important problems for which the sources are permitted to have arbitrary coordinates and surface normals arbitrary direction cosines. With these equations in this form, it becomes possible to calculate the total flux falling on the elementary receiving area in the presence of an array of semicircular, circular, and rectangular surface sources having arbitrary coordinates and surface normals with arbitrary direction cosines. One numerical example is included for each of these three sources showing how these equations may be used to determine the total flux falling on the elementary receiving area when it is limited to particular planes.

© 1954 Optical Society of America

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References

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  1. J. H. Lambert, Photometrie (1760) (German Translation, E. Anding, Wilhelm Engelmann, Leipzig, 1892), Vol. I, p. 22.
  2. See reference 1, pp. 67 and 80.
  3. E. P. Hyde, Bull. Natl. Bur. Standards 3, 81 (1907);K. Norden, Electrotech. Z. 28, 757 (1907);Electrotech. Z. 29, 883 (1908);U. Bordoni, Assoc. Elect. Ital. 12, 265 (1908);B. Jones, Trans. Ilium. Eng. Soc. (New York) 4, 216 (1909);Trans. Ilium. Eng. Soc. (New York) 5, 281 (1910);Trans. Ilium. Eng. Soc. (New York) 6, 365 (1911);C. Hillebrand, Akad. Wien, Berichte IIA, 118, 1399 (1909);E. B. Rosa, Trans. Illum. Eng. Soc. (N. Y.) 5, 473 (1910);P. D. Foote, Bull. Natl. Bur. Standards 12, 583 (1915);J. W. T. Walsh, Proc. Phys. Soc. (London) 32, 59 (1919);U. Bordoni, Riv. otticca e meccanica precisione 2, II, 1 (1921);P. J. Waldram, Illum. Eng. Soc. (London) 16, 90 (1923);J. Ondracek, Elektrotech. u. Maschinenban (Lichttechnik) 42, 336 (1924);Z. Yamanouti, Researches Electrotech. Lab. (Tokyo) No. 148 (1924);V. Fock, Z. Physik 28, 102 (1924);H. H. Higbie, Trans. Illum. Eng. Soc. (N. Y.) 20, 433 (1925);Trans. Illum. Eng. Soc. (N. Y.) 21, 273 (1926);J. W. T. Walsh, Photometry (Constable & Company, London, 1926), pp. 102, 103;O. Seibert, Arch. Warmwirtsch. 9, 180 (1928);W. Nusselt, Z. Ver. deut. Ing. 72, 673 (1928);P. Moon, J. Opt. Soc. Am. 29, 108 (1939);D. E. Spencer, J. Opt. Soc. Am. 32, 539 (1942);G. Bethe, Optik 8, 485 (1951);Optik 8, 489 (1951).
    [CrossRef]
  4. E. Liebenthal, Praktische Photometrie (F. Vieweg, Braunschweig, 1907), p. 84;E. Gehrcke, Handbuch der Physikalischen Oplik (Barth, Leipzig, 1927), Vol. 1, p. 4;Handbuch der Physik (Verlag julius Springer, Berlin, 1928), Vol. 19, p. 470.
  5. J. H. Lambert, reference 1, p. 76.

1907 (1)

E. P. Hyde, Bull. Natl. Bur. Standards 3, 81 (1907);K. Norden, Electrotech. Z. 28, 757 (1907);Electrotech. Z. 29, 883 (1908);U. Bordoni, Assoc. Elect. Ital. 12, 265 (1908);B. Jones, Trans. Ilium. Eng. Soc. (New York) 4, 216 (1909);Trans. Ilium. Eng. Soc. (New York) 5, 281 (1910);Trans. Ilium. Eng. Soc. (New York) 6, 365 (1911);C. Hillebrand, Akad. Wien, Berichte IIA, 118, 1399 (1909);E. B. Rosa, Trans. Illum. Eng. Soc. (N. Y.) 5, 473 (1910);P. D. Foote, Bull. Natl. Bur. Standards 12, 583 (1915);J. W. T. Walsh, Proc. Phys. Soc. (London) 32, 59 (1919);U. Bordoni, Riv. otticca e meccanica precisione 2, II, 1 (1921);P. J. Waldram, Illum. Eng. Soc. (London) 16, 90 (1923);J. Ondracek, Elektrotech. u. Maschinenban (Lichttechnik) 42, 336 (1924);Z. Yamanouti, Researches Electrotech. Lab. (Tokyo) No. 148 (1924);V. Fock, Z. Physik 28, 102 (1924);H. H. Higbie, Trans. Illum. Eng. Soc. (N. Y.) 20, 433 (1925);Trans. Illum. Eng. Soc. (N. Y.) 21, 273 (1926);J. W. T. Walsh, Photometry (Constable & Company, London, 1926), pp. 102, 103;O. Seibert, Arch. Warmwirtsch. 9, 180 (1928);W. Nusselt, Z. Ver. deut. Ing. 72, 673 (1928);P. Moon, J. Opt. Soc. Am. 29, 108 (1939);D. E. Spencer, J. Opt. Soc. Am. 32, 539 (1942);G. Bethe, Optik 8, 485 (1951);Optik 8, 489 (1951).
[CrossRef]

Hyde, E. P.

E. P. Hyde, Bull. Natl. Bur. Standards 3, 81 (1907);K. Norden, Electrotech. Z. 28, 757 (1907);Electrotech. Z. 29, 883 (1908);U. Bordoni, Assoc. Elect. Ital. 12, 265 (1908);B. Jones, Trans. Ilium. Eng. Soc. (New York) 4, 216 (1909);Trans. Ilium. Eng. Soc. (New York) 5, 281 (1910);Trans. Ilium. Eng. Soc. (New York) 6, 365 (1911);C. Hillebrand, Akad. Wien, Berichte IIA, 118, 1399 (1909);E. B. Rosa, Trans. Illum. Eng. Soc. (N. Y.) 5, 473 (1910);P. D. Foote, Bull. Natl. Bur. Standards 12, 583 (1915);J. W. T. Walsh, Proc. Phys. Soc. (London) 32, 59 (1919);U. Bordoni, Riv. otticca e meccanica precisione 2, II, 1 (1921);P. J. Waldram, Illum. Eng. Soc. (London) 16, 90 (1923);J. Ondracek, Elektrotech. u. Maschinenban (Lichttechnik) 42, 336 (1924);Z. Yamanouti, Researches Electrotech. Lab. (Tokyo) No. 148 (1924);V. Fock, Z. Physik 28, 102 (1924);H. H. Higbie, Trans. Illum. Eng. Soc. (N. Y.) 20, 433 (1925);Trans. Illum. Eng. Soc. (N. Y.) 21, 273 (1926);J. W. T. Walsh, Photometry (Constable & Company, London, 1926), pp. 102, 103;O. Seibert, Arch. Warmwirtsch. 9, 180 (1928);W. Nusselt, Z. Ver. deut. Ing. 72, 673 (1928);P. Moon, J. Opt. Soc. Am. 29, 108 (1939);D. E. Spencer, J. Opt. Soc. Am. 32, 539 (1942);G. Bethe, Optik 8, 485 (1951);Optik 8, 489 (1951).
[CrossRef]

Lambert, J. H.

J. H. Lambert, Photometrie (1760) (German Translation, E. Anding, Wilhelm Engelmann, Leipzig, 1892), Vol. I, p. 22.

J. H. Lambert, reference 1, p. 76.

Liebenthal, E.

E. Liebenthal, Praktische Photometrie (F. Vieweg, Braunschweig, 1907), p. 84;E. Gehrcke, Handbuch der Physikalischen Oplik (Barth, Leipzig, 1927), Vol. 1, p. 4;Handbuch der Physik (Verlag julius Springer, Berlin, 1928), Vol. 19, p. 470.

Bull. Natl. Bur. Standards (1)

E. P. Hyde, Bull. Natl. Bur. Standards 3, 81 (1907);K. Norden, Electrotech. Z. 28, 757 (1907);Electrotech. Z. 29, 883 (1908);U. Bordoni, Assoc. Elect. Ital. 12, 265 (1908);B. Jones, Trans. Ilium. Eng. Soc. (New York) 4, 216 (1909);Trans. Ilium. Eng. Soc. (New York) 5, 281 (1910);Trans. Ilium. Eng. Soc. (New York) 6, 365 (1911);C. Hillebrand, Akad. Wien, Berichte IIA, 118, 1399 (1909);E. B. Rosa, Trans. Illum. Eng. Soc. (N. Y.) 5, 473 (1910);P. D. Foote, Bull. Natl. Bur. Standards 12, 583 (1915);J. W. T. Walsh, Proc. Phys. Soc. (London) 32, 59 (1919);U. Bordoni, Riv. otticca e meccanica precisione 2, II, 1 (1921);P. J. Waldram, Illum. Eng. Soc. (London) 16, 90 (1923);J. Ondracek, Elektrotech. u. Maschinenban (Lichttechnik) 42, 336 (1924);Z. Yamanouti, Researches Electrotech. Lab. (Tokyo) No. 148 (1924);V. Fock, Z. Physik 28, 102 (1924);H. H. Higbie, Trans. Illum. Eng. Soc. (N. Y.) 20, 433 (1925);Trans. Illum. Eng. Soc. (N. Y.) 21, 273 (1926);J. W. T. Walsh, Photometry (Constable & Company, London, 1926), pp. 102, 103;O. Seibert, Arch. Warmwirtsch. 9, 180 (1928);W. Nusselt, Z. Ver. deut. Ing. 72, 673 (1928);P. Moon, J. Opt. Soc. Am. 29, 108 (1939);D. E. Spencer, J. Opt. Soc. Am. 32, 539 (1942);G. Bethe, Optik 8, 485 (1951);Optik 8, 489 (1951).
[CrossRef]

Other (4)

E. Liebenthal, Praktische Photometrie (F. Vieweg, Braunschweig, 1907), p. 84;E. Gehrcke, Handbuch der Physikalischen Oplik (Barth, Leipzig, 1927), Vol. 1, p. 4;Handbuch der Physik (Verlag julius Springer, Berlin, 1928), Vol. 19, p. 470.

J. H. Lambert, reference 1, p. 76.

J. H. Lambert, Photometrie (1760) (German Translation, E. Anding, Wilhelm Engelmann, Leipzig, 1892), Vol. I, p. 22.

See reference 1, pp. 67 and 80.

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Figures (9)

F. 1
F. 1

The total flux radiated by a semicircular surface source to an elementary receiving area with arbitrary coordinates having a surface normal with arbitrary direction cosines.

F. 2
F. 2

The total flux radiated by a semicircular surface source to an elementary receiving area on the X0 axis having a surface normal with arbitrary direction cosines.

F. 3
F. 3

The total flux radiated by a semicircular surface source to an elementary receiving area in the (X0,Y0) plane having a surface normal with arbitrary direction cosines.

F. 4
F. 4

The total flux radiated by a semicircular surface source to an elementary receiving area in the (X0,Z0) plane having a surface normal with arbitrary direction cosines.

F. 5
F. 5

The total flux radiated by a semicircular surface source to an elementary receiving area in a plane normal to the semicircular surface and through its diameter when the elementary receiving area has arbitrary coordinates and a surface normal with arbitrary direction cosines.

F. 6
F. 6

The total flux radiated by a rectangular surface source to an elementary receiving area with arbitrary coordinates having a surface normal with arbitrary direction cosines.

F. 7
F. 7

Isophotopic lines for a semicircular surface source when the elementary receiving area is limited to a plane parallel to the source at a distance equal to its radius and has its surface, normal perpendicular to this plane.

F. 8
F. 8

Isophotopic lines for a circular surface source when the elementary receiving area is limited to a plane parallel to the source at a distance equal to its radius and has its surface normal perpendicular to this plane.

F. 9
F. 9

Isophotopic lines for a square surface source when the elementary receiving area is limited to a plane parallel to the source at a distance equal to half the length of the side and has its surface normal perpendicular to this plane.

Equations (20)

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d 2 F = B d s cos θ d s cos θ a 2 .
cos θ = X 0 / a .
cos θ α cos ( a , X ) + β cos ( a , Y ) + γ cos ( a , Z ) ,
cos ( a , X ) = X 0 / a , cos ( a , Y ) = Y 0 h sin φ a , cos ( a , Z ) = Z 0 h cos φ a .
a = [ X 0 2 + Y 0 2 + Z 0 2 + h 2 2 Y 0 h sin φ 2 Z 0 h cos φ ] 1 2 .
d F = B d s X 0 π / 2 + π / 2 R × ( α X 0 + β X 0 + γ Z 0 β h sin φ γ h cos φ ) h d h d φ ( X 0 2 + Y 0 2 + Z 0 2 + h 2 2 Y 0 h sin φ 2 Z 0 h cos φ ) 2 .
d F = B d s X 0 π / 2 π / 2 0 R × α X 0 β h sin φ γ h cos φ ( X 0 2 + h 2 ) 2 h d h d φ .
d F = B d s X 0 { π α X 0 0 R h d h ( X 0 2 + h 2 ) 2 2 γ 0 R h 2 d h ( X 0 2 + h 2 ) 2 } .
d F = B d s { π α R 2 2 ( X 0 2 + R 2 ) + γ [ X 0 R X 0 2 + R 2 tan 1 R X 0 ] } .
d F = B d s { π α R 2 2 ( X 0 2 + R 2 ) γ [ X 0 R X 0 2 + R 2 tan 1 R X 0 ] } .
d F = 2 B d s X 0 { 0 R A E G C F G + D F 2 D E 2 ( E 2 r 2 2 ) ( E 2 F 2 ) d h + 0 R A E C F D G ( E 2 F 2 ) 3 2 tan 1 [ E 2 r 2 2 ] 1 2 2 Z 0 h d h } ,
A = h ( α X 0 + β Y 0 + γ Z 0 ) , E = X 0 2 + Y 0 2 + Z 0 2 + h 2 , C = β h 2 , F = 2 Y 0 h , D = γ h 2 , G = 2 Z 0 h , r 2 = 2 h [ Y 0 2 + Z 0 2 ] 1 2 .
d F = B d s X 0 { π α 4 X 0 π ( β Y 0 + γ Z 0 ) 4 ( Y 0 2 + Z 0 2 ) + ( α Z 0 γ X 0 ) 2 X 0 [ X 0 2 + Z 0 2 ] 1 2 tan 1 2 R [ X 0 2 + Z 0 2 ] 1 2 a R 2 + γ Y 0 + β Z 0 4 ( Y 0 2 + Z 0 2 ) log e a + R 2 + 2 Y 0 R a + R 2 2 Y 0 R α ( Y 0 2 + Z 0 2 ) ( a R 2 ) X 0 ( β Y 0 + γ Z 0 ) ( a + R 2 ) 2 X 0 ( Y 0 2 + Z 0 2 ) [ ( a + R 2 ) 2 b R 2 ] 1 2 × tan 1 [ ( a + R 2 ) 2 b R 2 ] 1 2 2 Z 0 R } .
a = X 0 2 + Y 0 2 + Z 0 2 , b = 4 ( Y 0 2 + Z 0 2 ) .
d F = B d s X 0 { π α 4 X 0 π ( β Y 0 + γ Z 0 ) 4 ( Y 0 2 + Z 0 2 ) ( α Z 0 γ X 0 ) 2 X 0 [ X 0 2 + Z 0 2 ] 1 2 tan 1 2 R [ X 0 2 + Z 0 2 ] 1 2 a R 2 γ Y 0 + β Z 0 4 ( Y 0 2 + Z 0 2 ) log e a + R 2 + 2 Y 0 R a + R 2 2 Y 0 R α ( Y 0 2 + Z 0 2 ) ( a R 2 ) X 0 ( β Y 0 + γ Z 0 ) ( a + R 2 ) 2 X 0 ( Y 0 2 + Z 0 2 ) [ ( a + R 2 ) 2 b R 2 ] 1 2 × tan 1 [ ( a + R 2 ) 2 b R 2 ] 1 2 2 Z 0 R } ,
d F = π B d s X 0 2 { α X 0 [ 1 a R 2 [ ( a + R 2 ) 2 b R 2 ] 1 2 ] + β Y 0 + γ Z 0 Y 0 2 + Z 0 2 [ a + R 2 [ ( a + R 2 ) 2 b R 2 ] 1 2 1 ] }
d F = B d s X 0 4 { π α X 0 [ 1 a R 2 [ ( a + R 2 ) 2 b R 2 ] 1 2 ] } + π ( β Y 0 + γ Z 0 ) Y 0 2 + Z 0 2 [ a + R 2 [ ( a + R 2 ) 2 b R 2 ] 1 2 1 ] β Z 0 γ Y 0 Y 0 2 + Z 0 2 log e a + R 2 + 2 R [ Y 0 2 + Z 0 2 ] 1 2 a + R 2 2 R [ Y 0 2 + Z 0 2 ] 1 2 + 2 ( β Z 0 γ Y 0 ) X 0 [ Y 0 2 + Z 0 2 ] 1 2 tan 1 2 X 0 R a R 2 } ,
d F = B d s X 0 w + w l + l × α X 0 + β ( Y 0 Y ) + γ ( Z 0 Z ) [ X 0 2 + ( Y 0 Y ) 2 + ( Z 0 Z ) 2 ] 2 d Y d Z .
d F = B d s 2 { γ X 0 α ( Z 0 w ) [ X 0 2 + ( Z 0 w ) 2 ] 1 2 [ tan 1 Y 0 + l [ X 0 2 + ( Z 0 w ) 2 ] 1 2 tan 1 Y 0 l [ X 0 2 + ( Z 0 w ) 2 ] 1 2 ] γ X 0 α ( Z 0 + w ) [ X 0 2 + ( Z 0 + w ) 2 ] 1 2 [ tan 1 Y 0 + l [ X 0 2 + ( Z 0 + w ) 2 ] 1 2 tan 1 Y 0 l [ X 0 2 + ( Z 0 + w ) 2 ] 1 2 ] + β X 0 α ( Y 0 l ) [ X 0 2 + ( Y 0 l ) 2 ] 1 2 [ tan 1 Z 0 + w [ X 0 2 + ( Y 0 l ) 2 ] 1 2 tan 1 Z 0 w [ X 0 2 + ( Y 0 l ) 2 ] 1 2 ] β X 0 α ( Y 0 + l ) [ X 0 2 + ( Y 0 + l ) 2 ] 1 2 [ tan 1 Z 0 + w [ X 0 2 + ( Y 0 + l ) 2 ] 1 2 tan 1 Z 0 w [ X 0 2 + ( Y 0 + l ) 2 ] 1 2 ] } .
d F = B d s 2 { γ X 0 α ( Z 0 w ) [ X 0 2 + ( Z 0 w ) 2 ] 1 2 tan 1 2 l [ X 0 2 + ( Z 0 w ) 2 ] 1 2 X 0 2 + ( Z 0 w ) 2 + Y 0 2 l 2 γ X 0 α ( Z 0 + w ) [ X 0 2 + ( Z 0 + w ) 2 ] 1 2 tan 1 2 l [ X 0 2 + ( Z 0 + w ) 2 ] 1 2 X 0 2 + ( Z 0 + w ) 2 + Y 0 2 l 2 + β X 0 α ( Y 0 l ) [ X 0 2 + ( Y 0 l ) 2 ] 1 2 tan 1 2 w [ X 0 2 + ( Y 0 l ) 2 ] 1 2 X 0 2 + ( Y 0 l ) 2 + Z 0 2 w 2 β X 0 α ( Y 0 + l ) [ X 0 2 + ( Y 0 + l ) 2 ] 1 2 tan 1 2 w [ X 0 2 + ( Y 0 + l ) 2 ] 1 2 X 0 2 + ( Y 0 + l ) 2 + Z 0 2 w 2 } .