Illuminance calculation for an arbitrarily shaped flat surface source

—Modification of the contour integration method—

Abstract

The contour integration formula, E=L/2∫_s cosδ·dω for calculating the illuminance of a flat surface source is composed of cosδ and dω, which are not easy to solve. Based on analysis of a three dimensional model, the above formula was modified into the following formulae:
(1) When the surface source is parallel to the illuminated plane:
E'=L/2∫^b_an/l²)dx
(2) When the surface source is inclined to the illuminated plane by ∠β:
(E')=L/2cosβ∫^b_an/l²dx
(3) When the surface source is perpendicular to the illuminated plane:
((E'))=-L/2z∫^b_a1/l²dx
where E'=the illuminance component of the interval A to B on the boundary of the flat surface source, n=the length of an intercept on the y axis by the tangent to the boundary, l=the distance between the illuminated point and the minute segment AB on the boundary of the flat surface source. Since n and l² in these formulae can easily be obtained from x, y and z, all calculations of the illuminance of a flat surface source of an arbitrary shape that can be expressed by x and y can easily be obtained from the sum of the illuminance components E'.