Illuminance calculation for an arbitrarily shaped flat surface source

—Modification of the contour integration method— (Follow-up paper)

抄録

My preceding paper disclosed a simple method for calculating the illuminance of a flat surface source of arbitrary shape which could be expressed by x and y using the following new formulae:
(1) When the surface source is parallel to the illuminated plane:
E'=L/2∫^b_a(intercept on y axis)/(x²+y²+z²)dx
(2) When the surface source is inclined to the illuminated plane by ∠β:
(E')=L/2 cosβ∫^b_a(intercept on y axis)/(x²+y²+2yz sin β+z²)dx
(3) When the surface source is perpendicular to the illuminated plane:
((E'))=-L/2z∫^ba1/(x²+y²+z²)dx
where E'=the illuminance component for the interval A to B on the boundary of the flat surface source, L=luminance of the source, z=the distance from the illuminated point to the origin located just above the illuminated point for cases (1) and (2), and the distance from the origin to the illuminated point located on the normal to the origin for case (3). This note provides some calculation examples to find the illuminance of flat surface sources of various shapes by means of this new method, and discusses the case having the primitive function F(x).