In a cubic model in which the illuminated point is directly under the origin of the co-ordinate axes located on a plane including a flat surface source thereon.
Assuming the length of the intercept of the y axis defined by the tangent to the elementary curve of the boundary of surface source to be
n, the length of the orthogonal projection of the elementary curve on the
x axis to be
dx, the distance between the elementary curve and the illuminated point to be
l, and the distance between the origin and the illuminated point to be z, the formula
E=
L/2∫
Sdω cosδin the contour integration method was analized, and as a result, the following relations were discovered.
(1) In case the surface source is parallel to the illuminated plane,
dω cosδ=
n/l2dx,
(2) In case the surface source is inclined to the illuminated plane,
dω cosδ=
n/l2dx cos β, and
(3) In case the surface source is perpendicular to the illuminated plane,
dω cosδ=
z/l2dxHence, the formulae for obtaining the element of illuminance by the part AB of the boundary of the uniform brightness surface source are as follows:
for the case (1):
E'=
L/2ba
n/l2 dx (1)
for the case (2): (
E')=
L/2 cos β ba
n/l2 dx (2)
for the case (3): ((
E'))=
L/2
zba 1/
l2 dx(3)
The routine method of obtaining the illuminance of a flat surface source by using these formulae is as follows:
(1) obtaining y=f (x) and
y' depending on the shape of the boundary of surface source.
(2) obtaining the length of the intercept on the y axis by the formula
n=
y-
y'·
x.
(3) obtaining
l2.(in case of 1 and 3:=
x2+
y2+
z2, in case of 2:=
x2+
y2+2
yz sin β+
z2)
(4) substituting the above values into the above formulae (1) to (3).
(5) setting the lower limit of the interval of the denite integral at the point where
x is smaller.
(6) calculating the elements of illuminance using Simpson's formula, etc.
(7) obtaining the illuminance by subtracting the arithmetic sum of elements of illuminance belonging to the lower half of the boundary calculating from the arithmetic sum of the elements of illuminance which belongs to the upper half. In case the integral of parameter ψ or
y is to be substituted depending upon the shape of the surface source, the procedure for such a substitution should be inserted between (5) and (6).
The calculation of the illuminance of a fan-shaped source as shown in this paper is an example of a case where the integrals by
x,
y and ψ are mixed up.
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