Stage Lighting And Effects Equipment

Let
Ps = perimeter of the semicircle
Pr = perimeter of the rectangle
Pt = total perimeter of the window
r = radius of the semicircle
h = height of the rectangle
Ls = light transmittance of the semicircle
Lr = light transmittance of the rectangle
Lt = total light transmited through the window
As = area of the semicircle
Ar = area of the rectangle
PI = number pi

Let's start with the perimeter:
Perimiter of a semicircle has the formula
Ps = (PI)*r
For this rectangle the perimeter will be obtained by adding 3 sides, height + width + height, (in this case width is the same as 2 radii)
Pr = h + 2r + h = 2r + 2h
Total perimeter
Pt = Ps + Pr = (PI)*r + 2r + 2h = r(PI + 2) + 2h
Since the perimeter is fixed, we can express h in terms of r
Pt = r(PI + 2) + 2h
h = (Pt - (PI + 2)r) / 2

Now for the light transmited
semicircle transmits only half as much light as rectangle

Ls = (1/2)Lr
total light transmited
Lt = Ls*As + Lr*Ar
Lt = Ls*((PI*r^2)/2) + 2Ls*(h*2r)
Lt = Ls((PI*r^2)/2) + 4h*r)
And,
h = (Pt - (PI + 2)r) / 2
Thus,
Lt = Ls((PI*r^2)/2) + 4r*((Pt - (PI + 2)r) / 2))
Lt = Ls((PI*r^2)/2) + 2Pt*r - 2PI*r^2 - 4r^2)
Lt = Ls((-3PI*r^2)/2) + 2Pt*r - 4r^2)
Lt = Ls(2Pt*r - r^2((3/2)PI + 4))

to get the maximum d(Lt)/dr = 0
d(Lt)/dr = Ls(2Pt - 2r((3/2)PI + 4)) = 0
(2Pt - 2r((3/2)PI + 4)) = 0
r = 2Pt / (3PI +
And,
h = (Pt - (PI + 2)r) / 2
h = Pt/2 - (2Pt / (3PI + 8))(PI + 2)/2
h = Pt(1/2 - (PI + 2)/(3PI + 8))
h = Pt(PI + 4)/(6PI + 16)

Therefore the proportions of the window are a semicricle with radius
r = 2Pt / (3PI +
and a rectangle with a height
h = Pt(PI + 4)/(6PI + 16)

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