solid angle of rectangular pyramid∗
pahio†
2013-03-22 3:36:01
We calculate the apical solid angle of a right rectangular pyramid, as an
example of using the formula of van Oosterom and Strackee for determining the
solid angle Ω subtended at the origin by a triangle:
tan
Ω
~r1 ×~r2 ·~r3
=
2
(~r1 ·~r2 )r3 + (~r2 ·~r3 )r1 + (~r3 ·~r1 )r2 + r1 r2 r3
(1)
Here, ~r1 , ~r2 , ~r3 are the position vectors of the vertices of the triangle and
r1 , r2 , r3 their lengths.
Let the apex of the pyramid be in the origin and the vertices of the base
rectangle be
(±a, ±b, h)
where a, b and h are positive numbers. We take the half-triangle of the base
determined by the three vertices
(a, b, h), (−a, b, h), (a, −b, h),
with the position vectors ~r1 , ~r2 , ~r3 , respectively. Then we have in the numerator
of (1) the scalar triple product
a
b h
−a b h −a
b h
= 4abh.
+ h
~r1 ×~r2 ·~r3 = −a b h = a + b
a −b
−b h
h a
a −b h
√
The vectors have the common
length a2 +b2 +h2 , and the denominator of (1)
√
then attains the value 4h2 a2 +b2 +h2 . Thus the formula (1) gives
tan
Ω
ab
= √
2
2
h a +b2 +h2
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which result may be reformulated by using the goniometric formula
sin x = √
tan x
1 + tan2 x
as
sin
Ω
ab
.
= p
2
2
(a +h2 )(b2 +h2 )
(2)
Thus the whole apical solid angle of the right rectangular pyramid is
Ω = 4 arcsin p
ab
(a2 +h2 )(b2 +h2 )
.
(3)
A variant of (3) is found in [3].
In the special case of a regular pyramid we have simply
Ω = 4 arcsin
a2
(4)
a2 +h2
where 2a is the side of the base square.
Note that in (2), the quotients
in the pyramid.
√ a
a2 +h2
and
√ b
b2 +h2
are sines of certain angles
.
2b
2a
h
.
References
[1] A. van Oosterom & J. Strackee: A solid angle of a plane triangle. –
IEEE Trans. Biomed. Eng. 30:2 (1983); 125–126.
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[2] M. S. Gossman & A. J. Pahikkala & M. B. Rising & P. H. McGinley: Providing solid angle formalism for skyshine calculations. – Journal
of Applied Clinical Medical Physics 11:4 (2010); 278–282.
[3] M. S. Gossman & A. J. Pahikkala & M. B. Rising & P. H. McGinley: Letter to the editor. – Journal of Applied Clinical Medical Physics
12:1 (2011); 242–243.
[4] M. S. Gossman & M. B. Rising & P. H. McGinley & A. J.
Pahikkala: Radiation skyshine from a 6 MeV medical accelerator. – Journal of Applied Clinical Medical Physics 11:3 (2010); 259–264.
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