solid angle of rectangular pyramid

solid angle of rectangular pyramid

We calculate the apical

solid angle

of a rectangular

pyramid

, as an example of using the

http://planetmath.org/node/7266

formula

of van Oosterom and Strackee for determining the solid angle

Ω

subtended at the origin by a

triangle

:

tan

⁡

Ω

2

=

r

→

1

×

r

→

2

⋅

r

→

3

(

r

→

1

⋅

r

→

2

)

⁢

r

3

+

(

r

→

2

⋅

r

→

3

)

⁢

r

1

+

(

r

→

3

⋅

r

→

1

)

⁢

r

2

+

r

1

⁢

r

2

⁢

r

3

(1)

Here,

r

→

1

,

r

→

2

,

r

→

3

are the

position vectors

of the vertices of the triangle and

r

1

,

r

2

,

r

3

their .

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

r

→

1

,

r

→

2

,

r

→

3

, respectively.  Then we have in the

numerator

of (1) the

scalar triple product

r

→

1

×

r

→

2

⋅

r

→

3

=

|

a

b

h

-

a

b

h

a

-

b

h

|

=

a

⁢

|

b

h

-

b

h

|

+

b

⁢

|

h

-

a

h

a

|

+

h

⁢

|

-

a

b

a

-

b

|

=

4

⁢

a

⁢

b

⁢

h

.

The vectors have the common length

a

2

+

b

2

+

h

2

, and the denominator of (1) then attains the value

4

⁢

h

2

⁢

a

2

+

b

2

+

h

2

.  Thus the formula (1) gives

tan

⁡

Ω

2

=

a

⁢

b

h

⁢

a

2

+

b

2

+

h

2

which result may be reformulated by using the goniometric formula

sin

⁡

x

=

tan

⁡

x

1

+

tan

2

⁡

x

as

sin

⁡

Ω

2

=

a

⁢

b

(

a

2

+

h

2

)

⁢

(

b

2

+

h

2

)

.

(2)

Thus the whole apical solid angle of the

http://planetmath.org/node/7357

right rectangular pyramid is

Ω

=

4

⁢

arcsin

⁡

a

⁢

b

(

a

2

+

h

2

)

⁢

(

b

2

+

h

2

)

.

(3)

A variant of (3) is found in [3].

In the special case of a

regular pyramid

we have simply

Ω

=

4

⁢

arcsin

⁡

a

2

a

2

+

h

2

(4)

where

2

⁢

a

is the side (

http://planetmath.org/

Polygon

) of the base square.

Note that in (2), the quotients

a

a

2

+

h

2

and

b

b

2

+

h

2

are sines of certain angles in the pyramid.

References

1

A. van Oosterom & J. Strackee

:  A solid angle of a plane triangle.  –

IEEE Trans. Biomed. Eng.

30

:2 (1983); 125–126.

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.

Title

solid angle of rectangular pyramid

Canonical name

SolidAngleOfRectangularPyramid

Date of creation

2013-03-22 19:16:02

Last modified on

2013-03-22 19:16:02

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Example

Classification

msc 15A72

Classification

msc 51M25

Related topic

CyclometricFunctions

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