solid angle

solid angle

A

conical surface

may contain a certain portion

Ω

of the space

ℝ

3

.  This portion is called

solid angle

or

space angle

.  If the conical surface contains a portion

A

of a spherical surface with radius

R

and with centre (

http://planetmath.org/Sphere

)

P

in the of the solid angle, then the magnitude of the solid angle is given by

Ω

=

A

R

2

which is on the radius

R

. The spherical surface can be replaced by any surface

a

, through which all the

half-lines

originating from

P

and being contained in the solid angle go.  Then the solid angle may be computed from the

Ω

=

-

∫

a

d

⁢

a

→

⋅

∇

⁡

1

r

,

(1)

where

r

is the length of the

position vector

r

→

for the points on the surface

a

.
The

full solid angle

, consisting of all points of

ℝ

3

, has the magnitude

4

⁢

π

.

The SI of solid angle, analogous to the angle radian, is the

steradian

(

=

1

⁢

sr

).  The steradian takes a proportion

1

4

⁢

π

, or approximately 7.957747 %, of the

surface area

of a sphere.

If the solid angle is

bounded

by three planes having exactly one common point, it may be called a

trihedral angle

; cf. the example 2!

Example 1.

The solid angle determined by a

right circular cone

with the angle

α

between its axis and is equal to

2

⁢

π

⁢

(

1

-

cos

⁡

α

)

, i.e.

4

⁢

π

⁢

sin

2

⁡

α

2

.

Example 2.

Let

r

→

1

,

r

→

2

,

r

→

3

be the position vectors of three points in

ℝ

3

and

r

1

,

r

2

,

r

3

their lengths.  Then the solid angle

Ω

of the

tetrahedron

by the vectors

r

i

→

is obtained from the equation

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

,

(2)

where the

numerator

of the is the

triple scalar product

of the vectors.  This equation is expressed simplier using the

unit vectors

u

→

i

corresponding

r

→

i

:

tan

⁡

Ω

2

=

u

→

1

×

u

→

2

⋅

u

→

3

1

+

u

→

2

⋅

u

→

3

+

u

→

3

⋅

u

→

1

+

u

→

1

⋅

u

→

2

The result (2) is due to van Oosterom and Strackee 1983.

Example 3.

Using (2), one can easily get the apical (

http://planetmath.org/ConeInMathbbR3

) solid angle of a

right pyramid

(

http://planetmath.org/ConeInMathbbR3

) with square base:

Ω

=

4

⁢

arctan

⁡

a

2

2

⁢

h

⁢

2

⁢

a

2

+

4

⁢

h

2

=

4

⁢

arcsin

⁡

a

2

a

2

+

4

⁢

h

2

Here

a

is the side of the base square and

h

is the height (

http://planetmath.org/ConeInMathbbR3

) of the

pyramid

.  Cf. the

solid angle of rectangular 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.

Title

solid angle

Canonical name

SolidAngle

Date of creation

2014-03-12 21:18:06

Last modified on

2014-03-12 21:18:06

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

26

Author

pahio (2872)

Entry type

Definition

Classification

msc 15A72

Classification

msc 51M25

Related topic

ConvexAngle

Related topic

Radian

Related topic

AreaOfASphericalTriangle

Related topic

DihedralAngle

Defines

space angle

Defines

full solid angle

Defines

steradian

Defines

trihedral angle

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