[ ] arc length radian =1

sr

2 1

s 2 r pi

pi

=

unit circle

( )Consider a differential variation of the direction ,

d and d

What solid anlgle corresponds to this variation?

=

+ +

unit sphere

[ ] [ ] area on a unit sphere

steradian , sr =

rA

2

24 1

A 4 r pi

pi

=

r

d

dA

sphere ofradius r

s

r = 2

A

r = 2

dAd r

=

sphere 4 pi=

( )

( )22

1 1

cos

cos

d d

= =

=

( )( )2 1 1 2cos cos =

d

= 2 2

1 1

sin d d

=

( ) ( ) 2dA r sin d rd r sin d d =

( )12.8

1 2 1 2

Consider a finite solid anglebounded by the directions:

and

Sphere

2 2

hemisphere0 0

sin d d 2pi pi

pi

= =

= = 2

sphere0 0

sin d d 4pi pi

pi

= =

= = =

Hemisphere Useful Fact

2 2

2 0 0

cos d cos sin d d pi pi

pi pi

= =

= =

hemisphere2 pi=

cos

circle of radius r

( )Direction is defined by a pair of angles: ,

is an azimutal angle: 0 2 is a polar angle: 0

pi pi

=

0

( ), =

x

z

( ),

hemisphere2 pi=

substutioncos =

GEOMETRY OF RADIATION

2 2 2 2

x r cos siny r sin sinz r cos

x y z r

=

=

=

+ + =

r

0

1

1

2

2

solid angle

differential solid angle

( ), =

rd

dA

d +

rd d +

r

r sin

x

d

b r sin d =

d

b r sinrd r

=

s

r =

Plane Angle Solid Angle Differential Solid Angle

Differential Solid Anglein spherical coordinates

Finite Solid Anglein spherical coordinates

( )12.7

2

dAd sin d dr

= =