Advanced Physics for Brachytherapy 2018

The method

m: 0 1 2 3 … 0 1

Many orthogonal bases exist

l:

2 3 …

In example: ➢ Spherical harmonics, , are a series of functions defined on the surface of a sphere that exhibit orthogonality: ) ,(   m l Y

m

Y

' ) ,( l Y m  



) ,(' 



dφ dθ sin =

 

l

2

 

1

m

Y

' ) ,( l Y m   

) (cos ) ,('  d



=

δ dφ

  − 

ll' δ mm'

l

2

1

and therefore they can be used to expand any function defined on the surface of a sphere in an infinite series:

Various types of spherical harmonics are available. A particular set, of order l=N (orthogonal basis) + the corresponding weights of a function are called a quadrature set of order N .

l

  =

m

f

Y mlC





= ) ,(





, ) ,(



,

l

l

l m

0

−=

l Y f 

 d m

= mlC





( ) ,(



) ,

 

,