Spherical harmonic

Let us consider continuous functions that only depend on the orientation in space (θ,φ). The spherical harmonics are a basis of such functions.

The decomposition in spherical harmonics is used to represent these functions ; it is similar to the Fourier transform for periodic functions.

In the plane (circular harmonics)

A function is decomposed as

$f(\theta) = \sum_{l = 0}^\infty C_l \cdot Y_l (\theta)$

where Y_l is the circular harmonic. It is expressed as

Y_l(θ) = P_l(cosθ)

where P_l is the Legendre polynomial

The circular harmonics are represented in three ways:

• in cartesian coordinates: y = Y_l(θ)
• in polar coordinates: $r = r_0 + r_1 \cdot Y_l(\theta)$
• in polar coordinates: r = | Y_l(θ) | ²

$l\,$	Cartesian plot of $Y_l^0\left( \theta \right)$	Polar plot of $r_0+r_1Y_l^0\left( \theta \right)$	Polar plot of $|Y_l^0\left( \theta \right)|^2$
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2
3
4		250px	250px

In space

	m=0	m=1	m=2	m=3	m=4
l=0
l=1
l=2
l=3
l=4

Representation as ρ = ρ₀ + ρ₁·Y_l^m(θ,φ)
then the representative surface looks like a "battered" sphere;
Y_l^m is equal to 0 along circles (the representative surface intersects the ρ = ρ₀ sphere at these circles). Y_l^m is alternatively positive and negative between two circles.

the Y₃² with four sections