Visualization of The Spherical Harmonics
The Laplace spherical harmonics can be visualized by considering their "nodal lines", that is, the set of points on the sphere where, or alternatively where . Nodal lines of are composed of circles: some are latitudes and others are longitudes. One can determine the number of nodal lines of each type by counting the number of zeros of in the latitudinal and longitudinal directions independently. For the latitudinal direction, the real and imaginary components of the associated Legendre polynomials each possess ℓ−|m| zeros, whereas for the longitudinal direction, the trigonometric sin and cos functions possess 2|m| zeros.
When the spherical harmonic order m is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. Such spherical harmonics are a special case of zonal spherical functions. When ℓ = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. For the other cases, the functions checker the sphere, and they are referred to as tesseral.
More general spherical harmonics of degree ℓ are not necessarily those of the Laplace basis, and their nodal sets can be of a fairly general kind.
Read more about this topic: Spherical Harmonics