Visualization of Spherical Harmonics

This is an interactive web page (based on Javascript) to visualize Spherical Harmonics.

You can directly select the indices 'l' and 'm' or start an animation.
  • 'show all m' loops 'm' for fixed 'l' by 'm = 0,...,l'
  • 'show all l' loops 'l' for fixed 'm' by 'l = l0,...,50'
  • 'Rotate' rotates the Earth around the polar Axis
  • 'Stop' stops animation
  • 'Export' displays the image in a separate window with the option to save it into a file (right mouse click is also possible to open the dialogue).
Short Explanation
The image shows one specific surface spherical harmonic of degree 'l' and order 'm',
denoted as 'Ylm(φ,λ)' (they are solutions of ΔYlm = 0 ).
Any source-free field on the sphere can be computed with these terms as the double sum over 'l' (l = 1,..,lmax) and 'm' (m = 0,..,l) with:

G(φ,λ) = ∑lm Ylm(φ,λ) with Ylm(φ,λ) = Plm(sin(φ)) × [Clm cos(m λ) + Slm sin(m λ)]

where 'φ' = [-90°,+90°] is the latitude, 'λ' = [0°,360°] the longitude. 'Plm(x)' are the Legendre polynoms and 'Clm', 'Slm' are constant coefficients, which describe the actual field/model.
Here the cos-term 'ClmPlm(sin(φ))cos(m λ)' is displayed, arbitrarily scaled by Clm = 100⁄sqrt(l) . The sin-term is identical with a phase shift of π⁄m.
The blue lines indicate the zero-circles, where 'Ylm = 0'.

Select 'l'   Select 'm'          
plates/lines   Boost   Color   Background  
By clicking on the image you can Rotate and by scrolling with the mouse wheel over the image you can Zoom.