# 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.