Visualizing the spherical harmonics

The real spherical harmonics \(Y_l^m(\theta, \varphi)\) form an orthonormal basis on the sphere. This notebook constructs and plots them by applying the inverse SHT to unit basis vectors in spectral space.

Setup

import torch
import matplotlib.pyplot as plt

from torch_harmonics import RealSHT, InverseRealSHT
from torch_harmonics.plotting import plot_sphere

Forming the Vandermonde matrix

The Vandermonde matrix maps spectral coefficients to grid values. Each column corresponds to a single spherical harmonic \(Y_l^m\). We construct it by applying the ISHT to each canonical basis vector \(e_{l,m}\) in the coefficient space.

For negative orders \(m < 0\), the real spherical harmonics are recovered by placing the basis vector in the imaginary part of the \(|m|\) coefficient.

nlat = 60
nlon = 2*nlat
lmax = mmax = nlat

sht = RealSHT(nlat, nlon, lmax=lmax, mmax=mmax)
isht = InverseRealSHT(nlat, nlon, lmax=lmax, mmax=mmax)

# forming the Vandermonde matrix

nmodes = int(lmax * lmax)
e = torch.zeros(nmodes, lmax, mmax, dtype=torch.complex64)

midx = lambda l,m : l*(l+1) + m

for l in range(lmax):
    for m in range(-l, l+1):
        e[midx(l,m), l, abs(m)] = 1.0 if m >= 0 else 1.0j

vdm = isht(e)

Plotting the spherical harmonics

We plot the first few spherical harmonics on a grid indexed by degree \(l\) (rows) and order \(m\) (columns). Each subplot shows an orthographic projection of the sphere:

import os
os.makedirs("plots", exist_ok=True)

plt_lmax = 3

fig = plt.figure(layout='constrained', figsize=(12, 8))
subfigs = fig.subfigures(plt_lmax, 2*plt_lmax-1)

for l in range(plt_lmax):
    for m in range(-l, l+1):
        plot_sphere(vdm[midx(l,m)], fig=subfigs[l, m+plt_lmax-1], projection="orthographic", title=f"l={l}, m={m}", central_latitude=20, central_longitude=30, cmap="plasma")

plt.savefig("./plots/spherical_harmonics.png")

Animated version (optional)

The following cells generate a rotating animation of the spherical harmonics. Uncomment to run:

# import matplotlib.animation as animation

# fig = plt.figure(layout='constrained', figsize=(12, 8), dpi=72)
# subfigs = fig.subfigures(plt_lmax, plt_lmax)

# moviewriter = animation.writers['pillow'](fps=30)
# moviewriter.setup(fig, './plots/spherical_harmonics.gif', dpi=72)

# for frame in range(120):
#     # compute the rotation of the sphere
#     lon = -120 - 3 * frame

#     if frame % 10 == 0:
#         print(f"frame={frame}")

#     for l in range(plt_lmax):
#         for m in range(l+1):
#             plot_sphere(vdm[midx(l,m)], fig=subfigs[l, m], projection="orthographic", title=f"l={l}, m={m}", central_longitude=lon)

#     plt.draw()
#     moviewriter.grab_frame()

# moviewriter.finish()