It is intuitively obvious that a cow is much better approximated by an ellipsoid than by a
sphere. Less obvious, or at least less well known, is the method of ellipsoidal harmonics for
solving the ellipsoidal cow exactly1,2 . In an effort to increase the popularity and impact of
these fascinating functions, we present in this paper two open-source (BSD) implementations
for calculating the ellipsoidal harmonics and solving problems of potential theory
I've always wanted to solve cows.
We present two open-source (BSD) implementations of ellipsoidal harmonic expansions for solving problems of potential theory using separation of variables. Ellipsoidal harmonics are used surprisingly infrequently, considering their substantial value for problems ranging in scale from molecules to the entire solar system. In this article, we suggest two possible reasons for the paucity relative to spherical harmonics. The first is essentially historical---ellipsoidal harmonics developed during the late 19th century and early 20th, when it was found that only the lowest-order harmonics are expressible in closed form. Each higher-order term requires the solution of an eigenvalue problem, and tedious manual computation seems to have discouraged applications and theoretical studies. The second explanation is practical: even with modern computers and accurate eigenvalue algorithms, expansions in ellipsoidal harmonics are significantly more challenging to compute than those in Cartesian or spherical coordinates. The present implementations reduce the "barrier to entry" by providing an easy and free way for the community to begin using ellipsoidal harmonics in actual research. We demonstrate our implementation using the specific and physiologically crucial problem of how charged proteins interact with their environment, and ask: what other analytical tools await re-discovery in an era of inexpensive computation?http://arxiv.org/abs/1204.0267
Unsurprisingly, the more general shape allows ellipsoidal-harmonic expansions to be
more accurate than ones based on spherical harmonics7,12,13 . Successful applications cover
most of potential theory: gravity7,10 , electrostatics8,14–18 , electromagnetics19–22 , hydrody-
namics23–26 , and elasticity27 . Specific uses in molecular and biological sciences include solv-
ing the Schr ̈dinger equation28 , modeling van der Waals (close packing) interactions between
o
molecules12 , design of magnetic resonance imaging (MRI) devices29 and analysis of clinical
2
electroencephalography (EEG) and magnetoencephalography (MEG) data21,22 . Unfortu-
nately, despite the range of applications and their advantages over spheres, a comparatively
small number of publications actually use, or encourage the use of, ellipsoidal harmonics in
practice.
