2 Introduction to the Bispherical Coordinate System
In the bispherical coordinate system [8, 9] shown in Fig. 2 for a two-sphere system,
the position of any point X in space is described with reference to a pair of foci,
which are separated by a distance of 2a. The foci are defined as inverse points of
each other such that
d 1 c 1 ¼ a
2
1
ð1Þ
and
d 2 c 2 ¼ a
2
2 :
ð2Þ
The centre-to-centre separation h > 0 is given by
h ¼ s þ a 1 þ a 2 ¼ 2a þ c 1 þ c 2
ð3Þ
where s ≥ 0 is the surface-to-surface separation. The bispherical coordinates are
often denoted as (η, ξ, ϕ), where
Fig. 2 Schematic diagrams of the bispherical coordinate system. The diagram on the left
illustrates how the length quantities c 1 , c 2 , d 1 and d 2 are related to the inter-focal separation 2a and
to the centre-to-centre separation h, and the diagram on the right illustrates how the position of an
arbitrary point X in space is described with reference to the foci
A General Geometric Representation of Sphere-Sphere Interactions
31