Following my recent ignominious extraction from the Dupin node
quicksands with the assistance of a flying-squad who do know some
algebraic geometry, and muster a computer algebra system which can
solve polynomial equations, I remounted my cyclide and pedalled
off shakily to further adventures in neglected corners of this subfusc
mathematical suburb. Among the customary extensive selection
of technical lamposts, potholes and manhole-covers lying subsequently
in wait, there lurked the following conundrum.
To recap a recent post by Ralph, a Dupin cyclide in canonical pose
may be specified by three parameters (due apparently to Maxwell,
rather than to Cayley as earlier hazarded). The radius "a" specifies
a central circle at the origin; with centre on this, a perpendicular
cross-sectional circle sweeps out a tube with mean radius "m";
while in the course of a single revolution, the actual tube radius
is offset from m-c through m+c and back to m-c again.
In terms of c,a,m, the implicit equation is then a Cartesian
quartic
(x^2 + y^2 + z^2 - m^2 - a^2 + c^2)^2
- 4(a x - c m)^2 - 4(a^2 + c^2)y^2 = 0 .
The general shape of the surface will be
"horned" when 0 < m < c < a ;
"ring" when 0 < c < m < a ;
"spindle" when 0 < c < a < m ;
with various special cases at boundaries between. For example, when
the differential offset vanishes c = 0 we have a torus; when in
addition the tube radius equals the central radius m = a we have a
double sphere centred at the origin (and giving the average graphics
surface plotter a furry tongue).
Now consider the special case where offset equals central radius
c = a. It's easily established that the equation factors as the
product of
(x - 2a)^2 + y^2 + z^2 - (m - 2a)^2 ,
with
(x + 2a)^2 + y^2 + z^2 - (m + 2a)^2 ;
the surface comprises two tangent spheres with centres on the x-axis
(by the way, as oriented spheres properly tangent, not anti-tangent).
'Ang abaht tho' --- one definition of a Dupin cyclide is the envelope
(unique when spheres are oriented) swept out by a sphere moving tangent
to 3 fixed spheres. For what 3 spheres (including planes and points)
could such an envelope possibly be a pair of tangent spheres?
Fred Lunnon
