[math-fun] Things they never tell you about Dupin cyclides
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
Apology for minor typo --- and further ramblings ... WFL Dear Ralph, Thanks for checking that, but you were too trusting --- the actual howler occurs before the first line [always the best place to conceal them, I find!] Your (Forsyth/Maxwell's?) cyclide equation as originally stated was (x^2 + y^2 + z^2 - m^2 - b^2)^2 - 4(a x - c m)^2 - 4 b^2 y^2 which I had attempted manually to simplify via b^2 -> a^2 - c^2 --- my very first equation should then actually have read (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 and the remainder follows from that correctly as before. Incidentally, this exercise of my inimitable clerical skills prompted the observation that your / Maxwell's "b" is what I earlier denoted "r": the radius of the circle where a generator plane touches the cyclide, a Laguerre invariant. And I've since discovered that my alternative choice for the third parameter --- the Moebius invariant sine of the cone semi-angle "s" --- is what John Knapman (1987) calls the "eccentricity". Reassuring to find that both these were already recognised as significant. Now Groebner bases expose this simple relation, bilinear between their squares --- q^2 b^2 s^2 - p^2 q^2 s^2 + p^2 b^2 + p^2 q^2 = 0 . Since pairs of cyclide nodes are apparently symmetric, it's a bit of a shock that this expression is instead invariant under p -> q, q -> p, s -> 1/s, b -> i b; the exact geometrical interpretation of this subtle wrinkle in the symmetry at present eludes yours truly, Fred Lunnon On 7/29/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
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
I asked earlier whether it is possible that the envelope of a sphere moving tangent to to three fixed spheres can comprise two tangent spheres. The question is manifestly unfair: to begin with, how many people can visualise the envelope of a sphere moving tangent to three fixed spheres, anyway? Which exposes one major weakness of this oft-quoted synthetic definition of a Dupin cyclide --- elegant, but unusable. At the cost of a certain amount of algebraic machinery an alternative approach provides more insight, including a practical construction counterpointing this tangency constraint. The prerequisite is essentially "hexaspheric" coordinates, as adopted by XIX-th century geometers such as Coolidge for "oriented spheres" or "contact" geometry --- a modern reference is T. E. Cecil "Lie-sphere Geometry" (2008). The details are not relevant here: what matters is that it provides a framework representing extrinsically oriented 3-space points, spheres, planes, utilising coordinate 6-vectors. Now in classical line geometry, a 3-space quadric hyperboloid may be regarded in two ways as a linear system generated by 3 skew lines lying on the surface, lines being represented by Pluecker 6-vectors. Analogously, a Dupin cyclide is simply the linear regulus generated by any 3 distinct spheres along its envelope. Furthermore, a given cyclide is generated by two distinct systems: if one system is thought of as a path swept by a continuously moving sphere, the other comprises all spheres to which the motion remains tangent; and vice-versa. The isometries of contact geometry combine Euclidean (rigid), Moebius (conformal) and Laguerre (equilong) transformations; elementary continuous motions generalising rotation include dilation, offset, and many more intriguing things ... but I digress. A Dupin cyclide may be regarded as the envelope of a sphere under the action of one of these elementary rotors; and now at last we can get to the point. More precisely, to the point of tangency of two spheres proposed as a special case of a cyclide, when swing offset c equals central radius a. Under rotovation (as it were) subsequent motion is determined completely by current location, so each position along the orbit is visited (at most) once during the period; whereas a sphere sweeping the two-sphere envelope must shrink down twice over to pass through the tangency point. We conclude that from a synthetic aspect the class of cyclides is not compact, and is properly contained within the class defined analytically via their implicit Cartesian equation. In particular, a tangent sphere pair is analytically a cyclide, but synthetically only a limit of a cyclides. Incidentally the same objection can also be raised in connection with the double sphere with two double points, mentioned earlier as a special case of a torus when tube radius m equals central radius a. Although this case is easier to visualise, it seems harder to equip with a convincingly rigorous argument. Fred Lunnon
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Fred lunnon