* Allan Wechsler <acwacw@gmail.com> [Jul 23. 2011 08:04]:

Repair URL, please?

Just remove the space: http://jalape.no/math/dupin_s.gif

[...]

[and my reply to his reply] On 7/24/11, Ralph Martin <Ralph.Martin@cs.cardiff.ac.uk> wrote:

On 23 Jul 2011, at 11:35:59PM, Fred lunnon wrote:

...

... Your engineering notation for a canonical cyclide is rather different from mine: ...

It's not mine - it's Forsyth's, dating back 100 years at least. And that's assuming he did not take it from someone earlier still.

Probably Arthur Cayley, now you mention it --- this is the kind of intuitive, constructive notion I associate with the man (and attempt to emulate).

...

Considering (say) a ring cyclide, each pencil has a real axial or "limit" circle where all its spheres intersect. The circles lie in perpendicular planes (xz-plane and xy-plane), their centres [0,0,0] and [d,0,0] lying along the intersection (x-axis), at distance given by d^2 = p^2 + q^2 . [Warning: this usage differs from my earlier notation for a torus, when p,q corresponded to your a,m .]

Rewritten in my reference frame (note x-translation), your nodes become [x,y,z] = [0, +/- p i, 0], [d, 0, +/- q i] . *** What this is saying is that nodes and limit circles are virtually the same thing --- it's just that a real limit circle manifests a complex node-pair, and vice-versa --- brilliant!

OK, seems a reasonable way of doing it.

One application is the freedom-1 "conodal family" where the nodes are fixed, and the third parameter (cone angle / tangent radius) varies: when these are canonical tori (p = 1, q = oo), this is the foliation of tori whose Villarceau circles constitute the Hopf bundle which fibres inversive space.

...

The nodes fix only 2 of 3 shape/size parameters. One natural choice for the other parameter is cone angle at the node, invariant under the Moebius group, but real only for spindle / horned cases (when one limit radius is imaginary).

I prefer real parameters when possible, as I find them easier to understand! :-)

Me too; but you can always keep things real here by substituting for the cone semi-angle t its squared sign, given in a,m,c notation by sin^2 t = (m^2 - a^2)/(m^2 - c^2) . Alternatively, use i sin(t) = sinh(i t) to interpret the complex angle as a hyperbolic extent.

...

Another natural parameter is the radius r of the circle in which (either) plane generator is tangent to the cyclide, invariant under the Laguerre group. Surprisingly, this is real for both ring and spindle / horned cases: explicitly in engineering parameters, its square equals a^2 - c^2 .

Sounds a better choice.

Easier to visualise perhaps; but the Laguerre group is sadly less familiar. One nice transformation available therein is offset, incrementing the (signed) radius of every sphere by a constant: in this way for example, a ring torus can transmogrify into a spindle. This is apparently paradoxical, since complex points --- the nodes --- change into real points under a transformation which (in Lie-sphere, hexaspheric coordinates) is both real and linear!

Perhaps the most intuitive way for engineers is to think of the cyclide as a torus with varying radius as we "go round". In this view the parameters are max radius as we go round, min radius as we go round, and the radius of the "going round circle" which locates the centres of each "cross section" circle. Then horned cyclides have a negative minimum radius.

I didn't really appreciate this idea when I first encountered a,m,c notation in D. Dutta, R. R. Martin, M. J. Pratt "Cyclides in Surface and Solid Modelling" Computer Aided Geometric Design vol. 10, 53--59 (1993) --- but it is growing on me. [Maybe the gloss above and a better diagram would have helped!] Fred Lunnon On 7/23/11, Fred lunnon <fred.lunnon@gmail.com> wrote:

[original to Ralph Martin; copied to math-fun]

Dear Ralph,

Thanks for the Mathematica session --- I still have no idea why Maple found only one pair of roots, but that's entirely within character --- for me and for Maple 9.

Your engineering notation for a canonical cyclide is rather different from mine: instead of using major, minor, eccentricity parameters a,m,c and a central origin, I start from the perpendicular pair of pencils of spheres in which the cyclide is symmetric under inversion ("anallagmatic").

Considering (say) a ring cyclide, each pencil has a real axial or "limit" circle where all its spheres intersect. The circles lie in perpendicular planes (xz-plane and xy-plane), their centres [0,0,0] and [d,0,0] lying along the intersection (x-axis), at distance given by d^2 = p^2 + q^2 . [Warning: this usage differs from my earlier notation for a torus, when p,q corresponded to your a,m .]

Rewritten in my reference frame (note x-translation), your nodes become [x,y,z] = [0, +/- p i, 0], [d, 0, +/- q i] . *** What this is saying is that nodes and limit circles are virtually the same thing --- it's just that a real limit circle manifests a complex node-pair, and vice-versa --- brilliant!

Of course, Darboux or Maxwell or some XIX-th century mathematician probably knew all this perfectly well. As yet I haven't managed to consult for example Virgil Snyder, Annals of Math \bf 11 \rm 136--147 (1896).

The nodes fix only 2 of 3 shape/size parameters. One natural choice for the other parameter is cone angle at the node, invariant under the Moebius group, but real only for spindle / horned cases (when one limit radius is imaginary).

Another natural parameter is the radius of the circle in which (either) plane generator is tangent to the cyclide, invariant under the Laguerre group. Surprisingly, this is real for both ring and spindle / horned cases: explicitly in engineering parameters, its square equals a^2 - c^2 .

Fred