Re: [math-fun] Dupin cyclides --- another rhinoceros' pancreas?
Sorry, missed that embedded space. Here's a better URL, anyway: < http://jalape.no/math/mathgal.htm >. Click on the middle image in the second column to get an enlarged version of a Dupin cyclide with two conical singularities. Click instead on its ",description" and you get an explanation of how the image was obtained. ----- But come to think of it, should there be more singularities lurking in the complex version? --Dan P.S. In any case one way to regularize the calculations is by using a conformal transformation of space to make all 3 "generating" 2-spheres be of equal radius with. Then I think there are only two essentially distinct cases to consider. But I regret I don't have time at the moment to think this out carefully. -------------------------------------------------------- Allan wrote: << Repair URL, please? I wrote: << I think the complex singularities can be brought into the real domain by an appropriate transformation, yielding a surface like this: [bad URL was here].
Sometimes the brain has a mind of its own.
On 7/22/11, Dan Asimov <dasimov@earthlink.net> wrote:
Sorry, missed that embedded space.
Here's a better URL, anyway:
< http://jalape.no/math/mathgal.htm >.
Click on the middle image in the second column to get an enlarged version of a Dupin cyclide with two conical singularities.
Click instead on its ",description" and you get an explanation of how the image was obtained. -----
But come to think of it, should there be more singularities lurking in the complex version?
--Dan
As I said earlier, there are _4_ nodes, in two complex conjugate pairs (assuming real coefficients). In the general nontrivial case, one pair is always complex; the other pair may be real, as visible at Dan's URL, or complex, as at the Wikipedia page. It's traditional to split the real case into "spindle" where the nodes lie half within the surface, and "horned" fully on the exterior; the complex case is are "ring" cyclides. There are also numerous special cases to consider, such as coincident nodes (cuspidal), and the cubic or "parabolic" full-twist surface resembling nothing so much as nightmare knickers ... It's the interpretation of the always-complex pair as a separate real geometric feature that's currently baffling me. One possibility I've considered is that the associated length and angle are always equal for both pairs: but that doesn't work either, since there would then be only 2 parameters controlling size and shape, instead of 3 ... Fred Lunnon
P.S. In any case one way to regularize the calculations is by using a conformal transformation of space to make all 3 "generating" 2-spheres be of equal radius with.
Then I think there are only two essentially distinct cases to consider.
But I regret I don't have time at the moment to think this out carefully. --------------------------------------------------------
Allan wrote: << Repair URL, please?
I wrote: << I think the complex singularities can be brought into the real domain by an appropriate transformation, yielding a surface like this:
[bad URL was here].
Sometimes the brain has a mind of its own.
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