http://spacesymmetrystructure.wordpress.com/2009/02/06/rheotomic-surfaces/

I’ve also been wondering about the combination of these with some other stuff on rotations of the 3-sphere:

http://spacesymmetrystructure.wordpress.com/2008/12/11/4-dimensional-rotations/

Maybe you could take a look and tell me if I’m talking nonsense. hopefully you might enjoy some of the visualisations too

]]>The image is xz=y^2.. Making the change of coordinates x=(x’+z’), z=(x’-z’), this becomes (x’)^2=y^2+(z’)^2, which is parameterized by (x’,y,z’)=(t^2+u^2, 2tu, t^2-u^2). (Fans of recreational number theory may recognize this as the standard parametrization of pythagorean triples.) Throwing an i in front of the x’ coordinate to switch the sign of (x’)^2 gives your paramterization.

]]>See Hitchin’s papers on monopoles in the early 1980s for a very beautiful exposition.

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