Helicoid in H^3

Helicoid in H^3

Here are three views of a hyperbolic helicoid, an idea suggested by Saul Schleimer. The first puts the helicoid axis on the vertical line through the origin. The other two apply a hyperbolic isometry to move the axis towards the right. Whenever the axis does not go through the origin, it will be an arc in the ball model.

What curve does a single ideal edge of the surface trace out on the ball boundary? I suspect it is a loxodrome, but I haven't proved this. (Update: It is! see comments below.)

Relevant Links

Euclidean helicoids
en.wikipedia.org/wiki/Helicoid

Loxodromes
en.wikipedia.org/wiki/Rhumb_line


Comments

  1. Refurio AnachroNovember 19, 2015 at 5:26 AM

    Nice! Next: K3 and Calabi-Yau surfaces?

    ReplyDelete
  2. Saul SchleimerNovember 19, 2015 at 5:38 AM

    Pretty cool! The two boundaries at infinity are definitely loxodromes - this is easier to see in the upper half space model, using polar coordinates on the complex plane at infinity. (Moreover, by conformality, this is also a proof for the ball model.)

    ReplyDelete
  3. Matt McIrvinNovember 19, 2015 at 7:34 AM

    A loxodrome seems correct to me, because the angle between the helicoid and a plane parallel to the axis should be preserved by the conformal mapping to the sphere at infinity, so you should end up with a line at a constant angle to the meridians of longitude.

    ReplyDelete
  4. Matt McIrvinNovember 19, 2015 at 7:36 AM

    ...that is, for the first case, where the mapping to Euclidean space puts the axis at the center.

    ReplyDelete
  5. Roice NelsonNovember 19, 2015 at 8:50 AM

    Matt McIrvin, I think if you demonstrate the curve is a loxodrome when the helicoid axis is a line, you can consider the transformed curves loxodromes as well, at least from the perspective of the Mobius transformation formulas for a general loxodrome: roice3.blogspot.com/2008/11/loxodromes.html.

    But I see what you mean if you identify loxodromes with rhumb lines, like wikipedia does. I think it'd be nice if wikipedia had a separate article on loxodromes, more focused on them from the perspective of complex analysis.

    ReplyDelete
  6. Matt McIrvinNovember 19, 2015 at 10:23 AM

    Roice Nelson Yes, all the cases are going to be related by Mobius transformations, so that follows. I was thinking of a loxodrome as a rhumbline.

    ReplyDelete
  7. wendy kriegerNovember 23, 2015 at 2:43 AM

    A loxodrome in H3, would be essentially a straight line. Recall that a loxodrome strikes the same angles to perpendiculars to a plane.

    ReplyDelete

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