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Radii of curvature for pseudosphere

  1. Feb 23, 2009 #1
    This is a problem I'm having reading Visual Complex Analysis, page 295.
    If you look up "pseudosphere circles of curvature" on Google, it should be the first thing listed.

    On a point of a psuedosphere, there are 2 "circles of curvature", one with its center on the normal pointing out and the other with its center on the normal pointing in.
    1. Is that right?
    2. Does the circle with its center on the normal pointing inwards have to have its center on the axis of the psuedosphere?
    3. If the answer to 2. is yes, why is this? I do not know differential geometry and hope for an answer that is as intuitive as possible.

  2. jcsd
  3. Mar 7, 2009 #2
    Well, I don't know any Complex Analysis but I can give you the Differential Geometry side of the problem. Do you know what a principal curvature is?

    Take a point on a "nice" 2-dimensional surface and look at cross-sections of the surface at that point. Each cross-section will give you a 1-dimensional curve on the surface. The signed curvature of those curves will vary continuously as you rotate around the point, and in general there will be exactly one local maximum and one minimum curvature: those are the two principal curvatures of the surface at this point. Then two book draws the "osculating circles" for the two curves with extremal curvature: you want circles that are tangent to the curve, have the same curvature and lie in the plane spanned by the tangent and normal vector of the curve. (There's a third vector, called a torsion vector, that you want to be perpendicular to.) That the circles look a certain way is more or less a coincidence. The two circles being on different sides of the surface (relative to the normal vector) indicates that the surface has negative Gaussian curvature, but that's not generally the case.
    Last edited: Mar 7, 2009
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