Kuen Surface

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Kuen Surface

This surface of constant Gaussian curvature K = -1 was discovered early. It is very popular because of its beauty.

Kuen Surface Parametric Equations

  x = 2 * cosh(v) * (cos(u) + u * sin(u)) / (cosh(v) * cosh(v) + u * u)
  y = 2 * cosh(v) * (-u * cos(u) + sin(u)) / (cosh(v) * cosh(v) + u * u)
  z = v - (2 * sinh(v) * cosh(v)) / (cosh(v) * cosh(v) + u * u)

Note that K = -1 surfaces necessarily have the sharp singular rims.

There is a famous theorem by Hilbert which says: The complete hyperbolic plane (i.e. the plane with a complete metric that has Gaussian curvature K = -1) cannot be isometrically (i.e. arc length preserving) immersed (i.e. mapped so that the derivative has no singularities) into Euclidean R^3. The proof has a surprisingly constructive aspect: You pick any regular point of a (K = -1)-surface in R^3. Then you can compute a distance D such that within distance D from the chosen point the pair of asymptote lines (i.e. curves on the surface with normal curvature = 0) becomes tangential to each other. The points where the asymptote lines touch ( instead of intersect) each other are those sharp singular rims.

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Kuen Surface, plotted growing domain of the parametrizing map.
kuen half
Half of the Kuen surface is free of self-intersections because since the 19th century the surface is only computed until the seam at the back is reached.
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Kuen surface artwork, by Luc Benard
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kuen st
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kuen sw

Related Surface

  1. Dini Surface
  2. Breather Surface
  3. Kuen Surface
  4. Two-Soliton Surface
  5. Three-Soliton