Discrete K-Surfaces

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Discrete surfaces of constant negative Gaussian curvature (K-surfaces) are a good example to illustrate key features of Discrete Differential Geometry:

  • Discrete K-surfaces are derived from smooth surfaces of constant Gaussian curvature by a characterizing geometric property rather than constancy of some discretization of curvature.
  • Their integrable nature is preserved, in fact clearly exposed.
  • A fast and stable algorithm to produce such surfaces, which in particular leads to discrete surfaces that are visually indistinguishable from discretized smooth ones.

See Ulrich Pinkall: Designing Cylinders with Constant Negative Curvature and references therein for more details on the mathematical background.

Discrete K-surfaces with a planar strip in JRViewerVR


Start the discrete K-surfaces java web start application and try the following:

  • Investigate the initial surface: left mouse to rotate the surface, middle mouse to drag the surface, right mouse to look around, wasd-keys and space to walk around, g to turn of/on gravity.
  • Change the initial planar curve (red and yellow points connected by a blue tube) by dragging the yellow points (control points).
  • Change the curvature and length of the surface in the right hand panel labeled Discrete K-surface. This panel also allows to change parameters of the initial curve.
  • Change the visibility of lines in the "Content Appearance" panel. Turn of reflection or the tubes all together if the viewer gets to slow.
  • Go to application menu "Side Panels->Panels->Sky" and check the box. Then a new panel on the left hand side should appear. Select snow.
  • Uncheck the box "Side Panels->Panels->Sky". Now you should see an image similar to the screen shot.

See also the ViewerVR_User_Manual and User_Tutorial.

Here is a Tutorial which explains how to implement the discrete K-surfaces application with jReality.

Two older applications with evolution of the Gauss map

DkSurfacesPlanarStrip.png    DkSurfacesPlanarStripGauss.png

There are two older java web start applications to investigate discrete K-surfaces :

They allow to investigate the evolution of the Gauss map (rubber band on the sphere) in addition to the surface.

Try the following:

  • Rotate and move the surface with left and middle mouse.
  • Change the initial curve on the left hand side by dragging the bigger yellow points. In the planar strip application this curve represents the planar curve, in the cone point application it represents the

initial velocity of the Gauss map.

  • Open the "Gauss map" tab on the left to investigate the evolution of the Gauss map, use the play, stop, pause buttons or the jog wheel.

There is a small mpeg (2MiB) video or animated gif (11MiB) showing the evolution of the Gauss map.