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In what sense is it true that a NURBS surface can only have the topology of a plane, cylinder or torus?

For example I can do a NURBS sphere.

Is the sphere homeomorphic to one of the above surfaces?

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  • $\begingroup$ "..of a plane, cylinder or torus" Klein bottle and real-projective plane feels left out :( Have a look at en.wikipedia.org/wiki/Fundamental_polygon $\endgroup$ – tom Nov 29 '15 at 22:58
  • $\begingroup$ Mobius band is now crying... $\endgroup$ – tom Nov 29 '15 at 23:06
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Before we begin, let us differentiate between two things:

  • The shape of a sphere,
  • and the topology of a sphere.

A NURBS surface can make the shape of a sphere. In a typical configuration, it will be 'open' at the poles. That is the mathematical function of the surface does not wrap over the pole in that it is not a true sphere (it has the shape of a sphere). Topologically the sphere is in this typical configuration a cylinder.

It is possible to leave the other direction open too in which case it is topologically a plane. Several CAD applications choose this approach. A torus is also possible if you allow a thin no volume sliver at the center of your sphere.

enter image description here

Image 1: Turning cylinder to sphere. Note this is not a topological sphere it is still a cylinder as the top is open (even though infinitesimally small).

Is the sphere homeomorphic to one of the above surfaces?

No, but you can still have a spherical shape even if it does not satisfy the topology condition of mathematics.

Why only 3 topological families?

Simply, a NURBS surface has only 4 possible configurations of wrapping around the parameter space:

  • It does not wrap at all. Topology: Plane.
  • It wraps around the U direction. Topology: Cylinder.
  • It wraps around V direction. Same as above. Topology: Cylinder.
  • It wraps around both U and V. Topology: Torus.

A wrap is always periodic so it goes from - direction to + direction. It can not arbitrarily connect (on a mathematical level).

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    $\begingroup$ I found this while searching around: neil-strickland.staff.shef.ac.uk/courses/algtop/pictures/sphere I think the animations help picture the homeomorphism between a sphere without poles and the plane/cylinder, so I thought you might want to include it $\endgroup$ – Martin Ender Nov 26 '15 at 9:02
  • $\begingroup$ Yeah i was planning on drawing a picture once in front of a computer. $\endgroup$ – joojaa Nov 26 '15 at 9:10
  • $\begingroup$ Well your answer is not very consistent. If the sphere is not a sphere but a cylinder, because you are missing those two points, than the cylinder is not a cylinder but it is a plane, because you are missing the whole edge. And there are more than 3 topological families, by gluing different edges you can get sphere, cylinder, torus, mobius band, klein bottle, real-projective plane. Have a look at wiki page about Fundamental polygon. $\endgroup$ – tom Nov 29 '15 at 23:13

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