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 $\endgroup$– tomCommented Nov 29, 2015 at 22:58
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$\begingroup$ Mobius band is now crying... $\endgroup$– tomCommented Nov 29, 2015 at 23:06
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1 Answer
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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.
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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2$\begingroup$ 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 $\endgroup$ Commented Nov 26, 2015 at 9:02
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$\begingroup$ Yeah i was planning on drawing a picture once in front of a computer. $\endgroup$– joojaaCommented Nov 26, 2015 at 9:10
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$\begingroup$ 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. $\endgroup$– tomCommented Nov 29, 2015 at 23:13