<p>Before we begin, let us differentiate between two things:</p>

<ul>
<li>The shape of a sphere,</li>
<li>and the topology of a sphere.</li>
</ul>

<p>A NURBS surface can make the <em>shape</em> 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. </p>

<p>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.</p>

<p><a href="https://i.sstatic.net/kP30Z.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/kP30Z.gif" alt="enter image description here"></a></p>

<p><strong>Image 1</strong>: 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).</p>

<blockquote>
  <p>Is the sphere homeomorphic to one of the above surfaces?</p>
</blockquote>

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

<h3>Why only 3 topological families?</h3>

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

<ul>
<li>It does not wrap at all. Topology: Plane.</li>
<li>It wraps around the U direction. Topology: Cylinder.</li>
<li>It wraps around V direction. Same as above. Topology: Cylinder.</li>
<li>It wraps around both U and V. Topology: Torus.</li>
</ul>

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

CC BY-SA 3.0
2015-11-26T09:02:46.857Z
I found this while searching around: http://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
Martin Ender
CC BY-SA 3.0
2015-11-26T09:10:07.183Z
Yeah i was planning on drawing a picture once in front of a computer.
joojaa
CC BY-SA 3.0
2015-11-29T23:13:41.173Z
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.
tom
CC BY-SA 3.0
2015-11-26T08:04:16.070Z
2015-11-28T16:48:14.650Z
2015-11-28T16:48:14.650Z
joojaa
<p>In what sense is it true that a NURBS surface can only have the topology of a plane, cylinder or torus?</p>

<p>For example I can do a NURBS sphere.</p>

<p>Is the sphere homeomorphic to one of the above surfaces?</p>

CC BY-SA 3.0
2015-11-29T22:58:50.367Z
"..of a plane, cylinder or torus" Klein bottle and real-projective plane feels left out :( Have a look at https://en.wikipedia.org/wiki/Fundamental_polygon
tom
CC BY-SA 3.0
2015-11-29T23:06:49.073Z
Mobius band is now crying...
tom
<3d><nurbs>
NURBS topology classification
Valerio
# NURBS topology classification

(Asked by: username_0 on 2015-11-26)

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?

**Tags:** 3d, nurbs

> **Comments:**
> **username_1**: "..of a plane, cylinder or torus" Klein bottle and real-projective plane feels left out :( Have a look at https://en.wikipedia.org/wiki/Fundamental_polygon  
> **username_1**: Mobius band is now crying...  

---

## 1 answers

### Answer by username_2 on 2015-11-26. Score: 5 | Accepted answer

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](https://i.sstatic.net/kP30Z.gif)](https://i.sstatic.net/kP30Z.gif)

**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).

> **Comments:**
> **username_3**: I found this while searching around: http://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  
> **username_2**: Yeah i was planning on drawing a picture once in front of a computer.  
> **username_1**: 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.  

---

