Offers routines for constructing a variety of sample 2-dimensional triangulations.
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#include <triangulation/example2.h>
Offers routines for constructing a variety of sample 2-dimensional triangulations.
This is a specialisation of the generic Example class template; see the Example template documentation for a general overview of how the example triangulation classes work.
This 2-dimensional specialisation offers significant extra functionality, by providing several more hard-coded constructions.
◆ annulus()
Returns a two-triangle annulus.
This is isomorphic to the triangulation returned by the generic routine ballBundle().
- Returns
- the annulus.
◆ ball()
Bounded Triangulations.
Returns a one-simplex triangulation of the dim-ball.
- Returns
- a one-simplex dim-ball.
◆ ballBundle()
Returns a triangulation of the product space B^(dim-1) x S¹
.
This will use one simplex in odd dimensions, or two simplices in even dimensions.
- Returns
- the product
B^(dim-1) x S¹
.
◆ disc()
Returns a one-triangle disc.
This is isomorphic to the triangulation returned by the generic routine ball().
- Returns
- the disc.
◆ doubleCone()
Returns a double cone over the given (dim-1)-dimensional triangulation.
If the given triangulation represents the manifold M
, then this returns an ideal triangulation of the product M x I
(with two ideal boundary components). A copy of the original triangulation base can be found at the centre of this construction, formed from the dim-simplices that sit between the two ideal vertices.
Note that, as a special case, if M
is either a sphere or a ball, then this routine returns a (dim)-sphere or a (dim)-ball (since "ideal spheres" and "ideal balls" just become regular internal and boundary vertices respectively).
This construction is essentially the suspension of the triangulation base. We do not call it this however, since from a topological point of view, to form the ideal triangulation of M x I
we "remove" the vertices at the apex of each cone.
- Warning
- If the given (dim-1)-dimensional triangulation has any boundary whatsoever (either real or ideal), then unless it is a (dim-1)-ball, you will obtain an invalid dim-manifold triangulation as a result.
- Returns
- a double cone over the given triangulation.
◆ kb()
Returns a two-triangle Klein bottle.
This is isomorphic to the triangulation returned by the generic routine twistedSphereBundle().
- Returns
- the Klein bottle.
◆ mobius()
Returns a one-triangle Mobius band.
This is isomorphic to the triangulation returned by the generic routine twistedBallBundle().
- Returns
- the Mobius band.
◆ nonOrientable()
Returns a triangulation of the given non-orientable surface.
If the number of punctures is 0 or 1, then the resulting triangulation will be minimal (which, with the exception of the projective plane, means there is exactly one vertex).
- Parameters
-
genus | the non-orientable genus of the surface, i.e., the number of crosscaps that it contains; this must be greater than or equal to one. |
punctures | the number of punctures in the surface; this must be greater than or equal to zero. |
- Returns
- the requested non-orientable surface.
- Author
- Alex He, B.B.
◆ orientable()
Returns a triangulation of the given orientable surface.
If the number of punctures is 0, then the resulting triangulation will be minimal (which, for positive genus, means there is exactly one vertex).
- Parameters
-
genus | the genus of the surface; this must be greater than or equal to zero. |
punctures | the number of punctures in the surface; this must be greater than or equal to zero. |
- Returns
- the requested orientable surface.
◆ rp2()
Returns a two-triangle projective plane.
- Returns
- the projective plane.
◆ simplicialSphere()
Returns the standard (dim+2)-simplex triangulation of the dim-sphere as the boundary of a (dim+1)-simplex.
- Returns
- the standard simplicial dim-sphere.
◆ singleCone()
Returns a single cone over the given (dim-1)-dimensional triangulation.
If the given triangulation represents the manifold M
, then this returns a triangulation of the product M x I
that has one real boundary component and one ideal boundary component. The triangulation of the real boundary component will be identical to the original (dim-1)-dimensional triangulation base.
- Warning
- If the given (dim-1)-dimensional triangulation has any boundary whatsoever (either real or ideal), then unless it is a (dim-1)-ball, you will obtain an invalid dim-manifold triangulation as a result.
- Returns
- a single cone over the given triangulation.
◆ sphere()
Closed Triangulations.
Returns a two-simplex triangulation of the dim-sphere.
- Returns
- a two-simplex dim-sphere.
◆ sphereBundle()
Returns a two-simplex triangulation of the product space S^(dim-1) x S¹
.
- Returns
- the product
S^(dim-1) x S¹
.
◆ sphereOctahedron()
Returns the eight-triangle 2-sphere formed from the boundary of an octahedron.
- Returns
- the octahedral sphere.
◆ sphereTetrahedron()
Returns the four-triangle 2-sphere formed from the boundary of a tetrahedron.
This is isomorphic to the triangulation returned by the generic routine simplicialSphere().
- Returns
- the tetrahedral sphere.
◆ torus()
Returns a two-triangle torus.
This is isomorphic to the triangulation returned by the generic routine sphereBundle().
- Returns
- the torus.
◆ twistedBallBundle()
Returns a triangulation of the twisted product space B^(dim-1) x~ S¹
.
This will use one simplex in even dimensions, or two simplices in odd dimensions.
- Returns
- the twisted product
B^(dim-1) x~ S¹
.
◆ twistedSphereBundle()
Returns a two-simplex triangulation of the twisted product space S^(dim-1) x~ S¹
.
- Returns
- the twisted product
S^(dim-1) x~ S¹
.
The documentation for this class was generated from the following file:
Copyright © 1999-2025, The Regina development team
This software is released under the GNU General Public License,
with some additional permissions; see the source code for details.
For further information, or to submit a bug or other problem, please contact
Ben Burton (
bab@maths.uq.edu.au).