Regina Calculation Engine
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regina::Face< 3, 2 > Class Reference

Represents a triangle in the skeleton of a 3-manifold triangulation. More...

#include <triangulation/dim3.h>

Inheritance diagram for regina::Face< 3, 2 >:
regina::detail::FaceBase< 3, 2 > regina::detail::FaceStorage< dim, dim - subdim > regina::detail::FaceValidity< allowsInvalidFaces(dim, subdim), standardDim(dim)> regina::detail::FaceOrientability< allowsNonOrientableLinks(dim, subdim)> regina::FaceNumbering< dim, subdim > regina::MarkedElement regina::alias::FaceOfSimplex< FaceBase< dim, subdim >, dim, subdim - 1 > regina::Output< Face< dim, subdim > > regina::detail::FaceNumberingImpl< dim, subdim,((dim+1) >=2 *(subdim+1))> regina::detail::FaceNumberingAPI< dim, subdim >

Public Types

enum  Type {
  UNKNOWN_TYPE = 0, TRIANGLE = 1, SCARF = 2, PARACHUTE = 3,
  CONE = 4, MOBIUS = 5, HORN = 6, DUNCEHAT = 7,
  L31 = 8
}
 The type of a triangle, which indicates how the vertices and edges of the triangle are identified together. More...
 

Public Member Functions

Type type ()
 Returns a description of the triangle type. More...
 
int subtype ()
 Return the triangle vertex or triangle edge that plays a special role for the triangle type of this triangle. More...
 
bool isMobiusBand ()
 Determines whether this triangle is wrapped up to form a Mobius band. More...
 
bool isCone ()
 Determines whether this triangle is wrapped up to form a cone. More...
 
size_t index () const
 Returns the index of this face within the underlying triangulation. More...
 
Triangulation< dim > * triangulation () const
 Returns the triangulation to which this face belongs. More...
 
Component< dim > * component () const
 Returns the component of the triangulation to which this face belongs. More...
 
BoundaryComponent< dim > * boundaryComponent () const
 Returns the boundary component of the triangulation to which this face belongs. More...
 
bool isBoundary () const
 Determines if this face lies entirely on the boundary of the triangulation. More...
 
Face< dim, lowerdim > * face (int face) const
 Returns the lowerdim-face of the underlying triangulation that appears as the given lowerdim-dimensional subface of this face. More...
 
Perm< dim+1 > faceMapping (int face) const
 Examines the given lowerdim-dimensional subface of this face, and returns the mapping between the underlying lowerdim-face of the triangulation and the individual vertices of this face. More...
 
void writeTextShort (std::ostream &out) const
 Writes a short text representation of this object to the given output stream. More...
 
void writeTextLong (std::ostream &out) const
 Writes a detailed text representation of this object to the given output stream. More...
 
size_t degree () const
 Returns the degree of this face. More...
 
const FaceEmbedding< dim, dim - codim > & embedding (size_t index) const
 Returns one of the ways in which this face appears within a top-dimensional simplex of the underlying triangluation. More...
 
std::vector< FaceEmbedding< dim, dim - codim > >::const_iterator begin () const
 A begin function for iterating through all appearances of this face within the various top-dimensional simplices of the underlying triangulation. More...
 
std::vector< FaceEmbedding< dim, dim - codim > >::const_iterator end () const
 An end function for iterating through all appearances of this face within the various top-dimensional simplices of the underlying triangulation. More...
 
const FaceEmbedding< dim, dim - codim > & front () const
 Returns the first appearance of this face within a top-dimensional simplex of the underlying triangluation. More...
 
const FaceEmbedding< dim, dim - codim > & back () const
 Returns the last appearance of this face within a top-dimensional simplex of the underlying triangluation. More...
 
bool inMaximalForest () const
 Determines whether a codimension-1-face represents a dual edge in the maximal forest that has been chosen for the dual 1-skeleton of the triangulation. More...
 
bool isValid () const
 Determines if this face is valid. More...
 
bool hasBadIdentification () const
 Determines if this face is identified with itself under a non-identity permutation. More...
 
bool hasBadLink () const
 Determines if this face does not have an appropriate link. More...
 
bool isLinkOrientable () const
 Determines if the link of this face is orientable. More...
 
size_t markedIndex () const
 Returns the index at which this object is stored in an MarkedVector. More...
 
std::string str () const
 Returns a short text representation of this object. More...
 
std::string utf8 () const
 Returns a short text representation of this object using unicode characters. More...
 
std::string detail () const
 Returns a detailed text representation of this object. More...
 

Static Public Member Functions

static Perm< dim+1 > ordering (unsigned face)
 Given a subdim-face number within a dim-dimensional simplex, returns the corresponding canonical ordering of the simplex vertices. More...
 
static unsigned faceNumber (Perm< dim+1 > vertices)
 Identifies which subdim-face in a dim-dimensional simplex is represented by the first (subdim + 1) elements of the given permutation. More...
 
static bool containsVertex (unsigned face, unsigned vertex)
 Tests whether the given subdim-face of a dim-dimensional simplex contains the given vertex of the simplex. More...
 

Static Public Attributes

static constexpr int nFaces
 The total number of subdim-dimensional faces in each dim-dimensional simplex. More...
 

Protected Member Functions

void push_back (const FaceEmbedding< dim, dim - codim > &emb)
 Internal routine to help build the skeleton of a triangulation. More...
 
void markBadIdentification ()
 Marks this face as having a non-identity self-identification. More...
 
void markBadLink ()
 Marks this face as having a bad link. More...
 
void markLinkNonorientable ()
 Marks the link of this face as non-orientable. More...
 

Friends

class Triangulation< 3 >
 
class detail::TriangulationBase< 3 >
 

Detailed Description

Represents a triangle in the skeleton of a 3-manifold triangulation.

This is a specialisation of the generic Face class template; see the documentation for Face for a general overview of how this class works.

These specialisations for Regina's standard dimensions offer significant extra functionality.

Member Function Documentation

◆ back()

const FaceEmbedding< dim, dim - codim > & regina::detail::FaceStorage< dim, codim >::back
inlineinherited

Returns the last appearance of this face within a top-dimensional simplex of the underlying triangluation.

This is equivalent to calling embedding(degree()-1).

In most cases, the ordering of appearances is arbitrary. The exception is for codimension 2, where the appearances of a face are ordered in a way that follows the link around the face (which in codimension 2 is always a path or a cycle). In particular, for a boundary face of codimension 2, both front() and back() will refer to the two appearances of this face on the (dim-1)-dimensional boundary.

Returns
details of the last appearance.

◆ begin()

std::vector< FaceEmbedding< dim, dim - codim > >::const_iterator regina::detail::FaceStorage< dim, codim >::begin
inlineinherited

A begin function for iterating through all appearances of this face within the various top-dimensional simplices of the underlying triangulation.

In most cases, the ordering of appearances is arbitrary. The exception is for codimension 2, where these appearances are ordered in a way that follows the link around the face (which in codimension 2 is always a path or a cycle).

An iteration from begin() to end() will run through degree() appearances in total.

Python:\n Not present. However, Python users can call
the Python-only routine embeddings(), which will return all appearances (from begin() through to end()) in a Python sequence.
Returns
a iterator that points to the first appearance.

◆ boundaryComponent()

BoundaryComponent< dim > * regina::detail::FaceBase< dim, subdim >::boundaryComponent
inlineinherited

Returns the boundary component of the triangulation to which this face belongs.

See the note in the BoundaryComponent overview regarding what happens if the link of the face itself has more than one boundary component. Note that such a link makes both the face and the underlying triangulation invalid.

For dimensions in which ideal and/or invalid vertices are both possible and recognised: an ideal vertex will have its own individual boundary component to which it belongs, and so will an invalid vertex boundary component if the invalid vertex does not already belong to some real boundary component.

Returns
the boundary component containing this face, or 0 if this face does not lie entirely within the boundary of the triangulation.

◆ component()

Component< dim > * regina::detail::FaceBase< dim, subdim >::component
inlineinherited

Returns the component of the triangulation to which this face belongs.

Returns
the component containing this face.

◆ degree()

size_t regina::detail::FaceStorage< dim, codim >::degree
inlineinherited

Returns the degree of this face.

This is the number of different ways in which the face appears within the various top-dimensional simplices of the underlying triangulation.

Note that if this face appears multiple times within the same top-dimensional simplex, then it will be counted multiple times by this routine.

Returns
the degree of this face.

◆ detail()

std::string regina::Output< Face< dim, subdim > , false >::detail ( ) const
inherited

Returns a detailed text representation of this object.

This text may span many lines, and should provide the user with all the information they could want. It should be human-readable, should not contain extremely long lines (which cause problems for users reading the output in a terminal), and should end with a final newline. There are no restrictions on the underlying character set.

Returns
a detailed text representation of this object.

◆ embedding()

const FaceEmbedding< dim, dim - codim > & regina::detail::FaceStorage< dim, codim >::embedding ( size_t  index) const
inlineinherited

Returns one of the ways in which this face appears within a top-dimensional simplex of the underlying triangluation.

For convenience, you can also use begin() and end() to iterate through all such appearances.

In most cases, the ordering of appearances is arbitrary. The exception is for codimension 2, where these appearances are ordered in a way that follows the link around the face (which in codimension 2 is always a path or a cycle).

Parameters
indexthe index of the requested appearance. This must be between 0 and degree()-1 inclusive.
Returns
details of the requested appearance.

◆ end()

std::vector< FaceEmbedding< dim, dim - codim > >::const_iterator regina::detail::FaceStorage< dim, codim >::end
inlineinherited

An end function for iterating through all appearances of this face within the various top-dimensional simplices of the underlying triangulation.

In most cases, the ordering of appearances is arbitrary. The exception is for codimension 2, where these appearances are ordered in a way that follows the link around the face (which in codimension 2 is always a path or a cycle).

An iteration from begin() to end() will run through degree() appearances in total.

Python:\n Not present. However, Python users can call
the Python-only routine embeddings(), which will return all appearances (from begin() through to end()) in a Python sequence.
Returns
a "beyond the end" iterator that comes immediately after the last appearance.

◆ face()

Face< dim, lowerdim > * regina::detail::FaceBase< dim, subdim >::face ( int  face) const
inlineinherited

Returns the lowerdim-face of the underlying triangulation that appears as the given lowerdim-dimensional subface of this face.

The argument face must represent a lowerdim-face number within a subdim-simplex. This lowerdim-face number will be interpreted with respect to the inherent labelling (0, ..., subdim) of the vertices of this subdim-face. See FaceEmbedding<dim, subdim>::vertices() for details on how these map to the vertex numbers of the dim-dimensional simplices that contain this face in the overall triangulation.

See FaceNumbering<subdim, lowerdim> for the conventions of how lowerdim-faces are numbered within a subdim-simplex.

Python:\n Python does not support templates. Instead,
Python users should call this function in the form face(lowerdim, face); that is, the template parameter lowerdim becomes the first argument of the function.
Parameters
facethe lowerdim-face of this subdim-face to examine. This should be between 0 and (subdim+1 choose lowerdim+1)-1 inclusive.
Returns
the corresponding lowerdim-face of the triangulation.

◆ faceMapping()

Perm< dim+1 > regina::detail::FaceBase< dim, subdim >::faceMapping ( int  face) const
inherited

Examines the given lowerdim-dimensional subface of this face, and returns the mapping between the underlying lowerdim-face of the triangulation and the individual vertices of this face.

The argument face must represent a lowerdim-face number within a subdim-simplex. This lowerdim-face number will be interpreted with respect to the inherent labelling (0, ..., subdim) of the vertices of this subdim-face. See FaceEmbedding<dim, subdim>::vertices() for details on how these map to the vertex numbers of the dim-dimensional simplices that contain this face in the overall triangulation.

Let F denote this subdim-face of the triangulation, and let L denote the lowerdim-face of the triangulation that corresponds to the given subface of F. Then the permutation returned by this routine maps the vertex numbers (0, ..., lowerdim) of L to the corresponding vertex numbers of F. This is with respect to the inherent labellings (0, ..., lowerdim) and (0, ..., subdim) of the vertices of L and F respectively.

In particular, if this routine returns the permutation p, then the images p[0,...,lowerdim] will be some permutation of the vertices Face<subdim, lowerdim>::ordering(face)[0,...,lowerdim].

This routine differs from Simplex<dim>::faceMapping<lowerdim>() in how it handles the images of (lowerdim+1, ..., dim):

  • This routine will map (lowerdim+1, ..., subdim) to the remaining vertices of this face in an arbitrary order, and will map (subdim+1, ..., dim) to (subdim+1, ..., dim) again in an arbitrary order.
  • In contrast, Simplex<dim>::faceMapping<lowerdim>() chooses the images of (lowerdim+1, ..., dim) to satisfy an additional orientability constraint.

See FaceNumbering<subdim, lowerdim> for the conventions of how lowerdim-faces are numbered within a subdim-simplex.

Python:\n Python does not support templates. Instead,
Python users should call this function in the form faceMapping(lowerdim, face); that is, the template parameter lowerdim becomes the first argument of the function.
Parameters
facethe lowerdim-face of this subdim-face to examine. This should be between 0 and (subdim+1 choose lowerdim+1)-1 inclusive.
Returns
a mapping from the vertices of the underlying lowerdim-face of the triangulation to the vertices of this subdim-face.

◆ front()

const FaceEmbedding< dim, dim - codim > & regina::detail::FaceStorage< dim, codim >::front
inlineinherited

Returns the first appearance of this face within a top-dimensional simplex of the underlying triangluation.

This is equivalent to calling *begin(), or embedding(0).

In most cases, the ordering of appearances is arbitrary. The exception is for codimension 2, where the appearances of a face are ordered in a way that follows the link around the face (which in codimension 2 is always a path or a cycle). In particular, for a boundary face of codimension 2, both front() and back() will refer to the two appearances of this face on the (dim-1)-dimensional boundary.

Returns
details of the first appearance.

◆ hasBadIdentification()

bool regina::detail::FaceValidity< allowsInvalid, testLinks >::hasBadIdentification ( ) const
inherited

Determines if this face is identified with itself under a non-identity permutation.

For example, if this face is an edge then this routine tests whether the edge is identified with itself in reverse.

Such a face will always be marked as invalid. Note that, for standard dimensions dim, there are other types of invalid faces also. See isValid() for a full discussion of what it means for a face to be valid.

Returns
true if and only if this face is identified with itself under a non-identity permutation.

◆ hasBadLink()

bool regina::detail::FaceValidity< allowsInvalid, testLinks >::hasBadLink ( ) const
inherited

Determines if this face does not have an appropriate link.

See condition (2) in the documentation for isValid() for a full description of what "appropriate" means.

This routine is only available where dim is one of Regina's standard dimensions, since testing this condition in arbitrary dimensions is undecidable. For higher dimensions dim, this routine is not present.

A face whose link is not appropriate will always be marked as invalid. Note that there are other types of invalid faces also. See isValid() for a full discussion of what it means for a face to be valid.

Returns
true if and only if the link of this face is not appropriate.

◆ index()

size_t regina::detail::FaceBase< dim, subdim >::index
inlineinherited

Returns the index of this face within the underlying triangulation.

Returns
the index of this face.

◆ inMaximalForest()

bool regina::detail::FaceStorage< dim, codim >::inMaximalForest ( ) const
inherited

Determines whether a codimension-1-face represents a dual edge in the maximal forest that has been chosen for the dual 1-skeleton of the triangulation.

This routine is only available for faces of codimension 1; that is, (dim-1)-faces of a dim-dimensional triangulation.

When the skeletal structure of a triangulation is first computed, a maximal forest in the dual 1-skeleton of the triangulation is also constructed. Each dual edge in this maximal forest represents a (dim-1)-face of the (primal) triangulation.

This maximal forest will remain fixed until the triangulation changes, at which point it will be recomputed (as will all other skeletal objects, such as connected components and so on). There is no guarantee that, when it is recomputed, the maximal forest will use the same dual edges as before.

This routine identifies whether this (dim-1)-face belongs to the dual forest. In this sense it performs a similar role to Simplex::facetInMaximalForest(), but this routine is typically easier to use.

If the skeleton has already been computed, then this routine is very fast (since it just returns a precomputed answer).

Returns
true if and only if this (dim-1)-face represents a dual edge in the maximal forest.

◆ isBoundary()

bool regina::detail::FaceBase< dim, subdim >::isBoundary
inlineinherited

Determines if this face lies entirely on the boundary of the triangulation.

For dimensions in which ideal and/or invalid vertices are both possible and recognised: both ideal and invalid vertices are considered to be on the boundary.

Returns
true if and only if this face lies on the boundary.

◆ isLinkOrientable()

bool regina::detail::FaceOrientability< allowsNonorientable >::isLinkOrientable ( ) const
inherited

Determines if the link of this face is orientable.

This routine is fast: it uses pre-computed information, and does not need to build a full triangulation of the link.

Warning
If this face is identified with itself under a non-identity permutation (which makes the face invalid), then the return value of this routine is undefined.
Returns
true if and only if the link is orientable.

◆ isValid()

bool regina::detail::FaceValidity< allowsInvalid, testLinks >::isValid ( ) const
inherited

Determines if this face is valid.

There are several conditions that might make a subdim-face of a dim-dimensional triangulation invalid:

  1. if the face is identified with itself under a non-identity permutation (e.g., an edge is identified with itself in reverse, or a triangle is identified with itself under a rotation);
  2. if the face does not have an appropriate link. Here the meaning of "appropriate" depends upon the type of face:
    • for a face that belongs to some boundary facet(s) of the triangulation, its link must be a topological ball;
    • for a vertex that does not belong to any boundary facets, its link must be a closed (dim - 1)-manifold;
    • for a (subdim ≥ 1)-face that does not belong to any boundary facets, its link must be a topological sphere.

Condition (1) is tested for all dimensions subdim and dim. Condition (2) is more difficult, since it relies on undecidable problems. As a result, (2) is only tested when dim is one of Regina's standard dimensions.

If this face is invalid, then it is possible to find out why. In non-standard dimensions, this must mean that the face fails condition (1) above. In standard dimensions, you can call the functions hasBadIdentification() and/or hasBadLink() to determine whether the failure is due to conditions (1) or (2) respectively.

Returns
for standard dimensions dim, returns true if and only if this face is valid according to both conditions (1) and (2) above; for non-standard dimensions dim, returns true if and only if this face is valid according to condition (1).

◆ markBadIdentification()

void regina::detail::FaceValidity< allowsInvalid, testLinks >::markBadIdentification ( )
protectedinherited

Marks this face as having a non-identity self-identification.

◆ markBadLink()

void regina::detail::FaceValidity< allowsInvalid, testLinks >::markBadLink ( )
protectedinherited

Marks this face as having a bad link.

◆ markLinkNonorientable()

void regina::detail::FaceOrientability< allowsNonorientable >::markLinkNonorientable ( )
protectedinherited

Marks the link of this face as non-orientable.

◆ push_back()

void regina::detail::FaceStorage< dim, codim >::push_back ( const FaceEmbedding< dim, dim - codim > &  emb)
inlineprotectedinherited

Internal routine to help build the skeleton of a triangulation.

This routine pushes the given object onto the end of the internal list of appearances of this face within top-dimensional simplices.

Parameters
embthe appearance to push onto the end of the internal list.

◆ str()

std::string regina::Output< Face< dim, subdim > , false >::str ( ) const
inherited

Returns a short text representation of this object.

This text should be human-readable, should fit on a single line, and should not end with a newline. Where possible, it should use plain ASCII characters.

Python:\n In addition to str(), this is also used as the
Python "stringification" function str().
Returns
a short text representation of this object.

◆ triangulation()

Triangulation< dim > * regina::detail::FaceBase< dim, subdim >::triangulation
inlineinherited

Returns the triangulation to which this face belongs.

Returns
the triangulation containing this face.

◆ utf8()

std::string regina::Output< Face< dim, subdim > , false >::utf8 ( ) const
inherited

Returns a short text representation of this object using unicode characters.

Like str(), this text should be human-readable, should fit on a single line, and should not end with a newline. In addition, it may use unicode characters to make the output more pleasant to read. This string will be encoded in UTF-8.

Returns
a short text representation of this object.

◆ writeTextLong()

void regina::detail::FaceBase< dim, subdim >::writeTextLong ( std::ostream &  out) const
inherited

Writes a detailed text representation of this object to the given output stream.

The class Face<dim, subdim> may safely override this function, since the output routines cast down to Face<dim, subdim> before calling it.

Python:\n Not present.
Parameters
outthe output stream to which to write.

◆ writeTextShort()

void regina::detail::FaceBase< dim, subdim >::writeTextShort ( std::ostream &  out) const
inlineinherited

Writes a short text representation of this object to the given output stream.

The class Face<dim, subdim> may safely override this function, since the output routines cast down to Face<dim, subdim> before calling it.

Python:\n Not present.
Parameters
outthe output stream to which to write.

Member Data Documentation

◆ nFaces

constexpr int regina::detail::FaceNumberingImpl< dim, subdim, lex >::nFaces
staticconstexprinherited

The total number of subdim-dimensional faces in each dim-dimensional simplex.


The documentation for this class was generated from the following file:

Copyright © 1999-2020, 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).