regina::detail::FaceBase< dim, subdim > Class Template Reference

Helper class that provides core functionality for a subdim-face in the skeleton of a dim-dimensional triangulation. More...

#include <triangulation/detail/face.h>

Inheritance diagram for regina::detail::FaceBase< dim, subdim >:

Public Member Functions
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...
size_t	degree () const
	Returns the degree of this face. More...
const FaceEmbedding< dim, subdim > &	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...
auto	embeddings () const
	Returns an object that allows iteration through and random access to all of the ways in which this face appears within a top-dimensional simplex of the underlying triangluation. More...
auto	begin () const
	A begin function for iterating through all appearances of this face within the various top-dimensional simplices of the underlying triangulation. More...
auto	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, subdim > &	front () const
	Returns the first appearance of this face within a top-dimensional simplex of the underlying triangluation. More...
const FaceEmbedding< dim, subdim > &	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	isLinkOrientable () const
	Determines if the link of this face is orientable. 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...
template<int lowerdim>
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...
Face< dim, 0 > *	vertex (int i) const
	A dimension-specific alias for face<0>(). More...
Face< dim, 1 > *	edge (int i) const
	A dimension-specific alias for face<1>(). More...
Face< dim, 2 > *	triangle (int i) const
	A dimension-specific alias for face<2>(). More...
Face< dim, 3 > *	tetrahedron (int i) const
	A dimension-specific alias for face<3>(). More...
Face< dim, 4 > *	pentachoron (int i) const
	A dimension-specific alias for face<4>(). More...
template<int lowerdim>
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...
Perm< dim+1 >	vertexMapping (int face) const
	A dimension-specific alias for faceMapping<0>(). More...
Perm< dim+1 >	edgeMapping (int face) const
	A dimension-specific alias for faceMapping<1>(). More...
Perm< dim+1 >	triangleMapping (int face) const
	A dimension-specific alias for faceMapping<2>(). More...
Perm< dim+1 >	tetrahedronMapping (int face) const
	A dimension-specific alias for faceMapping<3>(). More...
Perm< dim+1 >	pentachoronMapping (int face) const
	A dimension-specific alias for faceMapping<4>(). More...
void	writeTextShort (std::ostream &out) const
	Writes a short text representation of this object to the given output stream. More...
	FaceBase (const FaceBase &)=delete
FaceBase &	operator= (const FaceBase &)=delete
size_t	markedIndex () const
	Returns the index at which this object is stored in an MarkedVector. More...
void	writeTextLong (std::ostream &out) const
	A default implementation for detailed output. 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 constexpr 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 constexpr 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 constexpr 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	dimension = dim
	A compile-time constant that gives the dimension of the triangulation containing this face. More...
static constexpr int	subdimension = subdim
	A compile-time constant that gives the dimension of this face. More...
static constexpr bool	allowsNonOrientableLinks = (subdim <= dim - 3)
	Indicates whether it is possible for a face of this dimension to have a non-orientable link. More...
static constexpr bool	allowsInvalidFaces
	Indicates whether it is possible for a face of this dimension to be invalid. More...
static constexpr int	oppositeDim = (dim - 1 - subdim)
	The dimension of the faces opposite these in a top-dimensional simplex of a dim-dimensional triangulation. More...
static constexpr bool	lexNumbering = (subdim <= oppositeDim)
	true if faces are numbered in lexicographical order according to their vertices, or false if faces are numbered in reverse lexicographical order. More...
static constexpr int	nFaces = binomSmall(dim + 1, lexDim + 1)
	The total number of subdim-dimensional faces in each dim-dimensional simplex. More...

Protected Member Functions
	FaceBase (Component< dim > *component)
	Creates a new face. More...

Static Protected Attributes
static constexpr int	lexDim = (lexNumbering ? subdim : oppositeDim)
	Whichever of subdim or oppositeDim uses lexicographical face numbering. More...

Friends
class	Triangulation< dim >
class	TriangulationBase< dim >

Detailed Description

template<int dim, int subdim>
class regina::detail::FaceBase< dim, subdim >

Helper class that provides core functionality for a subdim-face in the skeleton of a dim-dimensional triangulation.

Each subdim-face is described by a Face<dim, subdim> object, which uses this as a base class. End users should not need to refer to FaceBase directly.

See the Face template class notes for further information.

Neither this class nor the "end user" class Face<dim> support value semantics: they cannot be copied, swapped, or manually constructed. Their memory is managed by the Triangulation class, and their locations in memory define them. See Face<dim> for further details.

Python

This base class is not present, but the "end user" class Face<dim, subdim> is.

Template Parameters

dim	the dimension of the underlying triangulation. This must be between 2 and 15 inclusive.
subdim	the dimension of the faces that this class represents. This must be between 0 and dim-1 inclusive.

Constructor & Destructor Documentation

FaceBase()

template<int dim, int subdim>

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

inlineprotected

Creates a new face.

The face will be initialised as belong to no boundary component.

Parameters

component

the component of the underlying triangulation to which the new face belongs.

Member Function Documentation

back()

template<int dim, int subdim>

const FaceEmbedding< dim, subdim > & regina::detail::FaceBase< dim, subdim >::back

inline

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()

template<int dim, int subdim>

auto regina::detail::FaceBase< dim, subdim >::begin

inline

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.

The type that is returned will be a lightweight iterator type, guaranteed to satisfy the C++ LegacyRandomAccessIterator requirement. The precise C++ type of the iterator is subject to change, so C++ users should use auto (just like this declaration does).

Iterating from begin() to end() (that is, iterating directly over a face) is equivalent to iterating over embeddings(). Iterating directly over a face generates a tiny bit less overhead, but you may also find it to be less readable. In particular, the following three blocks of code are all equivalent:

Face<dim, subdim>* face = ...;
for (const auto& emb : *face) { ... }

Face<dim, subdim>* face = ...;
for (const auto& emb : face->embeddings()) { ... }

Face<dim, subdim>* face = ...;
for (size_t i = 0; i < face->degree(); ++i) {
    const auto& emb = face->embedding(i);
    ...
}

Python

Not present; Python users can iterate over embeddings() instead.

Returns

a iterator that points to the first appearance.

boundaryComponent()

template<int dim, int subdim>

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

inline

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 null if this face does not lie entirely within the boundary of the triangulation.

component()

template<int dim, int subdim>

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

inline

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

Returns

the component containing this face.

containsVertex()

template<int dim, int subdim>

static constexpr bool regina::detail::FaceNumberingAPI< dim, subdim >::containsVertex ( unsigned  face, unsigned  vertex  )

staticconstexprinherited

Tests whether the given subdim-face of a dim-dimensional simplex contains the given vertex of the simplex.

Parameters

face	a subdim-face number in a dim-simplex; this must be between 0 and (dim+1 choose subdim+1)-1 inclusive.
vertex	a vertex number in a dim-simplex; this must be between 0 and dim inclusive.

Returns

true if and only if the given subdim-face contains the given vertex.

degree()

template<int dim, int subdim>

size_t regina::detail::FaceBase< dim, subdim >::degree

inline

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()

template<class T , bool supportsUtf8 = false>

std::string regina::Output< T, supportsUtf8 >::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.

edge()

template<int dim, int subdim>

Face< dim, 1 > * regina::detail::FaceBase< dim, subdim >::edge ( int  i ) const

inline

A dimension-specific alias for face<1>().

This alias is available for all facial dimensions subdim.

See face() for further information.

edgeMapping()

template<int dim, int subdim>

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

inline

A dimension-specific alias for faceMapping<1>().

This alias is available for all facial dimensions subdim.

See faceMapping() for further information.

embedding()

template<int dim, int subdim>

const FaceEmbedding< dim, subdim > & regina::detail::FaceBase< dim, subdim >::embedding ( size_t  index ) const

inline

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, or equivalently you can iterate over the lightweight object returned by embeddings().

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

index

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

Returns

details of the requested appearance.

embeddings()

template<int dim, int subdim>

auto regina::detail::FaceBase< dim, subdim >::embeddings

inline

Returns an object that allows iteration through and random access to all of the ways in which this face appears within a top-dimensional simplex of the underlying triangluation.

The object that is returned is lightweight, and can be happily copied by value. The C++ type of the object is subject to change, so C++ users should use auto (just like this declaration does).

The returned object is guaranteed to be an instance of ListView, which means it offers basic container-like functions and supports C++11 range-based for loops. The elements of the list will be read-only objects of type FaceEmbedding<dim, subdim>. For example, your code might look like:

Face<dim, subdim>* face = ...;
for (const auto& emb : face->embeddings()) { ... }

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).

Using embeddings() is equivalent to iterating over the face itself. It generates a tiny amount of extra overhead, but also may be considered more readable. In particular, the code above is equivalent to both of the following alternatives:

Face<dim, subdim>* face = ...;
for (const auto& emb : *face) { ... }

Face<dim, subdim>* face = ...;
for (size_t i = 0; i < face->degree(); ++i) {
    const auto& emb = face->embedding(i);
    ...
}

Returns

access to the list of all appearances of this face within a top-dimensional simplex of the underlying triangulation.

end()

template<int dim, int subdim>

auto regina::detail::FaceBase< dim, subdim >::end

inline

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

See begin() for full details on how this iteration works, and what other alternatives are available.

The type that is returned will be a lightweight iterator type, guaranteed to satisfy the C++ LegacyRandomAccessIterator requirement. The precise C++ type of the iterator is subject to change, so C++ users should use auto (just like this declaration does).

Python

Not present; Python users can iterate over embeddings() instead.

Returns

a "beyond the end" iterator that comes immediately after the last appearance.

face()

template<int dim, int subdim>

template<int lowerdim>

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

inline

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

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

face	the 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()

template<int dim, int subdim>

template<int lowerdim>

Perm< dim+1 > regina::detail::FaceBase< dim, subdim >::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.

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

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

face	the 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.

faceNumber()

template<int dim, int subdim>

static constexpr unsigned regina::detail::FaceNumberingAPI< dim, subdim >::faceNumber ( Perm< dim+1 >  vertices )

staticconstexprinherited

Identifies which subdim-face in a dim-dimensional simplex is represented by the first (subdim + 1) elements of the given permutation.

In other words, this routine identifies which subdim-face number within a dim-dimensional simplex spans vertices vertices[0, ..., subdim].

Parameters

vertices

a permutation whose first (subdim + 1) elements represent some vertex numbers in a dim-simplex.

Returns

the corresponding subdim-face number in the dim-simplex. This will be between 0 and (dim+1 choose subdim+1)-1 inclusive.

front()

template<int dim, int subdim>

const FaceEmbedding< dim, subdim > & regina::detail::FaceBase< dim, subdim >::front

inline

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()

template<int dim, int subdim>

bool regina::detail::FaceBase< dim, subdim >::hasBadIdentification

inline

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()

template<int dim, int subdim>

bool regina::detail::FaceBase< dim, subdim >::hasBadLink

inline

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.

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.

Precondition

The dimension dim is one of Regina's standard dimensions. Any attempt to use this routine in higher dimensions dim will result in a compile-time error. This is because testing for bad links in higher dimensions can require solutions to problems that are proven to be undecidable.

Returns

true if the link of this face is not appropriate (thereby making the face invalid), or false if the link is appropriate.

index()

template<int dim, int subdim>

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

inline

Returns the index of this face within the underlying triangulation.

Returns

the index of this face.

inMaximalForest()

template<int dim, int subdim>

bool regina::detail::FaceBase< dim, subdim >::inMaximalForest

inline

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).

Precondition

The facial dimension subdim is precisely dim-1.

Returns

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

isBoundary()

template<int dim, int subdim>

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

inline

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()

template<int dim, int subdim>

bool regina::detail::FaceBase< dim, subdim >::isLinkOrientable

inline

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()

template<int dim, int subdim>

bool regina::detail::FaceBase< dim, subdim >::isValid

inline

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).

markedIndex()

size_t regina::MarkedElement::markedIndex ( ) const

inlineinherited

Returns the index at which this object is stored in an MarkedVector.

If this object does not belong to an MarkedVector, the return value is undefined.

Returns

the index at which this object is stored.

ordering()

template<int dim, int subdim>

static constexpr Perm< dim+1 > regina::detail::FaceNumberingAPI< dim, subdim >::ordering ( unsigned  face )

staticconstexprinherited

Given a subdim-face number within a dim-dimensional simplex, returns the corresponding canonical ordering of the simplex vertices.

If this canonical ordering is c, then c[0,...,subdim] will be the vertices of the given face in increasing numerical order. That is, c[0] < ... < c[subdim]. The remaining images c[(subdim + 1),...,dim] will be ordered arbitrarily.

Note that this is not the same permutation as returned by Simplex<dim>::faceMapping<subdim>():

• ordering() is a static function, which returns the same permutation for the same face number, regardless of which dim-simplex we are looking at. The images of 0,...,subdim will always appear in increasing order, and the images of (subdim + 1),...,dim will be arbitrary.
• faceMapping() examines the underlying face F of the triangulation and, across all appearances of F in different dim-simplices: (i) chooses the images of 0,...,subdim to map to the same respective vertices of F; and (ii) chooses the images of (subdim + 1),...,dim to maintain a "consistent orientation" constraint.

Parameters

face	identifies which subdim-face of a dim-dimensional simplex to query. This must be between 0 and (dim+1 choose subdim+1)-1 inclusive.

Returns

the corresponding canonical ordering of the simplex vertices.

pentachoron()

template<int dim, int subdim>

Face< dim, 4 > * regina::detail::FaceBase< dim, subdim >::pentachoron ( int  i ) const

inline

A dimension-specific alias for face<4>().

This alias is available for facial dimensions subdim ≥ 5.

See face() for further information.

pentachoronMapping()

template<int dim, int subdim>

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

inline

A dimension-specific alias for faceMapping<4>().

This alias is available for facial dimensions subdim ≥ 5.

See faceMapping() for further information.

str()

template<class T , bool supportsUtf8 = false>

std::string regina::Output< T, supportsUtf8 >::str ( ) const

inherited

Returns a short text representation of this object.

This text should be human-readable, should use plain ASCII characters where possible, and should not contain any newlines.

Within these limits, this short text ouptut should be as information-rich as possible, since in most cases this forms the basis for the Python str() and repr() functions.

Python

The Python "stringification" function str() will use precisely this function, and for most classes the Python repr() function will incorporate this into its output.

Returns

a short text representation of this object.

tetrahedron()

template<int dim, int subdim>

Face< dim, 3 > * regina::detail::FaceBase< dim, subdim >::tetrahedron ( int  i ) const

inline

A dimension-specific alias for face<3>().

This alias is available for facial dimensions subdim ≥ 4.

See face() for further information.

tetrahedronMapping()

template<int dim, int subdim>

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

inline

A dimension-specific alias for faceMapping<3>().

This alias is available for facial dimensions subdim ≥ 4.

See faceMapping() for further information.

triangle()

template<int dim, int subdim>

Face< dim, 2 > * regina::detail::FaceBase< dim, subdim >::triangle ( int  i ) const

inline

A dimension-specific alias for face<2>().

This alias is available for facial dimensions subdim ≥ 3.

See face() for further information.

triangleMapping()

template<int dim, int subdim>

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

inline

A dimension-specific alias for faceMapping<2>().

This alias is available for facial dimensions subdim ≥ 3.

See faceMapping() for further information.

triangulation()

template<int dim, int subdim>

Triangulation< dim > & regina::detail::FaceBase< dim, subdim >::triangulation

inline

Returns the triangulation to which this face belongs.

Returns

a reference to the triangulation containing this face.

utf8()

template<class T , bool supportsUtf8 = false>

std::string regina::Output< T, supportsUtf8 >::utf8 ( ) const

inherited

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

Like str(), this text should be human-readable, should not contain any newlines, and (within these constraints) should be as information-rich as is reasonable.

Unlike str(), this function may use unicode characters to make the output more pleasant to read. The string that is returned will be encoded in UTF-8.

Returns

a short text representation of this object.

vertex()

template<int dim, int subdim>

Face< dim, 0 > * regina::detail::FaceBase< dim, subdim >::vertex ( int  i ) const

inline

A dimension-specific alias for face<0>().

This alias is available for all facial dimensions subdim.

See face() for further information.

vertexMapping()

template<int dim, int subdim>

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

inline

A dimension-specific alias for faceMapping<0>().

This alias is available for all facial dimensions subdim.

See faceMapping() for further information.

writeTextLong()

void regina::ShortOutput< Face< dim, subdim > , false >::writeTextLong ( std::ostream &  out ) const

inlineinherited

A default implementation for detailed output.

This routine simply calls T::writeTextShort() and appends a final newline.

Python

Not present; instead you can call detail() from the subclass T, which returns this output as a string.

Parameters

out	the output stream to which to write.

writeTextShort()

template<int dim, int subdim>

void regina::detail::FaceBase< dim, subdim >::writeTextShort

(

std::ostream &

out

)

const

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

Not present; use str() instead.

Parameters

out	the output stream to which to write.

Member Data Documentation

allowsInvalidFaces

template<int dim, int subdim>

constexpr bool regina::detail::FaceBase< dim, subdim >::allowsInvalidFaces

staticconstexpr

Initial value:

=
                (dim >= 3 && subdim <= dim - 2)

Indicates whether it is possible for a face of this dimension to be invalid.

allowsNonOrientableLinks

template<int dim, int subdim>

constexpr bool regina::detail::FaceBase< dim, subdim >::allowsNonOrientableLinks = (subdim <= dim - 3)

staticconstexpr

Indicates whether it is possible for a face of this dimension to have a non-orientable link.

dimension

template<int dim, int subdim>

constexpr int regina::detail::FaceBase< dim, subdim >::dimension = dim

staticconstexpr

A compile-time constant that gives the dimension of the triangulation containing this face.

lexDim

template<int dim, int subdim>

constexpr int regina::detail::FaceNumberingAPI< dim, subdim >::lexDim = (lexNumbering ? subdim : oppositeDim)

staticconstexprprotectedinherited

Whichever of subdim or oppositeDim uses lexicographical face numbering.

lexNumbering

template<int dim, int subdim>

constexpr bool regina::detail::FaceNumberingAPI< dim, subdim >::lexNumbering = (subdim <= oppositeDim)

staticconstexprinherited

true if faces are numbered in lexicographical order according to their vertices, or false if faces are numbered in reverse lexicographical order.

nFaces

template<int dim, int subdim>

constexpr int regina::detail::FaceNumberingAPI< dim, subdim >::nFaces = binomSmall(dim + 1, lexDim + 1)

staticconstexprinherited

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

oppositeDim

template<int dim, int subdim>

constexpr int regina::detail::FaceNumberingAPI< dim, subdim >::oppositeDim = (dim - 1 - subdim)

staticconstexprinherited

The dimension of the faces opposite these in a top-dimensional simplex of a dim-dimensional triangulation.

subdimension

template<int dim, int subdim>

constexpr int regina::detail::FaceBase< dim, subdim >::subdimension = subdim

staticconstexpr

A compile-time constant that gives the dimension of this face.

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

• triangulation/detail/face.h