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size_t | index () const |
| Returns the index of this boundary component in the underlying triangulation. More...
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bool | isOrientable () const |
| Determines if this boundary component is orientable. More...
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std::string | str () const |
| Returns a short text representation of this object. More...
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std::string | utf8 () const |
| Returns a short text representation of this object using unicode characters. More...
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std::string | detail () const |
| Returns a detailed text representation of this object. More...
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const Triangulation< dim-1 > * | build () const |
| Returns the full (dim-1)-dimensional triangulation of this boundary component. More...
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bool | isReal () const |
| Determines if this boundary component is real. More...
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bool | isIdeal () const |
| Determines if this boundary component is ideal. More...
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bool | isInvalidVertex () const |
| Determines if this boundary component consists of a single invalid vertex and nothing else. More...
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void | writeTextShort (std::ostream &out) const |
| Writes a short text representation of this object to the given output stream. More...
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void | writeTextLong (std::ostream &out) const |
| Writes a detailed text representation of this object to the given output stream. More...
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size_t | size () const |
| Returns the number of (dim-1)-faces in this boundary component. More...
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size_t | countRidges () const |
| Returns the number of (dim-2)-faces in this boundary component. More...
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size_t | countFaces () const |
| Returns the number of subdim-faces in this boundary component. More...
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const std::vector< Face< dim, dim-1 > *> & | facets () const |
| Returns all (dim-1)-faces in this boundary component. More...
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const std::vector< Face< dim, subdim > *> & | faces () const |
| Returns all subdim-faces in this boundary component. More...
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Face< dim, dim-1 > * | facet (size_t index) const |
| Returns the requested (dim-1)-face in this boundary component. More...
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Face< dim, subdim > * | face (size_t index) const |
| Returns the requested subdim-face in this boundary component. More...
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Triangulation< dim > * | triangulation () const |
| Returns the triangulation to which this boundary component belongs. More...
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Component< dim > * | component () const |
| Returns the connected component of the triangulation to which this boundary component belongs. More...
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size_t | markedIndex () const |
| Returns the index at which this object is stored in an MarkedVector. More...
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const Triangulation< dim-1 > * | buildVertexLink () const |
| Triangulates the vertex link for an ideal or invalid vertex boundary component. More...
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void | push_back (Face< dim, subdim > *face) |
| Pushes the given face onto the end of the list of subdim-faces of this boundary component. More...
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void | reorderAndRelabelFaces (Triangulation< dim-1 > *tri) const |
| Reorders all lower-dimensional faces of the given triangulation so that they appear in the same order as the corresponding faces of this boundary component, and relabels these faces so that their vertices are numbered in a corresponding way. More...
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void | reorderAndRelabelFaces (Triangulation< tridim > *tri, const std::vector< Face< dim, tridim > * > &tridimFaces) const |
| Reorders and relabels all faces of all dimensions 0,...,subdim of the given triangulation, so that for each k, the k-faces of the given triangulation appear in the same order as the corresponding k-faces in this suite, and have their vertices numbered in a corresponding way. More...
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template<int dim>
class regina::BoundaryComponent< dim >
A component of the boundary of a dim-manifold triangulation.
Regina recognises three types of boundary components:
- A real boundary component is formed from unglued facets of top-dimensional simplices of the triangulation. For example, a real boundary component of a 3-manifold triangulation is formed from boundary triangles in the 2-skeleton.
- An ideal boundary component consists of a single ideal vertex; that is, a vertex whose link is a closed (dim-1)-manifold but not a sphere. This is only possible for dimensions dim ≥ 3.
- An invalid vertex boundary component consists of a single invalid invalid vertex that does not belong to a real boundary component. This means the vertex link must be an ideal (dim-1)-manifold, or an invalid (dim-1)-manifold with no real boundary. Invalid vertex boundary components are only possible for dimensions dim ≥ 4.
Ideal and invalid vertex boundary components are only recognised when dim is one of Regina's standard dimensions. This is because, in higher dimensions, the relevant conditions rely on undecidable problems.
Regina stores different skeletal information for different types of boundary components:
- For real boundary components in standard dimensions, Regina stores all boundary faces of all dimensions 0,1,...,dim-1.
- For real boundary components in higher (non-standard) dimensions, Regina only stores the boundary (dim-1)-faces (i.e., the unglued facets of top-dimensional simplices that make up the boundary component). This is simply to avoid excessive memory usage.
- For ideal and invalid vertex boundary components, Regina stores the relevant vertex. Such boundary components do not contain any faces of any dimension greater than zero.
We can encounter some interesting cases with invalid triangulations. Consider some face whose link has more than one boundary component (which makes the face invalid). This means that different parts of the (dim)-manifold boundary are effectively "pinched" together. If this happens, the different parts of the boundary that are pinched might or might not be listed as part of the same boundary component; if not then the offending face will be included in all of these boundary components. Nevertheless, only one of these can be considered the "official" boundary component of the face as returned by Face::boundaryComponent(). This is all a bit of a mess, but then again the entire triangulation is invalid and so you almost certainly have bigger problems to deal with.
Boundary components are highly temporary: whenever a triangulation changes, all of its boundary component objects will be deleted and new ones will be created in their place.
Boundary component objects are all created, managed and destroyed by the class Triangulation<dim>. See the Triangulation notes for further information on working with dim-dimensional triangulations.
For dimension dim = 3, this template is specialised and offers more functionality. In order to use this specialised class, you will need to include the corresponding header triangulation/dim3.h.
- Python:
- Python does not support templates. Instead this class can be used by appending the dimension as a suffix (e.g., BoundaryComponent2 and BoundaryComponent3 for dimensions 2 and 3).
- Template Parameters
-
dim | the dimension of the underlying triangulation. This must be between 2 and 15 inclusive. |
Returns the full (dim-1)-dimensional triangulation of this boundary component.
Note that this triangulation is read-only (though of course you can clone it and then operate upon the clone).
If this is a real boundary component (i.e., it is built from one or more (dim-1)-faces), then the triangulation of this boundary component is as follows:
- Let i lie between 0 and size()-1 inclusive. Then simplex i of the returned (dim-1)-dimensional triangulation is a copy of
facet(i)
of this boundary component, and its vertices 0,...,dim-1 are numbered in the same way. To relate these (dim-1)-face vertex numbers to the vertex numbers of top-dimensional simplices in the overall dim-dimensional triangulation, see Simplex<dim>::faceMapping<dim-1>().
- If this boundary component stores lower-dimensional faces (i.e., if the template argument allFaces is
true
), then a similar correspondence holds for these lower-dimensional faces also: for each i, k-face i of the returned triangulation is a copy of face<k>(i)
of this boundary component, and its vertices are numbered in the same way.
If this boundary component consists only of a single vertex (i.e., this is an ideal or invalid vertex boundary component), then this routine returns the triangulation of the corresponding vertex link. See Vertex::link() for details.
This routine is fast, since it caches the boundary triangulation. Moreover, it is guaranteed that the full skeleton of this (dim-1)-dimensional triangulation will have been generated already.
- Returns
- the triangulation of this boundary component.
Determines if this boundary component is ideal.
This is the case if and only if it consists of a single ideal vertex and no faces of any other dimensions.
See the BoundaryComponent class notes for an overview of ideal boundary components, which can only occur in dimensions ≥ 3, and which are only recognised where dim is one of Regina's standard dimensions.
Note that a boundary component formed from a single invalid vertex is not considered to be ideal. This means that, if a boundary component contains no faces of positive dimension, then one and only one of isIdeal() and isInvalidVertex() will return true
.
This routine is only available where dim is at least 3 and is one of Regina's standard dimensions.
- Returns
true
if and only if this boundary component is ideal.
Determines if this boundary component consists of a single invalid vertex and nothing else.
In particular, such a boundary component must contain no faces of any positive dimension.
See the BoundaryComponent class notes for an overview of invalid vertex boundary components, which can only occur in dimensions ≥ 4, and which are only recognised where dim is one of Regina's standard dimensions.
An invalid vertex is only placed in its own boundary component if it does not already belong to some larger boundary component (for instance, if its link is an ideal (dim-1)-manifold triangulation). This means that, for a boundary component consisting of one or more (dim-1)-faces, this routine will return false
even if the boundary component also includes one or more invalid vertices.
Note that, if a boundary component contains no faces of positive dimension, then one and only one of isIdeal() and isInvalidVertex() will return true
.
This routine is only available where dim is at least 3 and is one of Regina's standard dimensions.
- Returns
true
if and only if this boundary component consists of a single invalid vertex and nothing else.