Regina Calculation Engine
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Represents a single normal surface in a 3-manifold. More...
#include <surfaces/nnormalsurface.h>
Public Member Functions | |
NNormalSurface (const NTriangulation *triang, NNormalSurfaceVector *newVector) | |
Creates a new normal surface inside the given triangulation with the given coordinate vector. More... | |
NNormalSurface (const NTriangulation *triang, NormalCoords coordSystem, List allCoords) | |
A Python-only routine that creates a new normal surface inside the given triangulation with the given coordinate vector. More... | |
virtual | ~NNormalSurface () |
Destroys this normal surface. More... | |
NNormalSurface * | clone () const |
Creates a newly allocated clone of this normal surface. More... | |
NNormalSurface * | doubleSurface () const |
Creates a newly allocated surface that is the double of this surface. More... | |
NLargeInteger | getTriangleCoord (unsigned long tetIndex, int vertex) const |
Returns the number of triangular discs of the given type in this normal surface. More... | |
NLargeInteger | getOrientedTriangleCoord (unsigned long tetIndex, int vertex, bool orientation) const |
Returns the number of oriented triangular discs of the given type in this normal surface. More... | |
NLargeInteger | getQuadCoord (unsigned long tetIndex, int quadType) const |
Returns the number of quadrilateral discs of the given type in this normal surface. More... | |
NLargeInteger | getOrientedQuadCoord (unsigned long tetIndex, int quadType, bool orientation) const |
Returns the number of oriented quadrilateral discs of the given type in this normal surface. More... | |
NLargeInteger | getOctCoord (unsigned long tetIndex, int octType) const |
Returns the number of octagonal discs of the given type in this normal surface. More... | |
NLargeInteger | getEdgeWeight (unsigned long edgeIndex) const |
Returns the number of times this normal surface crosses the given edge. More... | |
NLargeInteger | getTriangleArcs (unsigned long triIndex, int triVertex) const |
Returns the number of arcs in which this normal surface intersects the given triangle in the given direction. More... | |
NLargeInteger | getFaceArcs (unsigned long triIndex, int triVertex) const |
A deprecated alias for getTriangleArcs(). More... | |
NDiscType | getOctPosition () const |
Determines the first coordinate position at which this surface has a non-zero octagonal coordinate. More... | |
size_t | getNumberOfCoords () const |
Returns the number of coordinates in the specific underlying coordinate system being used. More... | |
const NTriangulation * | getTriangulation () const |
Returns the triangulation in which this normal surface resides. More... | |
const std::string & | getName () const |
Returns the name associated with this normal surface. More... | |
void | setName (const std::string &newName) |
Sets the name associated with this normal surface. More... | |
void | writeTextShort (std::ostream &out) const |
The text representation will be in standard triangle-quad-oct coordinates. More... | |
void | writeRawVector (std::ostream &out) const |
Writes the underlying coordinate vector to the given output stream in text format. More... | |
virtual void | writeXMLData (std::ostream &out) const |
Writes a chunk of XML containing this normal surface and all of its properties. More... | |
bool | isEmpty () const |
Determines if this normal surface is empty (has no discs whatsoever). More... | |
bool | isCompact () const |
Determines if this normal surface is compact (has finitely many discs). More... | |
NLargeInteger | getEulerChar () const |
Returns the Euler characteristic of this surface. More... | |
NLargeInteger | getEulerCharacteristic () const |
A deprecated alias for getEulerChar(). More... | |
bool | isOrientable () const |
Returns whether or not this surface is orientable. More... | |
bool | isTwoSided () const |
Returns whether or not this surface is two-sided. More... | |
bool | isConnected () const |
Returns whether or not this surface is connected. More... | |
bool | hasRealBoundary () const |
Determines if this surface has any real boundary, that is, whether it meets any boundary triangles of the triangulation. More... | |
bool | isVertexLinking () const |
Determines whether or not this surface is vertex linking. More... | |
virtual const NVertex * | isVertexLink () const |
Determines whether or not a rational multiple of this surface is the link of a single vertex. More... | |
virtual std::pair< const NEdge *, const NEdge * > | isThinEdgeLink () const |
Determines whether or not a rational multiple of this surface is the thin link of a single edge. More... | |
bool | isSplitting () const |
Determines whether or not this surface is a splitting surface. More... | |
NLargeInteger | isCentral () const |
Determines whether or not this surface is a central surface. More... | |
bool | isCompressingDisc (bool knownConnected=false) const |
Determines whether this surface represents a compressing disc in the underlying 3-manifold. More... | |
bool | isIncompressible () const |
Determines whether this is an incompressible surface within the surrounding 3-manifold. More... | |
NTriangulation * | cutAlong () const |
Cuts the associated triangulation along this surface and returns a newly created resulting triangulation. More... | |
NTriangulation * | crush () const |
Crushes this surface to a point in the associated triangulation and returns a newly created resulting triangulation. More... | |
bool | sameSurface (const NNormalSurface &other) const |
Determines whether this and the given surface in fact represent the same normal (or almost normal) surface. More... | |
bool | locallyCompatible (const NNormalSurface &other) const |
Determines whether this and the given surface are locally compatible. More... | |
bool | disjoint (const NNormalSurface &other) const |
Determines whether this and the given surface can be placed within the surrounding triangulation so that they do not intersect anywhere at all, without changing either normal isotopy class. More... | |
NMatrixInt * | boundaryIntersections () const |
Computes the information about the boundary slopes of this surface at each cusp of the triangulation. More... | |
const NNormalSurfaceVector * | rawVector () const |
Gives read-only access to the raw vector that sits beneath this normal surface. More... | |
Input and Output | |
virtual void | writeTextLong (std::ostream &out) const |
Writes this object in long text format to the given output stream. More... | |
std::string | str () const |
Returns the output from writeTextShort() as a string. More... | |
std::string | toString () const |
A deprecated alias for str(), which returns the output from writeTextShort() as a string. More... | |
std::string | detail () const |
Returns the output from writeTextLong() as a string. More... | |
std::string | toStringLong () const |
A deprecated alias for detail(), which returns the output from writeTextLong() as a string. More... | |
Static Public Member Functions | |
static NNormalSurface * | findNonTrivialSphere (NTriangulation *tri) |
Searches for a non-vertex-linking normal 2-sphere within the given triangulation. More... | |
static NNormalSurface * | findVtxOctAlmostNormalSphere (NTriangulation *tri, bool quadOct=false) |
Searches the list of vertex octagonal almost normal surfaces for an almost normal 2-sphere within the given triangulation. More... | |
Protected Member Functions | |
void | calculateOctPosition () const |
Calculates the position of the first non-zero octagon coordinate and stores it as a property. More... | |
void | calculateEulerChar () const |
Calculates the Euler characteristic of this surface and stores it as a property. More... | |
void | calculateOrientable () const |
Calculates whether this surface is orientable and/or two-sided and stores the results as properties. More... | |
void | calculateRealBoundary () const |
Calculates whether this surface has any real boundary and stores the result as a property. More... | |
Protected Attributes | |
NNormalSurfaceVector * | vector |
Contains the coordinates of the normal surface in whichever space is appropriate. More... | |
const NTriangulation * | triangulation |
The triangulation in which this normal surface resides. More... | |
std::string | name |
An optional name associated with this surface. More... | |
NProperty< NDiscType > | octPosition |
The position of the first non-zero octagonal coordinate, or NDiscType::NONE if there is no non-zero octagonal coordinate. More... | |
NProperty< NLargeInteger > | eulerChar |
The Euler characteristic of this surface. More... | |
NProperty< bool > | orientable |
Is this surface orientable? More... | |
NProperty< bool > | twoSided |
Is this surface two-sided? More... | |
NProperty< bool > | connected |
Is this surface connected? More... | |
NProperty< bool > | realBoundary |
Does this surface have real boundary (i.e. More... | |
NProperty< bool > | compact |
Is this surface compact (i.e. More... | |
Friends | |
class | regina::NXMLNormalSurfaceReader |
Represents a single normal surface in a 3-manifold.
Once the underlying triangulation changes, this normal surface object is no longer valid.
The information provided by the various query methods is independent of the underlying coordinate system being used. See the NNormalSurfaceVector class notes for details of what to do when introducing a new coordinate system.
Note that non-compact surfaces (surfaces with infinitely many discs, such as spun-normal surfaces) are allowed; in these cases, the corresponding coordinate lookup routines will return NLargeInteger::infinity where appropriate.
Feature: Calculation of Euler characteristic and orientability for non-compact surfaces.
Feature (long-term): Determine which faces in the solution space a normal surface belongs to.
regina::NNormalSurface::NNormalSurface | ( | const NTriangulation * | triang, |
NNormalSurfaceVector * | newVector | ||
) |
Creates a new normal surface inside the given triangulation with the given coordinate vector.
This normal surface will claim ownership of the given vector (i.e., you should not change or delete the vector yourself afterwards).
triang | the triangulation in which this normal surface resides. |
newVector | a vector containing the coordinates of the normal surface in whichever space is appropriate. |
regina::NNormalSurface::NNormalSurface | ( | const NTriangulation * | triang, |
NormalCoords | coordSystem, | ||
List | allCoords | ||
) |
A Python-only routine that creates a new normal surface inside the given triangulation with the given coordinate vector.
triang | the triangulation in which this normal surface resides. |
coordSystem | the coordinate system used by this normal surface. |
allCoords | the corresponding vector of normal coordinates, expressed as a Python list. The list elements will be converted internally to NLargeInteger objects. |
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inlinevirtual |
Destroys this normal surface.
The underlying vector of coordinates will also be deallocated.
NMatrixInt* regina::NNormalSurface::boundaryIntersections | ( | ) | const |
Computes the information about the boundary slopes of this surface at each cusp of the triangulation.
This is for use with spun-normal surfaces (since for closed surfaces all boundary slopes are zero).
This routine is only available for use with SnapPea triangulations, since it needs to know the specific meridian and longitude on each cusp. This information is only available through the SnapPea kernel, since Regina does not use or store peripheral curves for its own NTriangulation class. Therefore, if the underlying triangulation (as returned by getTriangulation()) is not of the subclass NSnapPeaTriangulation, this routine will simply return 0.
All cusps are treated as complete. That is, any Dehn fillings stored in the SnapPea triangulation will be ignored.
The results are returned in a matrix with V rows and two columns, where V is the number of vertices in the triangulation. If row i of the matrix contains the integers M and L, this indicates that at the ith cusp, the boundary curves have algebraic intersection number M with the meridian and L with the longitude. Equivalently, the boundary curves pass L times around the meridian and -M times around the longitude. The rational boundary slope is therefore -L/M
, and there are gcd(L,M)
boundary curves with this slope.
The orientations of the boundary curves of a spun-normal surface are chosen so that if meridian and longitude are a positive basis as vieved from the cusp, then as one travels along an oriented boundary curve, the spun-normal surface spirals into the cusp to one's right and down into the manifold to one's left.
If the triangulation contains more than one vertex, the rows in the resulting matrix are ordered by cusp index (as stored by SnapPea). You can call NSnapPeaTriangulation::cuspVertex() to map these to Regina's vertex indices if needed.
At present, Regina can only compute boundary slopes if the triangulation is oriented, if every vertex link in the triangulation is a torus, and if the underlying coordinate system is for normal surfaces (not almost normal surfaces). If these conditions are not met, this routine will return 0.
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Calculates the Euler characteristic of this surface and stores it as a property.
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Calculates the position of the first non-zero octagon coordinate and stores it as a property.
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Calculates whether this surface is orientable and/or two-sided and stores the results as properties.
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Calculates whether this surface has any real boundary and stores the result as a property.
NNormalSurface* regina::NNormalSurface::clone | ( | ) | const |
Creates a newly allocated clone of this normal surface.
The name of the normal surface will not be copied to the clone; instead the clone will have an empty name.
NTriangulation* regina::NNormalSurface::crush | ( | ) | const |
Crushes this surface to a point in the associated triangulation and returns a newly created resulting triangulation.
The original triangulation is not changed.
Crushing the surface will produce a number of tetrahedra, triangular pillows and/or footballs. The pillows and footballs will then be flattened to triangles and edges respectively (resulting in the possible changes mentioned below) to produce a proper triangulation.
Note that the new triangulation will have at most the same number of tetrahedra as the old triangulation, and will have strictly fewer tetrahedra if this surface is not vertex linking.
The act of flattening pillows and footballs as described above can lead to unintended topological side-effects, beyond the effects of merely cutting along this surface and identifying the new boundary surface(s) to points. Examples of these unintended side-effects can include connected sum decompositions, removal of 3-spheres and small Lens spaces and so on; a full list of possible changes is beyond the scope of this API documentation.
NTriangulation* regina::NNormalSurface::cutAlong | ( | ) | const |
Cuts the associated triangulation along this surface and returns a newly created resulting triangulation.
The original triangulation is not changed.
Note that, unlike crushing a surface to a point, this operation will not change the topology of the underlying 3-manifold beyond simply slicing along this surface.
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inherited |
Returns the output from writeTextLong() as a string.
bool regina::NNormalSurface::disjoint | ( | const NNormalSurface & | other | ) | const |
Determines whether this and the given surface can be placed within the surrounding triangulation so that they do not intersect anywhere at all, without changing either normal isotopy class.
This is a global constraint, and therefore gives a stronger test than locallyCompatible(). However, this global constraint is also much slower to test; the running time is proportional to the total number of normal discs in both surfaces.
Note that this routine has a number of preconditions. Most importantly, it will only work if both this and the given surface use the same coordinate system. Running this test over two surfaces with different coordinate systems could give unpredictable results, and might crash the program entirely.
other | the other surface to test alongside this surface for potential intersections. |
true
if both surfaces can be embedded without intersecting anywhere, or false
if this and the given surface are forced to intersect at some point. NNormalSurface* regina::NNormalSurface::doubleSurface | ( | ) | const |
Creates a newly allocated surface that is the double of this surface.
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Searches for a non-vertex-linking normal 2-sphere within the given triangulation.
If a non-vertex linking normal 2-sphere exists anywhere at all within the triangulation, then this routine is guaranteed to find one.
Note that the surface returned (if any) depends upon the triangulation, and so must be destroyed before the triangulation itself.
tri | the triangulation in which to search. |
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Searches the list of vertex octagonal almost normal surfaces for an almost normal 2-sphere within the given triangulation.
This means that tubed almost normal 2-spheres or non-vertex octagonal almost normal 2-spheres will not be found.
This search can be done either in standard almost normal coordinates (with triangles, quadrilaterals and octagons), or in quadrilateral-octagon coordinates. This choice of coordinate system affects how we define "vertex". The default is to use standard coordinates (where the set of vertex surfaces is larger).
For "sufficiently nice" triangulations, if this routine fails to find an almost normal 2-sphere then we can be certain that the triangulation contains no almost normal 2-spheres at all. In particular, this is true for closed orientable one-vertex 0-efficient triangulations. For a proof in standard coordinates, see "0-efficient triangulations of 3-manifolds", William Jaco and J. Hyam Rubinstein, J. Differential Geom. 65 (2003), no. 1, 61–168. For a proof in quadrilateral-octagon coordinates, see "Quadrilateral-octagon coordinates for almost normal surfaces", Benjamin A. Burton, Experiment. Math. 19 (2010), 285-315.
Note that the surface that this routine returns (if any) depends upon the triangulation, and so this surface must be destroyed before the triangulation is destroyed.
tri | the triangulation in which to search. |
quadOct | true if we should search for vertex surfaces in quadrilateral-octagon coordinates, or false (the default) if we should search for surfaces in standard almost normal coordinates. |
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Returns the number of times this normal surface crosses the given edge.
edgeIndex | the index in the triangulation of the edge in which we are interested; this should be between 0 and NTriangulation::getNumberOfEdges()-1 inclusive. |
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Returns the Euler characteristic of this surface.
This routine caches its results, which means that once it has been called for a particular surface, subsequent calls return the answer immediately.
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A deprecated alias for getEulerChar().
Returns the Euler characteristic of this surface.
This routine caches its results, which means that once it has been called for a particular surface, subsequent calls return the answer immediately.
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A deprecated alias for getTriangleArcs().
This routine returns the number of arcs in which this normal surface intersects the given triangle in the given direction. See getTriangleArcs() for further details.
triIndex | the index in the triangulation of the triangle in which we are interested; this should be between 0 and NTriangulation::getNumberOfTriangles()-1 inclusive. |
triVertex | the vertex of the triangle (0, 1 or 2) around which the arcs of intersection that we are interested in lie; only these arcs will be counted. |
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Returns the name associated with this normal surface.
Names are optional and need not be unique. The default name for a surface is the empty string.
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Returns the number of coordinates in the specific underlying coordinate system being used.
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Returns the number of octagonal discs of the given type in this normal surface.
In each tetrahedron, there are three types of octagons, defined by how they separate the four tetrahedron vertices into two pairs. These types are labelled 0, 1 and 2, and they correspond exactly to the three vertex splittings described by regina::vertexSplit. Specifically:
An octagonal disc type is identified by specifying a tetrahedron and a vertex splitting 0, 1 or 2 as described above.
If you are using a coordinate system that adorns discs with additional information (such as orientation), this routine returns the total number of octagons in the given tetrahedron of the given type.
tetIndex | the index in the triangulation of the tetrahedron in which the requested octagons reside; this should be between 0 and NTriangulation::getNumberOfTetrahedra()-1 inclusive. |
octType | the number of the vertex splitting that this octagon type represents, as described above; this should be between 0 and 2 inclusive. |
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Determines the first coordinate position at which this surface has a non-zero octagonal coordinate.
In other words, if this routine returns the disc type t, then the octagonal coordinate returned by getOctCoord(t.tetIndex, t.type) is non-zero. Here NDiscType::type represents an octagon type within a tetrahedron, and takes values between 0 and 2 inclusive.
If this surface does not contain any octagons, this routine returns NDiscType::NONE instead.
This routine caches its results, which means that once it has been called for a particular surface, subsequent calls return the answer immediately. Moreover, if the underlying coordinate system does not support almost normal surfaces, then even the first call is fast (it returns NDiscType::NONE immediately).
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Returns the number of oriented quadrilateral discs of the given type in this normal surface.
This routine is for coordinate systems that support transversely oriented normal surfaces; for details see "The Thurston norm via normal surfaces", Stephan Tillmann and Daryl Cooper, Pacific Journal of Mathematics 239 (2009), 1-15.
An oriented quadrilateral disc type is identified by specifying a tetrahedron, a vertex splitting of that tetrahedron as described in getQuadCoord(), and a boolean orientation. The true
orientation indicates a transverse orientation pointing to the edge containing vertex 0 of the tetrahedron, and the false
orientation indicates a transverse orientation pointing to the opposite edge.
If the underlying coordinate system does not support transverse orientation, this routine will simply return zero.
tetIndex | the index in the triangulation of the tetrahedron in which the requested quadrilaterals reside; this should be between 0 and NTriangulation::getNumberOfTetrahedra()-1 inclusive. |
quadType | the number of the vertex splitting that this quad type represents; this should be between 0 and 2 inclusive. |
orientation | the orientation of the quadrilateral disc |
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Returns the number of oriented triangular discs of the given type in this normal surface.
This routine is for coordinate systems that support transversely oriented normal surfaces; for details see "The Thurston norm via normal surfaces", Stephan Tillmann and Daryl Cooper, Pacific Journal of Mathematics 239 (2009), 1-15.
An oriented triangular disc type is identified by specifying a tetrahedron, a vertex of that tetrahedron that the triangle surrounds, and a boolean orientation. The true
orientation indicates a transverse orientation pointing to the nearby vertex, and the false
orientation indicates a transverse orientation pointing to the opposite face.
If the underlying coordinate system does not support transverse orientation, this routine will simply return zero.
tetIndex | the index in the triangulation of the tetrahedron in which the requested triangles reside; this should be between 0 and NTriangulation::getNumberOfTetrahedra()-1 inclusive. |
vertex | the vertex of the given tetrahedron around which the requested triangles lie; this should be between 0 and 3 inclusive. |
orientation | the orientation of the triangle |
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Returns the number of quadrilateral discs of the given type in this normal surface.
In each tetrahedron, there are three types of quadrilaterals, defined by how they separate the four tetrahedron vertices into two pairs. These types are labelled 0, 1 and 2, and they correspond exactly to the three vertex splittings described by regina::vertexSplit. Specifically:
A quadrilateral disc type is identified by specifying a tetrahedron and a vertex splitting 0, 1 or 2 as described above.
If you are using a coordinate system that adorns discs with additional information (such as orientation), this routine returns the total number of quadrilaterals in the given tetrahedron of the given type.
tetIndex | the index in the triangulation of the tetrahedron in which the requested quadrilaterals reside; this should be between 0 and NTriangulation::getNumberOfTetrahedra()-1 inclusive. |
quadType | the number of the vertex splitting that this quad type represents, as described above; this should be between 0 and 2 inclusive. |
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Returns the number of arcs in which this normal surface intersects the given triangle in the given direction.
triIndex | the index in the triangulation of the triangle in which we are interested; this should be between 0 and NTriangulation::getNumberOfTriangles()-1 inclusive. |
triVertex | the vertex of the triangle (0, 1 or 2) around which the arcs of intersection that we are interested in lie; only these arcs will be counted. |
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Returns the number of triangular discs of the given type in this normal surface.
A triangular disc type is identified by specifying a tetrahedron and a vertex of that tetrahedron that the triangle surrounds.
If you are using a coordinate system that adorns discs with additional information (such as orientation), this routine returns the total number of triangles in the given tetrahedron of the given type.
tetIndex | the index in the triangulation of the tetrahedron in which the requested triangles reside; this should be between 0 and NTriangulation::getNumberOfTetrahedra()-1 inclusive. |
vertex | the vertex of the given tetrahedron around which the requested triangles lie; this should be between 0 and 3 inclusive. |
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Returns the triangulation in which this normal surface resides.
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Determines if this surface has any real boundary, that is, whether it meets any boundary triangles of the triangulation.
This routine caches its results, which means that once it has been called for a particular surface, subsequent calls return the answer immediately.
true
if and only if this surface has real boundary.
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Determines whether or not this surface is a central surface.
A central surface is a compact surface containing at most one normal or almost normal disc per tetrahedron. If this surface is central, the number of tetrahedra that it meets (i.e., the number of discs in the surface) will be returned.
Note that the results of this routine are not cached. Thus the results will be reevaluated every time this routine is called.
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Determines if this normal surface is compact (has finitely many discs).
This routine caches its results, which means that once it has been called for a particular surface, subsequent calls return the answer immediately.
true
if and only if this normal surface is compact. bool regina::NNormalSurface::isCompressingDisc | ( | bool | knownConnected = false | ) | const |
Determines whether this surface represents a compressing disc in the underlying 3-manifold.
Let this surface be D and let the underlying 3-manifold be M with boundary B. To be a compressing disc, D must be a properly embedded disc in M, and the boundary of D must not bound a disc in B.
The implementation of this routine is somewhat inefficient at present, since it cuts along the disc, retriangulates and then examines the resulting boundary components.
Optimise: Reimplement this to avoid cutting along surfaces.
Bug: Check for absurdly large numbers of discs and bail accordingly.
knownConnected | true if this normal surface is already known to be connected (for instance, if it came from an enumeration of vertex normal surfaces), or false if we should not assume any such information about this surface. |
true
if this surface is a compressing disc, or false
if this surface is not a compressing disc.
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Returns whether or not this surface is connected.
This routine caches its results, which means that once it has been called for a particular surface, subsequent calls return the answer immediately.
true
if this surface is connected, or false
if this surface is disconnected. bool regina::NNormalSurface::isEmpty | ( | ) | const |
Determines if this normal surface is empty (has no discs whatsoever).
bool regina::NNormalSurface::isIncompressible | ( | ) | const |
Determines whether this is an incompressible surface within the surrounding 3-manifold.
At present, this routine is only implemented for surfaces embedded within closed and irreducible 3-manifold triangulations.
Let D be some disc embedded in the underlying 3-manifold, and let B be the boundary of D. We call D a compressing disc for this surface if (i) the intersection of D with this surface is the boundary B, and (ii) although B bounds a disc within the 3-manifold, it does not bound a disc within this surface.
We declare this surface to be incompressible if there are no such compressing discs. For our purposes, spheres are never considered incompressible (so if this surface is a sphere then this routine will always return false
).
This test is designed exclusively for two-sided surfaces. If this surface is one-sided, the incompressibility test will be run on its two-sided double cover.
true
if this surface is incompressible, or false
if this surface is not incompressible (or if it is a sphere).
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Returns whether or not this surface is orientable.
This routine caches its results, which means that once it has been called for a particular surface, subsequent calls return the answer immediately.
true
if this surface is orientable, or false
if this surface is non-orientable.
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Determines whether or not this surface is a splitting surface.
A splitting surface is a compact surface containing precisely one quad per tetrahedron and no other normal (or almost normal) discs.
Note that the results of this routine are not cached. Thus the results will be reevaluated every time this routine is called.
true
if and only if this is a splitting surface.
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Determines whether or not a rational multiple of this surface is the thin link of a single edge.
If there are two different edges e1 and e2 for which this surface could be expressed as the thin link of either e1 or e2, the pair (e1,e2) will be returned. If this surface is the thin link of only one edge e, the pair (e,0) will be returned. If this surface is not the thin link of any edges, the pair (0,0) will be returned.
Note that the results of this routine are not cached. Thus the results will be reevaluated every time this routine is called.
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Returns whether or not this surface is two-sided.
This routine caches its results, which means that once it has been called for a particular surface, subsequent calls return the answer immediately.
true
if this surface is two-sided, or false
if this surface is one-sided.
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Determines whether or not a rational multiple of this surface is the link of a single vertex.
Note that the results of this routine are not cached. Thus the results will be reevaluated every time this routine is called.
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Determines whether or not this surface is vertex linking.
A vertex linking surface contains only triangles.
Note that the results of this routine are not cached. Thus the results will be reevaluated every time this routine is called.
true
if and only if this surface is vertex linking. bool regina::NNormalSurface::locallyCompatible | ( | const NNormalSurface & | other | ) | const |
Determines whether this and the given surface are locally compatible.
Local compatibility means that, within each individual tetrahedron of the triangulation, it is possible to arrange the normal discs of both surfaces so that none intersect.
This is a local constraint, not a global constraint. That is, we do not insist that we can avoid intersections within all tetrahedra simultaneously. To test the global constraint, see the (much slower) routine disjoint() instead.
Local compatibility can be formulated in terms of normal disc types. Two normal (or almost normal) surfaces are locally compatible if and only if they together have at most one quadrilateral or octagonal disc type per tetrahedron.
Note again that this is a local constraint only. In particular, for almost normal surfaces, it does not insist that there is at most one octagonal disc type anywhere within the triangulation.
If one of the two surfaces breaks the local compatibility constraints on its own (for instance, it contains two different quadrilateral disc types within the same tetrahedron), then this routine will return false
regardless of what the other surface contains.
other | the other surface to test for local compatibility with this surface. |
true
if the two surfaces are locally compatible, or false
if they are not.
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Gives read-only access to the raw vector that sits beneath this normal surface.
Generally users should not need this function. However, it is provided here in case the need should arise (e.g., for reasons of efficiency).
bool regina::NNormalSurface::sameSurface | ( | const NNormalSurface & | other | ) | const |
Determines whether this and the given surface in fact represent the same normal (or almost normal) surface.
Specifically, this routine examines (or computes) the number of normal or almost normal discs of each type, and returns true
if and only if these counts are the same for both surfaces.
It does not matter what coordinate systems the two surfaces use. In particular, it does not matter if this and the given surface use different coordinate systems, and it does not matter if one surface uses an almost normal coordinate system and the other does not.
other | the surface to be compared with this surface. |
true
if both surfaces represent the same normal or almost normal surface, or false
if not.
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inline |
Sets the name associated with this normal surface.
Names are optional and need not be unique. The default name for a surface is the empty string.
newName | the new name to associate with this surface. |
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inherited |
Returns the output from writeTextShort() as a string.
__str__()
function.
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inlineinherited |
A deprecated alias for str(), which returns the output from writeTextShort() as a string.
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inlineinherited |
A deprecated alias for detail(), which returns the output from writeTextLong() as a string.
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inline |
Writes the underlying coordinate vector to the given output stream in text format.
No indication will be given as to which coordinate system is being used or what each coordinate means. No newline will be written.
out | the output stream to which to write. |
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inlinevirtualinherited |
Writes this object in long text format to the given output stream.
The output should provide the user with all the information they could want. The output should be human-readable, should not contain extremely long lines (so users can read the output in a terminal), and should end with a final newline.
The default implementation of this routine merely calls writeTextShort() and adds a newline.
out | the output stream to which to write. |
Reimplemented in regina::NSnapPeaTriangulation, regina::NGroupPresentation, regina::NHomMarkedAbelianGroup, regina::NNormalSurfaceList, regina::NTetrahedron, regina::NSatRegion, regina::NVertex, regina::NEdge, regina::Dim2Triangle, regina::NTriangle, regina::NLayeredSolidTorus, regina::NHomGroupPresentation, regina::NGenericIsomorphism< dim >, regina::NGenericIsomorphism< 2 >, regina::NGenericIsomorphism< 3 >, regina::NTriangulation, regina::NComponent, regina::NTxICore, regina::NTriSolidTorus, regina::NAngleStructureList, regina::Dim2Edge, regina::NBoundaryComponent, regina::NLayeredChain, regina::Dim2Vertex, regina::Dim2Component, regina::NScript, regina::NAugTriSolidTorus, regina::NSpiralSolidTorus, regina::NSurfaceFilterProperties, regina::NLayeredTorusBundle, regina::NManifold, regina::NPlugTriSolidTorus, regina::NMatrixInt, regina::NBlockedSFSTriple, regina::NPluggedTorusBundle, regina::Dim2Triangulation, regina::NSurfaceSubset, regina::NLayeredLensSpace, regina::NLayeredLoop, regina::NFileInfo, regina::NBlockedSFSLoop, regina::NSnappedBall, regina::NBlockedSFSPair, regina::Dim2BoundaryComponent, regina::NTrivialTri, regina::NL31Pillow, regina::NLayeredChainPair, regina::NText, regina::NSurfaceFilterCombination, and regina::NBlockedSFS.
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virtual |
The text representation will be in standard triangle-quad-oct coordinates.
Octagonal coordinates will only be written if the surface is stored using a coordinate system that supports almost normal surfaces.
Implements regina::ShareableObject.
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virtual |
Writes a chunk of XML containing this normal surface and all of its properties.
This routine will be called from within NNormalSurfaceList::writeXMLPacketData().
out | the output stream to which the XML should be written. |
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mutableprotected |
Is this surface compact (i.e.
does it only contain finitely many discs)?
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mutableprotected |
Is this surface connected?
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mutableprotected |
The Euler characteristic of this surface.
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protected |
An optional name associated with this surface.
The position of the first non-zero octagonal coordinate, or NDiscType::NONE if there is no non-zero octagonal coordinate.
Here NDiscType::type is an octagon type between 0 and 2 inclusive.
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mutableprotected |
Is this surface orientable?
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mutableprotected |
Does this surface have real boundary (i.e.
does it meet any boundary triangles)?
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protected |
The triangulation in which this normal surface resides.
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mutableprotected |
Is this surface two-sided?
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protected |
Contains the coordinates of the normal surface in whichever space is appropriate.