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Normal Surfaces

Normal surfaces in triangulations. More...

Classes

struct  regina::NDiscSpec
 Specifies a single normal disc in a normal surface. More...
class  regina::NDiscSetTet
 Represents a set of normal discs inside a single tetrahedron. More...
class  regina::NDiscSetTetData< T >
 Stores data of type T for every normal disc inside a single tetrahedron. More...
class  regina::NDiscSetSurface
 Represents the set of all normal discs forming a normal surface. More...
class  regina::NDiscSetSurfaceData< T >
 Stores data of type T for every normal disc within a particular normal surface. More...
class  regina::NDiscSpecIterator
 An iterator used for running through all normal discs in a normal surface. More...
class  regina::NNormalSurfaceVector
 Stores the vector of a single normal surface in a 3-manifold. More...
class  regina::NNormalSurface
 Represents a single normal surface in a 3-manifold. More...
class  regina::NNormalSurfaceList
 A packet representing a collection of normal surfaces in a 3-manifold. More...
struct  regina::NPrismSpec
 Specifies a single triangular prism in a tetrahedron. More...
class  regina::NPrismSetSurface
 Represents the set of prisms defined by slicing along all the quads in a particular normal surface. More...
class  regina::NNormalSurfaceVectorANStandard
 An almost normal surface vector using standard triangle-quad-oct coordinates. More...
class  regina::NNormalSurfaceVectorMirrored
 A normal surface vector that is mirrored in another coordinate system to avoid frequent lengthy calculations. More...
class  regina::NNormalSurfaceVectorQuad
 A normal surface vector using quad coordinates. More...
class  regina::NNormalSurfaceVectorStandard
 A normal surface vector using standard triangle-quad coordinates. More...
class  regina::NSurfaceFilter
 A packet that accepts or rejects normal surfaces. More...
class  regina::NSurfaceSet
 Represents a collection of normal surfaces in a 3-manifold. More...
class  regina::NSurfaceSubset
 Represents a subset of a normal surface set. More...
class  regina::NXMLFilterReader
 An XML element reader that reads the specific details of a normal surface filter. More...
class  regina::NXMLFilterPacketReader
 An XML packet reader that reads a single surface filter. More...
class  regina::NXMLNormalSurfaceReader
 An XML element reader that reads a single normal surface. More...
class  regina::NXMLNormalSurfaceListReader
 An XML packet reader that reads a single normal surface list. More...
class  regina::NSurfaceFilterCombination
 A normal surface filter that simply combines other filters. More...
class  regina::NSurfaceFilterProperties
 A normal surface filter that filters by basic properties of the normal surface. More...

Functions

std::ostream & regina::operator<< (std::ostream &out, const NDiscSpec &spec)
 Writes the given disc specifier to the given output stream.
bool regina::numberDiscsAwayFromVertex (int discType, int vertex)
 Determines whether or not normal discs of the given type are numbered away from the given vertex.
bool regina::discOrientationFollowsEdge (int discType, int vertex, int edgeStart, int edgeEnd)
 Determines whether or not the natural boundary orientation of a normal disc of the given type follows the given directed normal arc.
template<class RayOutputIterator, class FaceOutputIterator>
void regina::createNonNegativeCone (NTriangulation *triangulation, int flavour, RayOutputIterator rays, FaceOutputIterator faces)
 Writes to the given output iterators newly allocated rays and faces representing the cone in the given flavour of coordinate system obtained by setting all coordinates non-negative.
NMatrixInt * regina::makeMatchingEquations (NTriangulation *triangulation, int flavour)
 Creates a new set of normal surface matching equations for the given triangulation using the given flavour of coordinate system.
std::ostream & regina::operator<< (std::ostream &out, const NPrismSpec &spec)
 Writes the given prism specifier to the given output stream.

Variables

const int regina::vertexSplit [4][4]
 Lists which vertex splits split which pairs of vertices.
const int regina::vertexSplitMeeting [4][4][2]
 Lists which vertex splits meet which edges.
const int regina::vertexSplitDefn [3][4]
 Lists the vertices which each vertex split splits.
const int regina::vertexSplitPartner [3][4]
 Lists the second vertex with which each vertex is paired under each vertex split.
const char regina::vertexSplitString [3][6]
 Contains strings describing which vertices each vertex split splits.
const NPerm regina::triDiscArcs [4][3]
 Lists in consecutive order the directed normal arcs that form the boundary of each type of triangular normal disc.
const NPerm regina::quadDiscArcs [3][4]
 Lists in consecutive order the directed normal arcs that form the boundary of each type of quadrilateral normal disc.
const NPerm regina::octDiscArcs [3][8]
 Lists in consecutive order the directed normal arcs that form the boundary of each type of octahedral normal disc.

Detailed Description

Normal surfaces in triangulations.


Function Documentation

template<class RayOutputIterator, class FaceOutputIterator>
void createNonNegativeCone NTriangulation *  triangulation,
int  flavour,
RayOutputIterator  rays,
FaceOutputIterator  faces
 

Writes to the given output iterators newly allocated rays and faces representing the cone in the given flavour of coordinate system obtained by setting all coordinates non-negative.

To rays will be written the extremal rays of this cone, each being a unit vector along a coordinate axis. They will be written in order from the unit vector along the 0th coordinate axis to the last, and will all be of the subclass of NNormalSurfaceVector corresponding to the given flavour of coordinate system.

To faces will be written the vectors perpendicular to the hyperplanes that make up the faces of this cone. Each of these vectors will also be a unit vector along a coordinate axis. These vectors will be written in the same order as the rays, and will all be of class NVectorUnit.

The resulting lists of extremal rays and faces are guaranteed not to contain any duplicates or redundancies.

Python:
Not present.
Parameters:
triangulation the triangulation upon which the underlying coordinate system is based.
flavour the flavour of coordinate system to be used; this must be one of the predefined coordinate system constants in NNormalSurfaceList.
rays the output iterator to which the newly allocated extremal rays will be written; this must accept objects of type NRay*.
faces the output iterator to which the newly allocated face perpendiculars will be written; this must accept objects of type NVector<NLargeInteger>*.

bool discOrientationFollowsEdge int  discType,
int  vertex,
int  edgeStart,
int  edgeEnd
 

Determines whether or not the natural boundary orientation of a normal disc of the given type follows the given directed normal arc.

Natural boundary orientation is defined by arrays triDiscArcs, quadDiscArcs and octDiscArcs.

Precondition:
The given normal arc lies on a normal disc of the given type.
Parameters:
discType the normal disc type under consideration; this should be between 0 and 9 inclusive, as described by the NDiscSpec class notes.
vertex the vertex about which the normal arc runs.
edgeStart the start vertex of the edge to which the normal arc is parallel.
edgeEnd the end vertex of the edge to which the normal arc is parallel.

NMatrixInt* makeMatchingEquations NTriangulation *  triangulation,
int  flavour
 

Creates a new set of normal surface matching equations for the given triangulation using the given flavour of coordinate system.

The returned matrix will be newly allocated and its destruction will be the responsibility of the caller of this routine.

Each equation will be represented as a row of the matrix. Each column of the matrix represents a coordinate in the given flavour of coordinate system.

Parameters:
triangulation the triangulation upon which these matching equations will be based.
flavour the flavour of coordinate system to be used; this must be one of the predefined coordinate system constants in NNormalSurfaceList.
Returns:
a newly allocated set of matching equations.

bool numberDiscsAwayFromVertex int  discType,
int  vertex
 

Determines whether or not normal discs of the given type are numbered away from the given vertex.

Parameters:
discType the normal disc type under consideration; this should be between 0 and 9 inclusive, as described by the NDiscSpec class notes.
vertex the vertex under consideration; this should be between 0 and 3 inclusive.
Returns:
true if normal discs of the given type are numbered away from the given vertex, or false if they are numbered towards the given vertex.

std::ostream& operator<< std::ostream &  out,
const NPrismSpec &  spec
 

Writes the given prism specifier to the given output stream.

The prism specifier will be written as a pair (tetIndex, edge).

Parameters:
out the output stream to which to write.
spec the prism specifier to write.
Returns:
a reference to out.

std::ostream& operator<< std::ostream &  out,
const NDiscSpec &  spec
 

Writes the given disc specifier to the given output stream.

The disc specifier will be written as a triple (tetIndex, type, number).

Parameters:
out the output stream to which to write.
spec the disc specifier to write.
Returns:
a reference to out.


Variable Documentation

const NPerm regina::octDiscArcs[3][8]
 

Lists in consecutive order the directed normal arcs that form the boundary of each type of octahedral normal disc.

Each permutation p represents an arc about vertex p[0] parallel to the directed edge from p[1] to p[2].

Array octDiscArcs[i] lists the boundary arcs of the octahedral disc of type i. See NNormalSurface::getOctCoord() for further details.

Note that permutation octDiscArcs[i][j] will be even precisely when j is 0, 1, 4 or 5.

C++:
This array is replaced by a macro octDiscArcs(discType, arcIndex) that essentially looks up the corresponding array. This is necessary because of a bug in gcc 2.95.

const NPerm regina::quadDiscArcs[3][4]
 

Lists in consecutive order the directed normal arcs that form the boundary of each type of quadrilateral normal disc.

Each permutation p represents an arc about vertex p[0] parallel to the directed edge from p[1] to p[2].

Array quadDiscArcs[i] lists the boundary arcs of the quadrilateral disc of type i. See NNormalSurface::getQuadCoord() for further details.

Note that permutation quadDiscArcs[i][j] will be even precisely when j is even.

C++:
This array is replaced by a macro quadDiscArcs(discType, arcIndex) that essentially looks up the corresponding array. This is necessary because of a bug in gcc 2.95.

const NPerm regina::triDiscArcs[4][3]
 

Lists in consecutive order the directed normal arcs that form the boundary of each type of triangular normal disc.

Each permutation p represents an arc about vertex p[0] parallel to the directed edge from p[1] to p[2].

Array triDiscArcs[i] lists the boundary arcs of the triangular disc of type i. See NNormalSurface::getTriangleCoord() for further details.

Note that every permutation in this array is even.

C++:
This array is replaced by a macro triDiscArcs(discType, arcIndex) that essentially looks up the corresponding array. This is necessary because of a bug in gcc 2.95.

const int regina::vertexSplit[4][4]
 

Lists which vertex splits split which pairs of vertices.

There are three vertex splits, numbered 0,1,2. Each vertex split split the four tetrahedron vertices 0,1,2,3 into two pairs. vertexSplit[i][j] is the number of the vertex split that keeps vertices i and j together.

const int regina::vertexSplitDefn[3][4]
 

Lists the vertices which each vertex split splits.

See vertexSplit for details on what a vertex split is. Vertex split number i splits the vertex pairs vertexSplitDefn[i][0,1] and vertexSplitDefn[i][2,3].

const int regina::vertexSplitMeeting[4][4][2]
 

Lists which vertex splits meet which edges.

See vertexSplit for details on what a vertex split is. vertexSplitMeeting[i][j][0,1] are the numbers of the two vertex splits that meet the edge joining tetrahedron vertices i and j.

const int regina::vertexSplitPartner[3][4]
 

Lists the second vertex with which each vertex is paired under each vertex split.

See vertexSplit for details on what a vertex split is. Vertex split number i pairs vertex v with vertex vertexSplitPartner[i][v].

const char regina::vertexSplitString[3][6]
 

Contains strings describing which vertices each vertex split splits.

See vertexSplit for details on what a vertex split is. The string describing vertex split number i is vertexSplitString[i] and is of the form 02/13, which in this case is the vertex split that splits vertices 0,2 from vertices 1,3.


Copyright © 1999-2004, Ben Burton
This software is released under the GNU General Public License.
For further information, or to submit a bug or other problem, please contact Ben Burton (bab@debian.org).