Documentation

Embedded, *connected*, planar graph

A *connected* Planar graph with bidirected edges. I.e. the edges (darts) are directed, however, for every directed edge, the edge in the oposite direction is also in the graph.

The types v, e, and f are the are the types of the data associated with the vertices, edges, and faces, respectively.

The orbits in the embedding are assumed to be in counterclockwise order. Therefore, every dart directly bounds the face to its right.

Note that the functor instance is in v

Construct a plane graph from a simple polygon. It is assumed that the polygon is given in counterclockwise order.

the interior of the polygon will have faceId 0

pre: the input polygon is given in counterclockwise order running time: \(O(n)\).

Constructs a connected plane graph

pre: The segments form a single connected component

running time: \(O(n\log n)\)

Construct a planar graph from a adjacency matrix. For every vertex, all vertices should be given in counter-clockwise order.

pre: No self-loops, and no multi-edges

running time: \(O(n)\).

Get the number of vertices

>>> numVertices smallG
4

Get the number of Edges

>>> numEdges smallG
5

Get the number of faces

>>> numFaces smallG
3

Get the number of Darts

>>> numDarts smallG
10

Get the dual graph of this graph.

Enumerate all vertices

>>> vertices' smallG
[VertexId 0,VertexId 1,VertexId 2,VertexId 3]

Enumerate all vertices, together with their vertex data

>>> mapM_ print $ vertices smallG
(VertexId 0,VertexData {_location = Point2 0 0, _vData = 0})
(VertexId 1,VertexData {_location = Point2 2 2, _vData = 1})
(VertexId 2,VertexData {_location = Point2 2 0, _vData = 2})
(VertexId 3,VertexData {_location = Point2 (-1) 4, _vData = 3})

Enumerate all edges. We report only the Positive darts

Enumerate all edges with their edge data. We report only the Positive darts.

>>> mapM_ print $ edges smallG
(Dart (Arc 0) +1,"0->2")
(Dart (Arc 1) +1,"0->1")
(Dart (Arc 2) +1,"0->3")
(Dart (Arc 4) +1,"1->2")
(Dart (Arc 3) +1,"1->3")

Enumerate all faces in the plane graph

All faces with their face data.

>>> mapM_ print $ faces smallG
(FaceId 0,"OuterFace")
(FaceId 1,"A")
(FaceId 2,"B")

Reports all internal faces. running time: \(O(n)\)

Reports the outerface and all internal faces separately. running time: \(O(n)\)

Enumerate all darts

Get all darts together with their data

Traverse the vertices

(^.vertexData) $ traverseVertices (i x -> Just (i,x)) smallG Just [(VertexId 0,0),(VertexId 1,1),(VertexId 2,2),(VertexId 3,3)] >>> traverseVertices (i x -> print (i,x)) smallG >> pure () (VertexId 0,0) (VertexId 1,1) (VertexId 2,2) (VertexId 3,3)

Traverses the darts

>>> traverseDarts (\d x -> print (d,x)) smallG >> pure ()
(Dart (Arc 0) +1,"0->2")
(Dart (Arc 0) -1,"2->0")
(Dart (Arc 1) +1,"0->1")
(Dart (Arc 1) -1,"1->0")
(Dart (Arc 2) +1,"0->3")
(Dart (Arc 2) -1,"3->0")
(Dart (Arc 3) +1,"1->3")
(Dart (Arc 3) -1,"3->1")
(Dart (Arc 4) +1,"1->2")
(Dart (Arc 4) -1,"2->1")

Traverses the faces

>>> traverseFaces (\i x -> print (i,x)) smallG >> pure ()
(FaceId 0,"OuterFace")
(FaceId 1,"A")
(FaceId 2,"B")

The vertex this dart is heading in to

running time: \(O(1)\)

>>> headOf (dart 0 "+1") smallG
VertexId 2

The tail of a dart, i.e. the vertex this dart is leaving from

running time: \(O(1)\)

>>> tailOf (dart 0 "+1") smallG
VertexId 0

Get the twin of this dart (edge)

>>> twin (dart 0 "+1")
Dart (Arc 0) -1
>>> twin (dart 0 "-1")
Dart (Arc 0) +1

endPoints d g = (tailOf d g, headOf d g)

running time: \(O(1)\)

>>> endPoints (dart 0 "+1") smallG
(VertexId 0,VertexId 2)

All edges incident to vertex v, in counterclockwise order around v.

running time: \(O(k)\), where \(k\) is the output size

>>> incidentEdges (VertexId 1) smallG
[Dart (Arc 1) -1,Dart (Arc 4) +1,Dart (Arc 3) +1]

All edges incident to vertex v in incoming direction (i.e. pointing into v) in counterclockwise order around v.

running time: \(O(k)\), where (k) is the total number of incident edges of v

>>> incomingEdges (VertexId 1) smallG
[Dart (Arc 1) +1,Dart (Arc 4) -1,Dart (Arc 3) -1]

All edges incident to vertex v in incoming direction (i.e. pointing into v) in counterclockwise order around v.

running time: \(O(k)\), where (k) is the total number of incident edges of v

>>> outgoingEdges (VertexId 1) smallG
[Dart (Arc 1) -1,Dart (Arc 4) +1,Dart (Arc 3) +1]

Gets the neighbours of a particular vertex, in counterclockwise order around the vertex.

running time: \(O(k)\), where \(k\) is the output size

>>> neighboursOf (VertexId 1) smallG
[VertexId 0,VertexId 2,VertexId 3]

Given a dart d that points into some vertex v, report the next dart in the cyclic order around v in clockwise direction.

running time: \(O(1)\)

>>> nextIncidentEdge (dart 1 "+1") smallG
Dart (Arc 2) +1

Given a dart d that points into some vertex v, report the next dart in the cyclic order around v (in clockwise order)

running time: \(O(1)\)

>>> prevIncidentEdge (dart 1 "+1") smallG
Dart (Arc 0) +1

The face to the left of the dart

running time: \(O(1)\).

>>> leftFace (dart 1 "+1") smallG
FaceId 2
>>> leftFace (dart 1 "-1") smallG
FaceId 1
>>> leftFace (dart 2 "+1") smallG
FaceId 0
>>> leftFace (dart 2 "-1") smallG
FaceId 2

The face to the right of the dart

running time: \(O(1)\).

>>> rightFace (dart 1 "+1") smallG
FaceId 1
>>> rightFace (dart 1 "-1") smallG
FaceId 2
>>> rightFace (dart 2 "+1") smallG
FaceId 2
>>> rightFace (dart 2 "-1") smallG
FaceId 0

Get the next edge along the face

running time: \(O(1)\).

>>> nextEdge (dart 1 "+1") smallG
Dart (Arc 4) +1

Get the previous edge along the face

running time: \(O(1)\).

>>> prevEdge (dart 1 "+1") smallG
Dart (Arc 0) -1

The darts bounding this face, for internal faces in clockwise order, for the outer face in counter clockwise order.

running time: \(O(k)\), where \(k\) is the output size.

Generates the darts incident to a face, starting with the given dart.

\(O(k)\), where \(k\) is the number of darts reported

Gets a dart bounding this face. I.e. a dart d such that the face lies to the right of the dart.

The vertices bounding this face, for internal faces in clockwise order, for the outer face in counter clockwise order.

running time: \(O(k)\), where \(k\) is the output size.

gets the id of the outer face

running time: \(O(n)\)

gets a dart incident to the outer face (in particular, that has the outerface on its left)

running time: \(O(n)\)

General interface to accessing vertex data, dart data, and face data.

Getter for the data at the endpoints of a dart

running time: \(O(1)\)

Data corresponding to the endpoints of the dart

running time: \(O(1)\)

Lens to access face data

lens to access the Dart Data

Lens to access the raw dart data, use at your own risk

Given a dart and the graph constructs the line segment representing the dart. The segment \(\overline{uv})\) is has \(u\) as its tail and \(v\) as its head.

\(O(1)\)

Reports all edges as line segments

>>> mapM_ print $ edgeSegments smallG
(Dart (Arc 0) +1,ClosedLineSegment (Point2 0 0 :+ 0) (Point2 2 0 :+ 2) :+ "0->2")
(Dart (Arc 1) +1,ClosedLineSegment (Point2 0 0 :+ 0) (Point2 2 2 :+ 1) :+ "0->1")
(Dart (Arc 2) +1,ClosedLineSegment (Point2 0 0 :+ 0) (Point2 (-1) 4 :+ 3) :+ "0->3")
(Dart (Arc 4) +1,ClosedLineSegment (Point2 2 2 :+ 1) (Point2 2 0 :+ 2) :+ "1->2")
(Dart (Arc 3) +1,ClosedLineSegment (Point2 2 2 :+ 1) (Point2 (-1) 4 :+ 3) :+ "1->3")

Alias for rawFace Boundary

runningtime: \(O(k)\), where \(k\) is the size of the face.

The polygon describing the face

runningtime: \(O(k)\), where \(k\) is the size of the face.

Lists all faces of the plane graph.

A vertex in a planar graph. A vertex is tied to a particular planar graph by the phantom type s, and to a particular world w.

The type to represent FaceId's

A dart represents a bi-directed edge. I.e. a dart has a direction, however the dart of the oposite direction is always present in the planar graph as well.

The world in which the graph lives

Shorthand for vertices in the primal.

Shorthand for FaceId's in the primal.

Labels the edges of a plane graph with their distances, as specified by the distance function.