Type Members

1.  case class BoundaryEdge(pb: PantsBoundary, top: Boolean, positivelyOriented: Boolean) extends OrientedEdge with Product with Serializable

   An edge in the boundary of a pair of pants which is not identified with a boundary component of another pair of pants, i.e., this is part of the boundary of the surface.

   An edge in the boundary of a pair of pants which is not identified with a boundary component of another pair of pants, i.e., this is part of the boundary of the surface.

   pb

   the pants boundary of which this is a part

   top

   whether this is in the top face

   positivelyOriented

   whether the edge is positively oriented

2.  case class BoundaryVertex(pb: PantsBoundary, isFirst: Boolean) extends Vertex with Product with Serializable

   Vertex on the boundary of a pair of pants decomposed into a pair of hexagons

   Vertex on the boundary of a pair of pants decomposed into a pair of hexagons

   pb

   the component of the boundary on which the vertex lies

   isFirst

   whether the vertex is the first vertex.

3.  case class ClosedSurface(n: Int) extends StandardSurface with Product with Serializable
4.  trait CollectionOfHomotopyClasses extends AnyRef
5.  case class Curve(left: PantsBoundary, right: PantsBoundary) extends Product with Serializable

   A curve in the curve system (i.e., the collection of terms) along which we split to get the pants decomposition.

   A curve in the curve system (i.e., the collection of terms) along which we split to get the pants decomposition. This is a quotient of two pants boundaries that have been identified, with orientation determining one as on the left.

   left

   the pants boundary on the left.

   right

   the pants boundary on the right.

6.  case class CurveEdge(curve: Curve, top: Boolean, positivelyOriented: Boolean) extends OrientedEdge with Product with Serializable

   An edge on a curve

   An edge on a curve

   curve

   the curve on which the edge lies

   top

   whether this is the top edge

   positivelyOriented

   whether the edge is positively oriented

7.  case class CurveVertex(curve: Curve, first: Boolean) extends Vertex with Product with Serializable

   vertex on a curve

   vertex on a curve

   curve

   the curve

   first

   whether this is the first vertex

8.  trait Edge extends AnyRef

   An oriented edge in a two-complex

9.  class EdgePair extends AnyRef
10.  sealed trait EdgePath extends AnyRef
11.  trait EquivalenceClass extends AnyRef
12.  case class FormalSum[A](coeffs: Map[A, Int]) extends Product with Serializable

    A formal sum of elements of A with coefficients integers.

13.  sealed trait Generator extends AnyRef

    Contribution of Ankit-Jaiswal and Soumyo Biswas, independent of the rest of the code.

    Contribution of Ankit-Jaiswal and Soumyo Biswas, independent of the rest of the code.

    This program basically abstracts : (i) OrientableSurface (i.e. a connected compact 2-manifold) parametrized by g and r. where, g is its genus and r is the number of components in its boundary. (ii) Surfaces' fundamental group Generators along with the 'relation' over them, (iii) Closed curves/Words (on the surfaces) in terms of Generators, And, verifies whether a given word is homotopic to a simple closed curve.

    NOTE: This program works for surfaces which are not sphere(where g = 0 as well as r = 0).

    Reference:: "Simple Closed Curves on Surfaces" by D.R.J. Chillingworth

14.  trait Hexagon extends Polygon

    A hexagon

15.  trait HomotopyClassesOfPaths extends AnyRef
16.  trait Intersection extends AnyRef
17.  trait NonPosQuad extends TwoComplex
18.  case class NormalArc(initial: Index, terminal: Index, face: Polygon) extends Product with Serializable

    A normal arc in a face

    A normal arc in a face

    initial

    the edge containing the initial point

    terminal

    the edge containing the final point

    face

    the face containing the arc

19.  case class NormalPath(edges: Vector[NormalArc]) extends Product with Serializable
20.  case class OrientableSurface(g: Int, r: Int) extends Product with Serializable

    Contribution of Ankit-Jaiswal and Soumyo Biswas, independent of the rest of the code.

    Contribution of Ankit-Jaiswal and Soumyo Biswas, independent of the rest of the code.

    'a' and 'b' are the two Generators of a OrientableSurface, parametrized by the genus number. And 's' is the generator corressponding to surface's boundary, parametrized by r, the number of components. Because their inverse can also generate the fundamental group; therefore they are also being considered as Generators.

21.  trait OrientedEdge extends Edge
22.  case class PantsBoundary(pants: Index, direction: Z3) extends Product with Serializable

    label for boundary of pants, may be a curve in the curve system or a component of the boundary of the surface; if part of the curve system it will be identified with another PantsBoundary

    label for boundary of pants, may be a curve in the curve system or a component of the boundary of the surface; if part of the curve system it will be identified with another PantsBoundary

    pants

    the index of the pair of pants

    direction

    the prong of the pair of pants

23.  case class PantsHexagon(pants: Index, top: Boolean, cs: Set[Curve]) extends Hexagon with Product with Serializable

    Hexagon in a pair of pants

    Hexagon in a pair of pants

    pants

    the index of the pair of pants

    top

    whether this is the top face

    cs

    system of curves giving the pants decomposition

24.  case class PantsSeam(pants: Index, initial: Vertex, terminal: Vertex, positivelyOriented: Boolean = true) extends OrientedEdge with Product with Serializable

    An edge that is the seam of a pair of pants

    An edge that is the seam of a pair of pants

    pants

    the index of the pair of pants of which this is a seam

    initial

    the initial vertex

    terminal

    the terminal vertex

    positivelyOriented

    whether the edge is positively oriented

25.  case class PantsSurface(numPants: Index, cs: Set[Curve]) extends PureTwoComplex with Product with Serializable

    Surface with a pants decomposition (compact, possibly with boundary and possibly disconnected)

    Surface with a pants decomposition (compact, possibly with boundary and possibly disconnected)

    numPants

    the number of pairs of pants

    cs

    the curve system giving the pants decomposition

26.  trait Polygon extends TwoComplex

    Abstract polygon, with given edges and vertices, i.e.

    Abstract polygon, with given edges and vertices, i.e. a two-complex with a single face.

27.  case class PureComplex(polys: Set[Polygon]) extends PureTwoComplex with Product with Serializable
28.  trait PureTwoComplex extends TwoComplex

    A two-complex with all vertices and edges contained in faces, hence determined by its faces.

29.  class QuadPlot extends AnyRef
30.  trait Quadrangulation extends TwoComplex
31.  trait Quadrilateral extends Polygon

    a quadrilateral is a polygon with four edges.

32.  case class QuotientEdge(edges: Set[Edge]) extends Edge with Product with Serializable

    An edge obtained by identifications

33.  case class QuotientVertex(vertices: Set[Vertex]) extends Vertex with Product with Serializable

    A vertex obtained by identifications

34.  case class SkewCurve(left: PantsBoundary, right: PantsBoundary, twist: Double, length: Double) extends Product with Serializable

    A curve in the curve system for a pants decomposition for a hyperbolic surface obtained by twisting and regluing along the pants system Twists and vertex positions are parametrized as fractions of the length, so are in [0, 1].

    A curve in the curve system for a pants decomposition for a hyperbolic surface obtained by twisting and regluing along the pants system Twists and vertex positions are parametrized as fractions of the length, so are in [0, 1].

    Generically, there are 4 vertices, two at 0 and 0.5 and the other two at twist and twist + 0.5 (mod 1).

    left

    the pants boundary to the left

    right

    the pants boundary to the right

    twist

    the twist, parametrized as a fraction of the length, so is in [0, 1]

    length

    the length of the curve in the hyperbolic structure.

35.  case class SkewCurveEdge(curve: SkewCurve, initialPosition: Double, positivelyOriented: Boolean) extends OrientedEdge with Product with Serializable

    An edge on a skew-curve, i.e., curve in a pants decomposition glued with a twist

    An edge on a skew-curve, i.e., curve in a pants decomposition glued with a twist

    curve

    the skew-curve in the pants decomposition

    initialPosition

    the position of the initial vertex

    positivelyOriented

    whether the edge is positively oriented

36.  case class SkewCurveVertex(curve: SkewCurve, position: Double) extends Vertex with Product with Serializable

    vertex on a skew curve

    vertex on a skew curve

    curve

    the curve

    position

    position on the curve in [0, 1)

37.  case class SkewPantsHexagon(pants: Index, top: Boolean, cs: Set[SkewCurve]) extends Polygon with Product with Serializable

    "hexagon" in a skew pair of pants; note that this is not in general a hexagon as some of the edges are subdivided

    "hexagon" in a skew pair of pants; note that this is not in general a hexagon as some of the edges are subdivided

    pants

    the index of the pair of (skew) pants

    top

    whether this is the top face

    cs

    system of curves giving the pants decomposition

38.  case class SkewPantsSurface(numPants: Index, cs: Set[SkewCurve]) extends PureTwoComplex with Product with Serializable
39.  class StandardSurface extends TwoComplex
40.  trait TwoComplex extends AnyRef

    A polyheadral two complex, with faces polygons, a collection of edges and

41.  class Vertex extends AnyRef

    A vertex in a two-complex

42.  final case class Z3(n: Int) extends AnyVal with Product with Serializable

    One of 0, 1 and 2, indexing the boundaries of a pair of pants.

    One of 0, 1 and 2, indexing the boundaries of a pair of pants. Class created to avoid accidentally mixing with integers

43.  trait sphereWithHoles extends AnyRef
44.  trait surfaceWithBoundary extends AnyRef