Quick Start

If you're already familiar with the concept of flip graphs, triangulations on closed surfaces, and Δ-complexes, and don't want to read the whole documentation, then here are some quick examples of what you can do with this package.

In any other case, please be sure to take a look at the rest of the documentation first.

Triangulated Convex Polygon

Create a triangulated convex 10-gon:

julia> g = triangulated_polygon(8)
TriangulatedPolygon with 8 vertices, and adjacency list:
 1  → [2, 8]
 2  → [1, 3, 8]
 3  → [2, 4, 8, 7]
 4  → [3, 5, 7, 6]
 5  → [4, 6]
 6  → [5, 7, 4]
 7  → [6, 8, 3, 4]
 8  → [7, 1, 2, 3]

Check if the edge going from vertex 2 to vertex 10 can be flipped:

julia> is_flippable(g, 2, 8)
true

Flip the edge connecting vertices 2 and 10:

julia> flip!(g, 2, 8)
TriangulatedPolygon with 8 vertices, and adjacency list:
 1  → [2, 8, 3]
 2  → [1, 3]
 3  → [2, 4, 8, 7, 1]
 4  → [3, 5, 7, 6]
 5  → [4, 6]
 6  → [5, 7, 4]
 7  → [6, 8, 3, 4]
 8  → [7, 1, 3]

Construct the flip graph of a convex octagon:

julia> G = flipgraph_planar(8)
FlipGraphPlanar with 132 vertices and 330 edges

Export the generated flip graph as a .gml file:

julia> export_gml("C:/Users/USERNAME/Desktop/FILENAME.gml", G);

Δ-Complex / Triangulation of closed surface

DeltaComplex

A DeltaComplex is the dual of a triangulation on a closed surface. It can be used to compute things like the diameter, but it does not offer a unique modelisation of a triangulation on a closed surface. Every DeltaComplex can be interpreted as the homeomorphism class of triangulations of points on a closed surface.

Create a DeltaComplex of a surface of genus 1 with 2 points:

julia> D = deltacomplex(1, 2)
DeltaComplex on orientable surface of genus 1 with 2 points
4 TriFaces:
 TriFace #1: Points(1 1 2) Neighbors(2 3 4)
 TriFace #2: Points(1 1 1) Neighbors(4 1 3)
 TriFace #3: Points(1 1 2) Neighbors(2 4 1)
 TriFace #4: Points(1 1 2) Neighbors(2 1 3)
6 DualEdges:
 DualEdge 1 : (Δ3)-(1)-------(3)-(Δ2)
 DualEdge 2 : (Δ1)-(1)-------(2)-(Δ2)
 DualEdge 3 : (Δ2)-(1)-------(1)-(Δ4)
 DualEdge 4 : (Δ4)-(2)-------(3)-(Δ1)
 DualEdge 5 : (Δ1)-(2)-------(3)-(Δ3)
 DualEdge 6 : (Δ3)-(2)-------(3)-(Δ4)

Check if the 4th edge (DualEdge 4) can be flipped:

julia> is_flippable(D, 4)
true

Flip said edge:

julia> flip!(D, 4)
DeltaComplex on orientable surface of genus 1 with 2 points
4 TriFaces:
 TriFace #1: Points(1 2 1) Neighbors(3 3 4)
 TriFace #2: Points(1 1 1) Neighbors(4 4 3)
 TriFace #3: Points(1 1 2) Neighbors(2 1 1)
 TriFace #4: Points(1 1 1) Neighbors(2 1 2)
6 DualEdges:
 DualEdge 1 : (Δ3)-(1)-------(3)-(Δ2)
 DualEdge 2 : (Δ4)-(1)-------(2)-(Δ2)
 DualEdge 3 : (Δ2)-(1)-------(3)-(Δ4)
 DualEdge 4 : (Δ4)-(2)-------(3)-(Δ1)
 DualEdge 5 : (Δ1)-(1)-------(3)-(Δ3)
 DualEdge 6 : (Δ3)-(2)-------(2)-(Δ1)

Randomly flip edges in D until the diameter stabilizes:

julia> randomize!(D)
10300000
julia> D
DeltaComplex on orientable surface of genus 1 with 2 points
4 TriFaces:
 TriFace #1: Points(1 1 1) Neighbors(3 2 2)
 TriFace #2: Points(1 1 1) Neighbors(1 3 1)
 TriFace #3: Points(1 1 1) Neighbors(1 2 4)
 TriFace #4: Points(2 1 1) Neighbors(4 3 4)
6 DualEdges:
 DualEdge 1 : (Δ3)-(3)-------(2)-(Δ4)
 DualEdge 2 : (Δ3)-(2)-------(2)-(Δ2)
 DualEdge 3 : (Δ4)-(3)-------(1)-(Δ4)
 DualEdge 4 : (Δ1)-(2)-------(3)-(Δ2)
 DualEdge 5 : (Δ1)-(1)-------(1)-(Δ3)
 DualEdge 6 : (Δ2)-(1)-------(3)-(Δ1)

Modular FlipGraph

Construct the modular flip graph of a torus with 2 labeled points on it:

julia> G = flipgraph_modular(1,2)
modular FlipGraph with 9 vertices and 8 edges

Construct the modular flip graph of a torus with 2 unlabeled points on it:

julia> G = flipgraph_modular(1,2;labeled_points=false)
modular FlipGraph with 5 vertices and 4 edges