Triangulating Convex Polygons

In this tutorial, we show how we can triangulate convex polygons. The function triangulate_convex is used for this. Let us start with a simple example.

using DelaunayTriangulation
using CairoMakie

points = [
    (10.0, 12.0), (7.0, 11.0), (8.0, 6.0),
    (10.0, 3.0), (14.0, 5.0), (15.0, 10.0),
    (13.0, 12.0),
]
S = 1:7
tri = triangulate_convex(points, 1:7)

Delaunay Triangulation.
   Number of vertices: 7
   Number of triangles: 5
   Number of edges: 11
   Has boundary nodes: false
   Has ghost triangles: true
   Curve-bounded: false
   Weighted: false
   Constrained: false

fig, ax, sc = triplot(tri)
fig

This tri is our triangulation of the convex polygon. The first input is the set of points, and S defines the vertices to take from these points and their order, which must be provided in a counter-clockwise order. Note that points does not have to contain only the points of the polygon, since S will be used to define the points needed.

Let us give a larger example. For simplicity, we triangulate a a discretised circle.

θ = LinRange(0, 2π, 5000) |> collect
pop!(θ)
x = cos.(θ)
y = sin.(θ)
points = tuple.(x, y)
S = 1:4999 # can also be [1:4999; 1], if you want the array to be circular
tri = triangulate_convex(points, S)

Delaunay Triangulation.
   Number of vertices: 4999
   Number of triangles: 4997
   Number of edges: 9995
   Has boundary nodes: false
   Has ghost triangles: true
   Curve-bounded: false
   Weighted: false
   Constrained: false

fig, ax, sc = triplot(tri)
fig

Here is a comparison of the time it takes to triangulate this using triangulate_convex or triangulate.

using BenchmarkTools
@benchmark triangulate_convex($points, $S)

BenchmarkTools.Trial: 120 samples with 1 evaluation.
 Range (min … max):  37.916 ms … 66.509 ms  ┊ GC (min … max): 0.00% … 35.06%
 Time  (median):     40.590 ms              ┊ GC (median):    0.00%
 Time  (mean ± σ):   41.852 ms ±  4.134 ms  ┊ GC (mean ± σ):  3.51% ±  6.45%

   ▃ █▆ ▂▁                                                     
  ▃████▅███▄▅▄▃▄▅▁▄▅▄▅▁▁▃▄▃▄▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃ ▃
  37.9 ms         Histogram: frequency by time        63.6 ms <

 Memory estimate: 16.03 MiB, allocs estimate: 59930.

@benchmark triangulate($points)

BenchmarkTools.Trial: 17 samples with 1 evaluation.
 Range (min … max):  293.368 ms … 301.396 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     297.683 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   297.725 ms ±   2.586 ms  ┊ GC (mean ± σ):  0.00% ± 0.00%

  █ █       █ █          █ ██  █  █  █  █ █           ██   █ ██  
  █▁█▁▁▁▁▁▁▁█▁█▁▁▁▁▁▁▁▁▁▁█▁██▁▁█▁▁█▁▁█▁▁█▁█▁▁▁▁▁▁▁▁▁▁▁██▁▁▁█▁██ ▁
  293 ms           Histogram: frequency by time          301 ms <

 Memory estimate: 16.91 MiB, allocs estimate: 44604.

For the smaller example that we started with above, triangulate_convex is also faster, although not by much (≈15.10 μs versus ≈10.7 μs).

Just the code

An uncommented version of this example is given below. You can view the source code for this file here.

using DelaunayTriangulation
using CairoMakie

points = [
    (10.0, 12.0), (7.0, 11.0), (8.0, 6.0),
    (10.0, 3.0), (14.0, 5.0), (15.0, 10.0),
    (13.0, 12.0),
]
S = 1:7
tri = triangulate_convex(points, 1:7)

fig, ax, sc = triplot(tri)
fig

θ = LinRange(0, 2π, 5000) |> collect
pop!(θ)
x = cos.(θ)
y = sin.(θ)
points = tuple.(x, y)
S = 1:4999 # can also be [1:4999; 1], if you want the array to be circular
tri = triangulate_convex(points, S)

fig, ax, sc = triplot(tri)
fig

using BenchmarkTools
@benchmark triangulate_convex($points, $S)

@benchmark triangulate($points)

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