Triangulation Operations

Vertex Insertion and Deletion

This tutorial demonstrates how to insert and delete vertices from a triangulation while maintaining the Delaunay property of the triangulation. First, load the packages we need:

using DelaunayTriangulation
using CairoMakie
using StableRNGs

Let us now define our initial triangulation.

points = [(0.0, 0.0), (2.0, 0.0), (1.0, 2.0)]
tri = triangulate(points)
fig, ax, sc = triplot(tri)
fig

Note that we use a structure for points that is mutable so that points can be pushed into it.

We now want to insert a vertex into the triangulation. To do this, we use add_point! and provide the coordinates of the new point. There is a method which uses the index of the vertex rather than the coordinates, but we don't use that here as the points to be added are not already in points. Here, we add a point at (1.0, 0.5).

add_point!(tri, 1.0, 0.5)
fig, ax, sc = triplot(tri)
fig

This is still a valid Delaunay triangulation, unsurprisingly due to the small number of points. We can also add points that are outside of the triangulation:

add_point!(tri, 0.0, 1.0)
fig, ax, sc = triplot(tri)
fig

One important thing to note here is that, if not for the ghost triangles inside tri, adding a point outside of the triangulation would not work. Here is an example of this failing.

delete_ghost_triangles!(tri)
add_point!(tri, 2.0, 1.5)

BoundsError([(0.0, 0.0), (2.0, 0.0), (1.0, 2.0), (1.0, 0.5), (0.0, 1.0)], (0,))

This is a BoundsError, because the triangulation has had to walk into the boundary but failed to find any ghost triangles. Anytime you need to perform some operation including a point outside of the boundary, you need to be sure that you have ghost triangles, which you can query using DelaunayTriangulation.has_ghost_triangles.

DelaunayTriangulation.has_ghost_triangles(tri)

false

add_ghost_triangles!(tri)

Delaunay Triangulation.
   Number of vertices: 5
   Number of triangles: 4
   Number of edges: 8
   Has boundary nodes: false
   Has ghost triangles: true
   Curve-bounded: false
   Weighted: false
   Constrained: false

DelaunayTriangulation.has_ghost_triangles(tri)

true

Another issue is that the convex hull is not updated as we add (or delete) points for performance reasons:

get_convex_hull_vertices(tri)

4-element Vector{Int64}:
 3
 1
 2
 3

If we do want to fix the convex hull, we can use convex_hull!(tri).

convex_hull!(tri)
fig, ax, sc = triplot(tri, show_convex_hull=true)
fig

We now have the same triangulation that we would have had if we had done triangulate on this set of points originally. To now push this further, let's add in a bunch of random points.

rng = StableRNG(123)
for _ in 1:1000
    new_point = 2rand(rng, 2)
    add_point!(tri, new_point)
end
fig, ax, sc = triplot(tri)
fig

Let us now demonstrate how to delete points. To do this, we use delete_point!. This function takes vertices rather than coordinates, identifying the corresponding point in points by its index. Let us demonstrate this operation by deleting all points within a distance of 1/2 around (1.0, 1.0).

vertices_to_delete = Iterators.filter(each_solid_vertex(tri)) do i
    p = get_point(tri, i)
    r2 = (getx(p) - 1.0)^2 + (gety(p) - 1.0)^2
    return r2 < 1 / 2^2
end
for i in vertices_to_delete
    delete_point!(tri, i)
end
fig, ax, sc = triplot(tri)
fig

Note that in this situation, points still contains those points that we have now deleted. This is the reason to be careful about using, say, DelaunayTriangulation.each_point rather than each_solid_vertex. This triangulation is also still Delaunay.

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
using StableRNGs

points = [(0.0, 0.0), (2.0, 0.0), (1.0, 2.0)]
tri = triangulate(points)
fig, ax, sc = triplot(tri)
fig

add_point!(tri, 1.0, 0.5)
fig, ax, sc = triplot(tri)
fig

add_point!(tri, 0.0, 1.0)
fig, ax, sc = triplot(tri)
fig

delete_ghost_triangles!(tri)
add_point!(tri, 2.0, 1.5)

DelaunayTriangulation.has_ghost_triangles(tri)

add_ghost_triangles!(tri)

DelaunayTriangulation.has_ghost_triangles(tri)

get_convex_hull_vertices(tri)

convex_hull!(tri)
fig, ax, sc = triplot(tri, show_convex_hull=true)
fig

rng = StableRNG(123)
for _ in 1:1000
    new_point = 2rand(rng, 2)
    add_point!(tri, new_point)
end
fig, ax, sc = triplot(tri)
fig

vertices_to_delete = Iterators.filter(each_solid_vertex(tri)) do i
    p = get_point(tri, i)
    r2 = (getx(p) - 1.0)^2 + (gety(p) - 1.0)^2
    return r2 < 1 / 2^2
end
for i in vertices_to_delete
    delete_point!(tri, i)
end
fig, ax, sc = triplot(tri)
fig

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