Triangulation Operations

DelaunayTriangulation.add_boundary_information! — Function

add_boundary_information!(tri::Triangulation)

Updates tri so that the ghost triangle information defined by the boundary nodes in tri is added to the triangulation.

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DelaunayTriangulation.add_ghost_triangles! — Function

add_ghost_triangles!(tri::Triangulation)

Adds all the ghost triangles to tri.

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DelaunayTriangulation.add_point! — Function

add_point!(c::RepresentativeCoordinates, p)

Treating c as an arithmetic average, updates the coordinates of c to include p.

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add_point!(tri::Triangulation, new_point; kwargs...) -> Triangle
add_point!(tri::Triangulation, x, y; kwargs...) -> Triangle

Arguments

tri::Triangulation: The Triangulation.
new_point: The point to be added to the triangulation. The second method uses (x, y) to represent the new point instead. If new_point is an integer, then the point added is get_point(tri, new_point).

Keyword Arguments

predicates::AbstractPredicateKernel=AdaptiveKernel(): Method to use for computing predicates. Can be one of FastKernel, ExactKernel, and AdaptiveKernel. See the documentation for a further discussion of these methods.
point_indices=each_solid_vertex(tri): The indices of the points to be used in the find_triangle algorithm for selecting the initial point.
m=default_num_samples(length(point_indices)): The number of samples (without replacement) to be used in the find_triangle algorithm for selecting the initial point.
try_points=(): Additional points to try for selecting the initial point, in addition to the m sampled.
rng::Random.AbstractRNG=Random.default_rng(): The random number generator to be used in find_triangle.
initial_search_point=integer_type(tri)(select_initial_point(tri, new_point; point_indices, m, try_points, rng)): The initial point to be used in find_triangle.
update_representative_point=false: Whether to update the representative point of the triangulation after adding the new point.
store_event_history=Val(false): Whether to store the event history of the triangulation from adding the new point.
event_history=nothing: The event history of the triangulation from adding the new point. Only updated if store_event_history is true, in which case it needs to be an InsertionEventHistory object.
concavity_protection=false: Whether to use concavity protection for finding V below. See concavity_protection_check. This is only needed if your triangulation is not convex.
V=find_triangle(tri, get_point(tri, new_point); m=nothing, point_indices=nothing, try_points=nothing, k=initial_search_point, concavity_protection, rng): The positively oriented triangle containing the point being added.

Non-convex domains

In cases where your triangulation is not convex and !concavity_protection, this V may not be correct, and you may encounter errors - errors either during add_point! or separately when you try to use the triangulation. In such cases, you should set concavity_protection=true to ensure that V is correct.

peek=Val(false): Whether the point should actually be added into the triangulation, or just 'peeked' at so that the events that would occur from its addition can be added into event_history.

Outputs

The triangulation is updated in-place, but we do return

V: The triangle containing the point being added.

Convex hull

In cases where (x, y) is outside of the triangulation, it will be added successfully but note that the convex_hull field of tri will no longer be accurate. You can use convex_hull! to fix it.

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add_point!(tri::Triangulation, x, y, w; kwargs...)

Adds the point (x, y) into tri with weight w. This function requires that add_weight! is defined on the weights stored in tri. The kwargs match those from add_point!(tri::Triangulation, ::Any).

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DelaunayTriangulation.add_segment! — Function

add_segment!(tri::Triangulation, segment; predicates::AbstractPredicateKernel=AdaptiveKernel(), rng::Random.AbstractRNG=Random.default_rng())
add_segment!(tri::Triangulation, i, j; predicates::AbstractPredicateKernel=AdaptiveKernel(), rng::Random.AbstractRNG=Random.default_rng())

Adds segment = (i, j) to tri.

Arguments

tri::Triangulation: The Triangulation.
segment: The segment to add. The second method uses (i, j) to represent the segment instead.

Keyword Arguments

predicates::AbstractPredicateKernel=AdaptiveKernel(): Method to use for computing predicates. Can be one of FastKernel, ExactKernel, and AdaptiveKernel. See the documentation for a further discussion of these methods.
rng::AbstractRNG=Random.default_rng(): The RNG object.

Outputs

There is no output, but tri will be updated so that it now contains segment.

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DelaunayTriangulation.add_triangle! — Function

add_triangle!(T, V...)
add_triangle!(T, i, j, k)

Add the triangles V... or V = (i, j, k) to the collection of triangles T.

Examples

julia> using DelaunayTriangulation

julia> T = Set(((1, 2, 3), (4, 5, 6)))
Set{Tuple{Int64, Int64, Int64}} with 2 elements:
  (4, 5, 6)
  (1, 2, 3)

julia> add_triangle!(T, (7, 8, 9));

julia> add_triangle!(T, (10, 11, 12), (13, 14, 15));

julia> add_triangle!(T, 16, 17, 18);

julia> T
Set{Tuple{Int64, Int64, Int64}} with 6 elements:
  (7, 8, 9)
  (10, 11, 12)
  (4, 5, 6)
  (13, 14, 15)
  (16, 17, 18)
  (1, 2, 3)

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DelaunayTriangulation.clear_empty_features! — Function

clear_empty_features!(tri::Triangulation)

Clears all empty features from the triangulation tri.

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DelaunayTriangulation.delete_ghost_triangles! — Function

delete_ghost_triangles!(tri::Triangulation)

Deletes all the ghost triangles from tri.

Ghost vertices

Ghost vertices are still used in the keys of the Adjacent2Vertex of tri, and are still present in the Graph. If you want to delete the ghost vertex keys from the Adjacent2Vertex, you need to use delete_adjacent2vertex!. For deleting the ghost vertices from the Graph, you need delete_ghost_vertices_from_graph!. Additionally, edges in Adjacent can still map to ghost vertices. If you also want to delete those, you need to filter through the values of the Adjacent map that are ghost vertices, and use delete_adjacent!.

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DelaunayTriangulation.delete_holes! — Function

delete_holes!(tri::Triangulation)

Deletes all the exterior faces to the boundary nodes specified in the triangulation tri.

Extended help

This function works in several stages:

First, find_all_points_to_delete is used to identify all points in the exterior faces.
Once all the points to delete have been found, all the associated triangles are found using find_all_triangles_to_delete, taking care for any incorrectly identified triangles and points.
Once the correct set of triangles to delete has been found, they are deleted using delete_all_exterior_triangles!.

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DelaunayTriangulation.get_surrounding_polygon — Function

get_surrounding_polygon(cache::TriangulationCache) -> Vector{Vertex}

Returns the polygon surrounding the triangulation stored in cache.

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get_surrounding_polygon(vor::VoronoiTessellation, i) -> Vector{Vertex}

Gets the polygon surrounding the generator with index i in vor.

You shouldn't need to use this, see get_polygon instead.

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get_surrounding_polygon(tri::Triangulation, u; skip_ghost_vertices=false) -> Vector

Returns the counter-clockwise sequence of neighbours of u in tri.

Arguments

tri::Triangulation: Triangulation.
u: The vertex.

Keyword Arguments

skip_ghost_vertices=false: Whether to skip ghost vertices in the returned polygon.

Outputs

S: The surrounding polygon. This will not be circular, meaning S[begin] ≠ S[end]. In case u is an exterior ghost vertex, the returned polygon is a clockwise list of vertices for the associated boundary curve. If you do not have ghost triangles and you try to get the surrounding polygon of a ghost vertex, then this function may return an invalid polygon.

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DelaunayTriangulation.delete_point! — Function

delete_point!(c::RepresentativeCoordinates, p)

Treating c as an arithmetic average, updates the coordinates of c to exclude p.

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delete_point!(tri::Triangulation, vertex; kwargs...)

Deletes the vertex of tri, retriangulating the cavity formed by the surrounding polygon of vertex using triangulate_convex.

It is not possible to delete vertices that are on the boundary, are ghost vertices, or adjoin a segment of tri. See also check_delete_point_args.

Point deletion

This function will not actually delete the corresponding coordinates from get_points(tri), nor will it remove the associated weight from get_weights(tri).

Arguments

tri::Triangulation: Triangulation.
vertex: The vertex to delete.

Keyword Arguments

predicates::AbstractPredicateKernel=AdaptiveKernel(): Method to use for computing predicates. Can be one of FastKernel, ExactKernel, and AdaptiveKernel. See the documentation for a further discussion of these methods.
store_event_history=Val(false): Whether to store the event history of the triangulation from deleting the point.
event_history=nothing: The event history of the triangulation from deleting the point. Only updated if store_event_history is true, in which case it needs to be an InsertionEventHistory object.
rng::Random.AbstractRNG=Random.default_rng(): The random number generator to use for the triangulation.

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DelaunayTriangulation.delete_triangle! — Function

delete_triangle!(V, T...)
delete_triangle!(V, i, j, k)

Delete the triangles T... from the collection of triangles V.

Examples

julia> using DelaunayTriangulation

julia> V = Set(((1, 2, 3), (4, 5, 6), (7, 8, 9), (10, 11, 12), (13, 14, 15)))
Set{Tuple{Int64, Int64, Int64}} with 5 elements:
  (7, 8, 9)
  (10, 11, 12)
  (4, 5, 6)
  (13, 14, 15)
  (1, 2, 3)

julia> delete_triangle!(V, (6, 4, 5))
Set{Tuple{Int64, Int64, Int64}} with 4 elements:
  (7, 8, 9)
  (10, 11, 12)
  (13, 14, 15)
  (1, 2, 3)

julia> delete_triangle!(V, (10, 11, 12), (1, 2, 3))
Set{Tuple{Int64, Int64, Int64}} with 2 elements:
  (7, 8, 9)
  (13, 14, 15)

julia> delete_triangle!(V, 8, 9, 7)
Set{Tuple{Int64, Int64, Int64}} with 1 element:
  (13, 14, 15)

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DelaunayTriangulation.flip_edge! — Function

flip_edge!(tri::Triangulation, i, j, store_event_history=Val(false), event_history=nothing)
flip_edge!(tri::Triangulation, i, j, k, ℓ, store_event_history=Val(false), event_history=nothing)

Flips the edge between vertices i and j in tri.

Arguments

tri::Triangulation: The Triangulation.
i: The first vertex of the edge to flip.
j: The second vertex of the edge to flip.
k: The vertex k = get_adjacent(tri, j, i). This is only used in the second method.
ℓ: The vertex ℓ = get_adjacent(tri, i, j). This is only used in the second method.
store_event_history=Val(false): Whether to store the event history of the flip.
event_history=nothing: The event history. Only updated if store_event_history is true, in which case it needs to be an InsertionEventHistory object. This storage is done using store_flip_edge_history!.

Outputs

There is no output, as tri is updated in-place.

Invalid flips

If (i, j, k, ℓ), where ℓ = get_adjacent(tri, i, j) and k = get_adjacent(tri, j, i), is not a convex quadrilateral, then this edge flip will make the triangulation non-planar.

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DelaunayTriangulation.legalise_edge! — Function

legalise_edge!(tri::Triangulation, i, j, r, store_event_history=Val(false), event_history=nothing; predicates::AbstractPredicateKernel=AdaptiveKernel())

Legalises the edge (i, j) and other neighbouring edges in tri if they are illegal, assuming the vertex r was just added into a triangle that contains (i, j). flip_edge! is used.

See also is_legal.

Arguments

tri::Triangulation: The Triangulation.
i: The first vertex of the edge to legalise.
j: The second vertex of the edge to legalise.
r: The vertex that was just added into a triangle that contains (i, j).
store_event_history=Val(false): Whether to store the event history of the flip.
event_history=nothing: The event history. Only updated if store_event_history is true, in which case it needs to be an InsertionEventHistory object.

Keyword Arguments

predicates::AbstractPredicateKernel=AdaptiveKernel(): Method to use for computing predicates. Can be one of FastKernel, ExactKernel, and AdaptiveKernel. See the documentation for a further discussion of these methods.

Outputs

There is no output, as tri is updated in-place.

Invalid event histories

Edge flipping can lead to event_history having triangles both in event_history.added_triangles and event_history.deleted_triangles. To get around this, we only store in these fields the triangles necessary to allow undo_insertion! to work, so that at a triangle that might have appeared in both will only appear in one.

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DelaunayTriangulation.lock_convex_hull! — Function

lock_convex_hull!(tri::Triangulation; rng=Random.default_rng(), predicates::AbstractPredicateKernel=AdaptiveKernel())

Locks the convex hull of the unconstrained triangulation tri so that it is now treated as a constrained triangulation with boundary given by its convex hull.

The random number generator (used inside add_segment! can be provided with the rng keyword argument, and similarly for predicates.

Warning

If an edge is encountered along the convex hull that contains a segment from tri.interior_segments, then this edge will be deleted from tri.interior_segments; this will be undone from unlock_convex_hull!, possibly splitting the segments in case they were split before unlocking.

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DelaunayTriangulation.unlock_convex_hull! — Function

unlock_convex_hull!(tri::Triangulation; reconstruct=false)

Unlocks the convex hull of the constrained triangulation tri so that it is now treated as an unconstrained triangulation, assuming that it was locked using lock_convex_hull!. If reconstruct = true, then the convex hull of tri will be reconstructed from the boundary nodes of tri. This is useful if, for example, you have split some of the boundary edges during mesh refinement.

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DelaunayTriangulation.split_edge! — Function

split_edge!(tree::BoundaryRTree, i, j, r)

Splits the diametral bounding box associated with (i, j) into two new boxes associated with the diametral circles of (i, r) and (j, r).

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split_edge!(enricher::BoundaryEnricher, i, j, r, update_boundary_nodes = Val(true), update_segments = Val(true), is_interior = is_segment(enricher, i, j))

Updates the fields of enricher after splitting an edge (i, j) at the rth vertex. The update_boundary_nodes argument can be used to avoid inserting an additional boundary node when boundary_nodes was already updated somewhere else (e.g., we need this for mesh refinement which already updates the boundary_nodes which is aliased with the same field in the enricher). The same point goes for update_segments which can be used to avoid inserting an additional segment when segments was already updated somewhere else. The is_interior argument can be used to specify whether the edge is an interior segment or a boundary edge.

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split_edge!(tri::Triangulation, i, j, r, store_event_history=Val(false), event_history=nothing)

Splits the edge (i, j) in tri at the vertex r. For the triangulation to be valid after this splitting, it is assumed that r is collinear with, or at least very close to collinear with, the edge (i, j).

Arguments

tri::Triangulation: The Triangulation.
i: The first vertex of the edge to split.
j: The second vertex of the edge to split.
r: The vertex to split the edge at.
store_event_history=Val(false): Whether to store the event history of the flip.
event_history=nothing: The event history. Only updated if store_event_history is true, in which case it needs to be an InsertionEventHistory object.

Outputs

There is no output, as tri is updated in-place.

Handling unoriented edges

The triangulation will only be updated as if (i, j) has been split rather than also (j, i). You will need to call split_edge! again with (j, i) if you want to split that edge as well.

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DelaunayTriangulation.complete_split_edge_and_legalise! — Function

complete_split_edge_and_legalise!(tri::Triangulation, i, j, r, store_event_history=Val(false), event_history=nothing; predicates::AbstractPredicateKernel=AdaptiveKernel())

Given a triangulation tri, an edge (i, j), and a point r, splits both (i, j) and (j, i) at r using split_edge! and then subsequently legalises the new edges with legalise_split_edge!.

Arguments

tri::Triangulation: The Triangulation.
i: The first vertex of the edge to split.
j: The second vertex of the edge to split.
r: The vertex to split the edge at.
store_event_history=Val(false): Whether to store the event history of the flip.
event_history=nothing: The event history. Only updated if store_event_history is true, in which case it needs to be an InsertionEventHistory object.

Keyword Arguments

predicates::AbstractPredicateKernel=AdaptiveKernel(): Method to use for computing predicates. Can be one of FastKernel, ExactKernel, and AdaptiveKernel. See the documentation for a further discussion of these methods.

Outputs

There is no output, as tri is updated in-place.

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DelaunayTriangulation.split_triangle! — Function

split_triangle!(tri::Triangulation, args::RefinementArguments, T) -> Certificate

Splits a bad triangle T of tri to improve its quality.

Arguments

tri::Triangulation: The Triangulation to split a triangle of.
args::RefinementArguments: The RefinementArguments for the refinement.
T: The triangle to split.

Output

cert: A Certificate indicating whether the split was successful or not. In particular, returns one of:
- Cert.SuccessfulInsertion: The triangle was split successfully.
- Cert.EncroachmentFailure: The triangle was not split successfully as the newly inserted point encroached upon a segment.
- Cert.PrecisionFailure: The triangle was not split successfully due to precision issues.

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split_triangle!(tri::Triangulation, i, j, k, r)

Splits the triangle (i, j, k) at the vertex r, assumed to be inside the triangle.

Arguments

tri::Triangulation: The Triangulation.
i: The first vertex of the triangle.
j: The second vertex of the triangle.
k: The third vertex of the triangle.
r: The vertex to split the triangle at.

Outputs

There is no output, but tri will be updated so that it now contains the triangles (i, j, r), (j, k, r), and (k, i, r).

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DelaunayTriangulation.complete_split_triangle_and_legalise! — Function

complete_split_triangle_and_legalise!(tri::Triangulation, i, j, k, r)

Splits the triangle (i, j, k) at the vertex r, assumed to be inside the triangle, and legalises the newly added edges in tri.

Arguments

tri::Triangulation: The Triangulation.
i: The first vertex of the triangle.
j: The second vertex of the triangle.
k: The third vertex of the triangle.
r: The vertex to split the triangle at.

Outputs

There is no output, as tri is updated in-place.

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