Burst Triangulator

A single-file package which provides simple Delaunay triangulation of the given set of points (float2) with mesh refinement. Implemented triangulation is based on Bowyer–Watson algorithm^{1 2} and refinement on Ruppert's algorithm³.

The package provides also constrained triangulation (with mesh refinement) which is based on Sloan's algorithm⁴.

Table of contents

• Burst Triangulator
  • Table of contents
  • Getting started
  • Example usage
    • Delaunay triangulation
    • Delaunay triangulation with mesh refinement
    • Constrained Delaunay triangulation
    • Constrained Delaunay triangulation with mesh refinement
    • Support for holes and boundaries
    • Summary
    • Input validation
    • Generating input in a job
    • Reduce the effect of roundoff error
  • Benchmark
  • Dependencies
  • Contributors
  • Bibliography

Getting started

Install the package using one of the following methods

Using scoped registry (recommended)

Use OpenUPM CLI or add corresponding entries to the project's manifest.json manually. Add or modify scoped registries in the manifest

  "scopedRegistries": [
    {
      "name": "OpenUPM",
      "url": "https://package.openupm.com/",
      "scopes": [
        "com.andywiecko"
      ]
    }
  ]

and in the dependencies provide selected version of the package

"dependencies": {
    "com.andywiecko.burst.triangulator": "1.4.0",
    ...

See Unity docs for more details https://docs.unity3d.com/2021.1/Documentation/Manual/upm-scoped.html

git install

Use package manager via git install: https://github.com/andywiecko/BurstTriangulator.git.

Manual instalation

Clone or download this repository and then select package.json using Package Manager (Window/Package Manager).

Copy Runtime/Triangulator.cs

Since the package is single-file only, one can put the file Runtime/Triangulator.cs somewhere in the project to use it independently.

Example usage

Below one can find example usage of the Triangulator with input set as four points that form the unit square:

var positions = new[]
{
    math.float2(0, 0),
    math.float2(1, 0),
    math.float2(1, 1),
    math.float2(0, 1)
};

using var triangulator = new Triangulator(capacity: 1024, Allocator.Persistent);
using var inputPositions = new NativeArray<float2>(positions, Allocator.Persistent);

triangulator.Input.Positions = inputPositions

triangulator.Run();

var outputTriangles = triangulator.Output.Triangles;
var outputPositions = triangulator.Output.Positions;

The result of the triangulation procedure will depend on selected settings. There are a few settings of the triangulation, shortly described below:

var settings = triangulator.Settings;

// Batch count used in parallel job.
settings.BatchCount = 64;
// Triangle is considered as bad if any of its angles is smaller than MinimumAngle. Note: radians.
settings.MinimumAngle = math.radians(33);
// Triangle is not considered as bad if its area is smaller than MinimumArea.
settings.MinimumArea = 0.015f
// Triangle is considered as bad if its area is greater than MaximumArea.
settings.MaximumArea = 0.5f;
// If true refines mesh using Ruppert's algorithm.
settings.RefineMesh = true;
// If true constrains edges defined in the Triangulator.Input.ConstraintEdges
settings.ConstrainEdges = false;
// If true and provided Triangulator.Input is not valid, it will throw an exception.
settings.ValidateInput = true;
// If true provided point positions and hole seeds are transformed into [-1, 1] box using "center of mass".
settings.UseLocalTransformation = false;

Below one can find the result of the triangulation for different selected options.

Note

To obtain the boundary from a texture, the UnityEngine.PolygonCollider was used. Generating the image boundary is certainly a separate task and is not considered in the project.

Delaunay triangulation

To use classic Delaunay triangulation make sure that constraint and refinement are disabled.

settings.RefineMesh = false;
settings.ConstrainEdges = false;

The result without mesh refinement (Delaunay triangulation):

Delaunay triangulation with mesh refinement

To proceed with triangulation with the mesh refinement one has to set a proper refinement option

settings.RefineMesh = true;
settings.ConstrainEdges = false;

Users can control the quality of the triangles by these options

// Triangle is considered as bad if any of its angles is smaller than MinimumAngle. Note: radians.
settings.MinimumAngle = math.radians(33);
// Triangle is not considered as bad if its area is smaller than MinimumArea.
settings.MinimumArea = 0.015f
// Triangle is considered as bad if its area is greater than MaximumArea.
settings.MaximumArea = 0.5f;

The result with mesh refinement:

Constrained Delaunay triangulation

It is not guaranteed that the boundary of the input will be present in the classic Delaunay triangulation result. One needs to specify the constraints to resolve this issue. To specify the edges which should be present in the final triangulation provide the additional input data

triangulator.Settings.RefineMesh = false;
triangulator.Settings.ConstrainEdges = true;

// Provided input of constraint edges
// (a0, a1), (b0, b1), (c0, c1), ...
// should be in the following form
// constraintEdges elements:
// [0]: a0, [1]: a1, [2]: b0, [3]: b1, ...
using var constraintEdges = new NativeArray<int>(64, Allocator.Persistent);

triangulator.Input.ConstraintEdges = constraintEdges;

In the following figure one can see the non-constrained triangulation result (with yellow), and user-specified constraints (with red).

After enabling Settings.ConstrainEdges = true and providing the corresponding input, the result of the constrained triangulation fully covers all specified edges by the user

Constrained Delaunay triangulation with mesh refinement

Constrained triangulation can be also refined in the same manner as non-constrained one, by enabling corresponding options in triangulation settings:

triangulator.Settings.RefineMesh = true;
triangulator.Settings.ConstrainEdges = true;

In the following figure one can see the non-constrained triangulation result (with yellow), and user-specified constraints (with red) with the refinement.

After enabling the refinement and the constraint and providing the input, the result of the constrained triangulation fully covers all specified edges by the user and the mesh is refined with the given refinement conditions.

Support for holes and boundaries

The package provides also an option for restoring the boundaries. One has to enable corresponding options and provide the constraints

settings.RestoreBoundary = true;
settings.ConstraintEdges = true;

In the following figure, one can see the constrained triangulation result (with yellow), and user-specified constraints (with red) with the disabled RestoreBoundary and refinement enabled.

After enabling the RestoreBoundary the result of the constrained triangulation fully covers all conditions and all invalid triangles are destroyed.

The package provides also an option for creating holes. Except for setting the ConstraintEdges, a user needs to provide positions of the holes in the same space as the Input.Positions. Enabling RestoringBoundary option is not mandatory, holes could be introduced independently of preserving the boundaries

settings.RestoreBoundary = true; // optional
settings.ConstraintEdges = true;

using var holes = new NativeArray<float2>(new[]{ math.float2(0.5f, 0.5f) }, Allocator.Persistent);
input.HoleSeeds = holes;

Summary

Below one can find the comparison of the results of all possible settings which are available in the package.

Input validation

If Triangulator.Settings.ValidateInput is set to true, the provided data will be validated before running the triangulation procedure. Input positions, as well as input constraints, have a few restrictions:

• Points count must be greater/equal 3.
• Points positions cannot be duplicated.
• Points cannot contain NaNs or infinities.
• Constraint edges cannot intersect with each other.
• Constraint edges cannot be duplicated or swapped duplicated.
• Zero-length constraint edges are forbidden.
• Constraint edges cannot intersect with points other than the points for which they are defined.

Note

Validation is limited only for Editor!

Generating input in a job

BurstTriangulation input can be generated with job pipeline. One has to use DeferredJobArrays, see the example snippet:

using var positions = new NativeList<float2>(64, Allocator.Persistent);
using var constraints = new NativeList<int>(64, Allocator.Persistent);
using var holes = new NativeList<float2>(64, Allocator.Persistent);
using var triangulator = new Triangulator(64, Allocator.Persistent)
{
  Input = 
  {
    Positions = positions.AsDeferredJobArray(),
    ConstraintEdges = constraints.AsDeferredJobArray(),
    HoleSeeds = holes.AsDeferredJobArray()
  }
}

var dependencies = new JobHandle();
dependencies = new GenerateInputJob(positions, constraints, holes).Schedule(dependencies); // Lists are fed here.
dependencies = triangulator.Schedule(dependencies);
dependencies.Complete();

Reduce the effect of roundoff error

BurstTriangulator has a built-in option for transforming input positions (as well as holes) into normalized local space, i.e. [-1, 1] box. To enable it use Settings.UseLocalTransformation = true. Converting points into normalized local space could increase numerical accuracy. The option is disabled by default.

Benchmark

The package uses Burst compiler, which produces highly optimized native code using LLVM. Below one can see a log-log plot of elapsed time as a function of the final triangles count after mesh refinement. Using Burst can provide more or less two order of magnitude faster computation.

Dependencies

• Unity.Burst
• Unity.Mathematics
• Unity.Collections
• Unity.Jobs

Contributors

• Andrzej Więckowski, Ph.D.

Bibliography

Footnotes

1. A. Bowyer. "Computing Dirichlet tessellations". Comput. J. 24 (2): 162–166 (1981). ↩

2. D.F. Watson. "Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes". Comput. J. 24 (2): 167–172 (1981). ↩

3. J. Ruppert. "A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation". J. Algorithms 18(3):548-585 (1995). ↩

4. S.W. Sloan. "A fast algorithm for generating constrained Delaunay triangulations." Comput. Struct. 47.3:441-450 (1993). ↩