Dynamic triangulation

Using the UnsafeTriangulator<T> API, you can perform dynamic triangulation. This feature is especially useful in scenarios like path-finding in RTS games, where recalculating only a small portion of the mesh, instead of the entire mesh, can significantly improve efficiency.

DynamicInsertPoint

The DynamicInsertPoint method allows you to insert a point into a specified triangle using barycentric coordinates. This method only supports point insertion within the original triangulation domain and cannot be used to insert points outside the existing mesh.

Inserting a point at specific T2 p coordinates can be computationally expensive since it requires locating the triangle that contains the point p. This package does not include acceleration structures, as it assumes the user will implement this based on their specific requirements. It is recommended to use structures such as bounding volume hierarchies, 2D trees, grids, or buckets for efficient point lookup (p \(\to \triangle\)). The most suitable acceleration structure may vary depending on the use case.

The DynamicInsertPoint method accepts the following parameters (in addition to output and allocator):

tId the index of the triangle where the point should be inserted.
bar the barycentric coordinates of the point inside triangle tId.

Important

All \(\texttt{bar}\) coordinates must be valid barycentric coordinates inside the triangle, meaning

\(\forall_i\,\,\texttt{bar}[i]\in(0, 1)\);
\(\sum_i \, \texttt{bar}[i] = 1\).

Here is an example of how to use the API:

var t = new UnsafeTriangulator<float2>();

using var positions = new NativeArray<float2>(..., Allocator.Persistent);
using var constraints = new NativeArray<int>(..., Allocator.Persistent);
var input = new NativeInputData<float2>
{
    Positions = positions,
    ConstraintEdges = constraints
};

using var outputPositions = new NativeList<float2>(Allocator.Persistent);
using var triangles = new NativeList<int>(Allocator.Persistent);
using var halfedges = new NativeList<int>(Allocator.Persistent);
using var constrainedHalfedges = new NativeList<bool>(Allocator.Persistent);
var output = new NativeOutputData<float2>
{
    Positions = outputPositions,
    Triangles = triangles,
    Halfedges = halfedges,
    ConstrainedHalfedges = constrainedHalfedges
};

t.Triangulate(input, output, args: Args.Default(autoHolesAndBoundary: true), Allocator.Persistent);

// Insert a new point in triangle with index 42 at the center (barycentric coordinates: [⅓, ⅓, ⅓]).
t.DynamicInsertPoint(output, tId: 42, bar: 1f / 3, allocator: Allocator.Persistent);

Note

To quickly calculate the corresponding bar for a given p inside a triangle (a, b, c), you can use the following utility function (not included in the package):

static float3 Bar(float2 p, float2 a, float2 b, float2 c)
{
  float cross(float2 x, float2 y) => x.x * y.y - x.y * y.x;
  var (v0, v1, v2) = (b - a, c - a, p - a);
  var v = cross(v2, v1) / cross(v0, v1);
  var w = cross(v0, v2) / cross(v0, v1);
  var u = 1 - v - w;
  return math.float3(u, v, w);
}

DynamicSplitHalfedge

The DynamicSplitHalfedge method allows you to split specified halfedge by inserting a point at a position determined by linear interpolation. The position is interpolated between the start and end points of the halfedge in the triangulation using \(\alpha\) as the interpolation parameter. This method preserves the "constrained" state of the halfedge, meaning that if the specified halfedge is constrained, the two resulting sub-segments will also be marked as constrained.

The DynamicSplitHalfedge method accepts the following parameters (in addition to output and allocator):

he the index of the halfedge to split.
alpha the interpolation parameter for positioning the new point between the start and end points of the halfedge, where \(p = (1 - \alpha) \, \text{start} + \alpha \, \text{end}\).

Here is an example of how to use the API:

var t = new UnsafeTriangulator<float2>();

using var positions = new NativeArray<float2>(..., Allocator.Persistent);
using var constraints = new NativeArray<int>(..., Allocator.Persistent);
var input = new NativeInputData<float2>
{
    Positions = positions,
    ConstraintEdges = constraints
};

using var outputPositions = new NativeList<float2>(Allocator.Persistent);
using var triangles = new NativeList<int>(Allocator.Persistent);
using var halfedges = new NativeList<int>(Allocator.Persistent);
using var constrainedHalfedges = new NativeList<bool>(Allocator.Persistent);
var output = new NativeOutputData<float2>
{
    Positions = outputPositions,
    Triangles = triangles,
    Halfedges = halfedges,
    ConstrainedHalfedges = constrainedHalfedges
};

t.Triangulate(input, output, args: Args.Default(autoHolesAndBoundary: true), Allocator.Persistent);

// Iteratively split random halfedges.
var random = new Unity.Mathematics.Random(seed: 42);
for(int i = 0; i < 32; i++)
{
    t.DynamicInsertPoint(output, he: random.NextInt(0, triangles.Length), alpha: 0.5f, allocator: Allocator.Persistent);
}

DynamicRemoveBulkPoint

The DynamicRemoveBulkPoint method allows you to remove bulk points, i.e. points which are not boundary ones, and re-triangulates the affected region to maintain a valid triangulation. In addition to output and allocator, the method requires the index of the point to be removed, specified as pId.

Below is an example demonstrating how to use the DynamicRemoveBulkPoint API:

using var inputPositions = new NativeArray<double2>(..., Allocator.Persistent);
using var outputPositions = new NativeList<double2>(Allocator.Persistent);
using var triangles = new NativeList<int>(Allocator.Persistent);
using var constrainedHalfedges = new NativeList<bool>(Allocator.Persistent);
using var halfedges = new NativeList<int>(Allocator.Persistent);

var t = new UnsafeTriangulator<double2>();
var input = new NativeInputData<double2> { Positions = inputPositions };
var output = new NativeOutputData<double2>
{
    Positions = outputPositions,
    Triangles = triangles,
    ConstrainedHalfedges = constrainedHalfedges,
    Halfedges = halfedges,
};

t.Triangulate(input, output, args: Args.Default(), allocator: Allocator.Persistent);
t.DynamicRemoveBulkPoint(output, pId: 8, allocator: Allocator.Persistent);