lukas-toenne / voronoi_tools

Experimental Blender addon for generating Delaunay triangulation and Voronoi meshes on spheres

Blender addon for generating Delaunay triangulation and Voronoi meshes.

Add Voronoi Cells operator

The "Add Voronoi Cells" operator generates a Delaunay triangulation or Voronoi mesh in the active object.

1. Select objects that contain the input points, such as mesh vertices or particles.

2. Select the target object that should contain the output triangulation or cell mesh.

   Warning: The active object's mesh data is replaced by the operator.

3. Run the operator from the search menu (F3).

Input Settings

• Sources: Mesh vertices and particles can be used as input points for the triangulation. All selected objects are used as sources, except the active object which is the output target mesh.

  Note that points are projected onto the XY plane for triangulation, the Z component is ignored.

• Bounds Mode:

  • None: No bounds are applied, point data is used as-is.
  • Limit: Bounds min/max values are use to prune the point set, points outside these bounds are ignored.
  • Repeat: The point set is repeated in the X and Y direction to create seamlessly tileable patterns.

Output Settings

• Output Graph:

  • Delaunay: Creates a Delaunay triangulation of the input points.

    Angles in this triangulation are maximised, skinny triangles are avoided as much as possible.

  • Voronoi: Creates Vornoi cells around each input point.

    Each location in the cell is closer to its cell point than any other input point.

• Output UV Layers: Selection of data to output as UV layers for use in shaders or modifiers.

  • Bounds: Position within the input bounds (Local space coordinates if bounds are disabled)
  • Cell Centered: Local space coordinates with circum-circle (Delaunay triangles) or cell center (Voronoi) at the origin
  • Edge Distance: Local space distance from the cell edge (Voronoi only, needs triangulated cells).
  • Point Index: Contiguous index of input point
  • Point ID: Unique ID of input point, if available
  • Random: Random value based on the id of the input point

• Triangulate Cells: If enabled, the Voronoi cells are created using triangle fans instead of ngons.

  This can be useful if the Edge Distance layer is used in shading, which requires the center vertex.

• Generate Debug Meshes: Debugging feature which generates intermediate meshes for each step in the algorithm.

  Warning: This can generate a lot of data and stall or even crash Blender if too many input points are used (less than 500 is advised). Should not be used in production.

Triangulator

The triangulator used for generating Delaunay or Voronoi meshes works in several stages:

1. Sweep-Hull triangulation

   

   Creates a base non-overlapping triangulation of the XY plane from the input points. Points are first sorted by distance from the center of the point distribution. Then a convex hull is generated by successively adding points.

   Note that the resulting triangulation does not generally satisfy the Delaunay condition yet.

2. Edge Flipping

   

   In this step the edges of triangles are tested for the Delaunay condition: No other point must lie within the circumcircle of either adjacent triangle. If the Delaunay condition is not satisfied, the edge is "flipped", i.e. it is replaced by the edge between the opposing vertices of the adjacent triangles.

   This procedure ensures the Delaunay condition for the edge in question, but requires re-examining the surrounding edges, leading to a O(n^2) algorithm in the worst case. However, the preceding sweep-hull stage tends to create well-conditioned triangles that lead to O(n*log(n)) runtime.

3. Voronoi Cells

   

   If the final output type is a Voronoi graph, the triangulation mesh is replaced by its dual graph, constructed from cells around each vertex. The cell boundaries are constructed by connecting the centers of triangle circumcircles. The resulting ngons contain all locations around a point that closest to that point.