Line-segment voronoi diagram notes

The next step for the opencamlib voronoi-diagram (vd) algorithm is to be able to add line-segment generators. Some notes and ideas on that then (this turned into a rather long post...)

First let's recall (from February) how the point-generator vd-algorithm works:

In (A) we want to insert a new point-generator (yellow). First we need to identify the closest point-generator among the existing generators. (B) Then among the vertices that bound this voronoi-face(region) we search for a seed vertex (pink). The seed vertex is the one that violates the inCircle-predicate the most, i.e. has the minimum distance to the new point-to-be-inserted. (C) Starting from the seed vertex, we then identify other vertices (red) that need to be deleted, because they are closer to the new generator than to any other generator. These vertices are called "IN" vertices, and the associated edges (also in red) should (i) form a tree (a connected acyclic graph) and (ii) for any face, the "IN"-vertices should be connected, and the "OUT" vertices should be connected. (D) IN-IN edges (red) will be deleted completely, while NEW vertices will be generated on each IN-OUT edge (green). (E) NEW-NEW edges are created and connected into a cycle which forms the face/region for the new generator. (F) Now all IN-vertices and all IN-IN as well as IN-NEW edges are deleted and we are done.

Now the case when we insert a line segment. We start as above by constructing the diagram for all point-generators as well as all the endpoints of our line-segments. Here is an image from the Sugihara&Iri 2000 paper ( which I have enhanced with colors and text.

The steps of the algorithm are similar to (A)-(F) above: In (A), since we have already inserted the two endpoints of the line-segment, we obviously know in which face to look for the seed-vertex. (B) when we search for a seed-vertex we now need both a dist(VDVertex, PointGenerator) function and a dist(VDVertex, LineGenerator) function. In (C) we again expand the set of "IN" vertices (red circles), while keeping the associated graph a tree, and not disconecting "IN"-vertices or "OUT"-vertices of any face (i and ii above) .

A new (iii) requirement is that the tree should connect the regions of the line-segment endpoints. A new situation also appears where the two endpoints of an edge are classified as "IN", but the complete edge should not be deleted. This is because edges between point and line-segment generators are parabolic arcs (curved!). In this case two NEW vertices are generated on the edge, so it splits into three parts: IN-NEW, NEW-NEW, and NEW-IN. This needs to be dealt with, but I'm not sure exactly how. I think one of the Held papers talks about avoiding this special case by splitting each parabolic edge in two, at the "vertex" of the parabola i.e. the point of greatest curvature (?).

"IN-IN" edges (magenta) will again be deleted. (D) NEW-vertices (green) should now be generated on each IN-OUT edge. This needs new functionality, since we need to deal with two types of edges: LineEdge and ParabolicEdge. The LineEdge occurs between two PointGenerators or between two LineGenerators, while the ParabolicEdge appears between a PointGenerator and a LineGenerator. The NEW vertex we need to generate will obviously be located on the IN-OUT edge (with associated generators gen1, gen2), and should be positioned so that it is equidistant from gen1, gen2, and the new line-segment generator. In contrast to the old point-generator diagram code where we only had one type of VDVertex, namely VDVertex( PointGenerator, PointGenerator, PointGenerator), we now have 6 (six!) different types of vertices in the diagram, depending on the three neighbouring generators, which can be of type LineGenerator, PointGenerator(endpoint), or PointGenerator:

  • VDVertex( PointGenerator, PointGenerator, PointGenerator), type V1 in Okabe et al.
  • VDVertex( PointGenerator, PointGenerator(endpoint) , LineGenerator), type V2
  • VDVertex( LineGenerator, PointGenerator, PointGenerator), type V3
  • VDVertex( LineGenerator, LineGenerator, PointGenerator(endpoint) ),  type V4
  • VDVertex( LineGenerator, LineGenerator, PointGenerator ), type V5
  • VDVertex( LineGenerator, LineGenerator, LineGenerator ), type V6

When these NEW-vertices (green) have been created, the next step is to connect them into a cycle with NEW-NEW edges (green). As we loop around the face we need to create either LineEdges or ParabolicEdges, depending on the neighbouring generators. Okabe lists the four different types of edges that appear in the diagram:

  • VDEdge( PointGenerator, PointGenerator), type E1 (LineEdge)
  • VDEdge( PointGenerator(endpoint), LineGenerator), type E2 (LineEdge)
  • VDEdge( PointGenerator, LineGenerator), type E3 (ParabolicEdge)
  • VDEdge( LineGenerator, LineGenerator), type E4 (LineEdge)

That's about it. It shouldn't be that hard to modify the current code to allow for six types of different VDVertex and four different types of VDEdge, and come up with a suitable design-pattern so that the correct type of vertex and edge are generated based on the type of the generators. Stay tuned...

Update1: here is another image with the same color-coding, from the Held VRONI-paper. The orange edge illustrates a case where the edge is marked "IN-IN" but in fact two new vertices should be generated on it, and the NEW-NEW portion of the edge is retained in the updated diagram (on the right).

Update2: Here is an image from the Held 2009 paper, which describes VDs for line-segment and circular arc generators. The circular arcs add a few additional special cases and some complexity, but the core of the algorithm remains the same: (1) find a seed vertex, (2) expand the set of 'marked' or "IN" vertices maximally, while keeping IN-IN edges (red) a tree, (3) generate NEW vertices (green) on IN-OUT edges (cyan), (4) hook up the NEW vertices with new edges (green) to form the new voronoi region, (5) delete IN-vertices, IN-IN edges, and IN-NEW edges.

Update3: Here's a drawing which shows the six different types of vertices.

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