![]() ![]() We may still decide to represent it by the path of edges connecting the points.Īnother nasty case is co-circular points. Also in this case, the dual graph is no longer a triangulation. In this case, the Voronoi diagram has all edges connecting two points at infinity, and the Voronoi diagram is disconnected (in all other cases it's connected). The first case is when all the points in $$$S$$$ are contained in a line. The Euclidean minimum spanning tree is a subset of Delaunay edges.īecause this is geometry, there are some degenerate cases I have hidden from you up to this point.The Delaunay triangulation maximizes the minimum angle in the triangles among all possible triangulations.Each point $$$p\in S$$$ is adjacent to its nearest neighbor with a Delaunay edge. ![]() The number of Delaunay edges is at most $$$3n-6$$$, so there is hope for an efficient construction.(In fact, you can also consider this the defining property of Delaunay triangulation) For each Delaunay triangle, its circumcircle does not strictly contain any points in $$$S$$$.Other than this, the Delaunay triangulation has many amazing properties of its own that can make it extremely useful for solving problems. Or it can be done online with persistent data structures. This can be done in $$$O((n + q)\log(n+q))$$$ offline by sweeping the Voronoi diagram and query points. Given a set $$$S$$$ of $$$n$$$ points and $$$m$$$ query points $$$p_1,\ldots, p_m$$$, we can answer for each query point, its nearest neighbor in $$$S$$$. The definition of the Voronoi diagram immediately shows signs of applications. For this reason, it is often more convenient to compute the Delaunay triangulation and only convert to the Voronoi diagram if needed. The Delaunay triangulation only involves edges between existing points, while the Voronoi diagram creates new vertices and needs a way to represent edges going infinitely in one direction. The black points and edges are the Delaunay triangulation. The red vertices and edges are the Voronoi diagram. In the following image, the set $$$S$$$ corresponds to the black points. A Delaunay edge connects two points if and only if their corresponding Voronoi regions share a border. ![]() That is, a Voronoi vertex is a Delaunay face and a Delaunay vertex is a Voronoi region. The Delaunay triangulation is the dual graph of the Voronoi diagram. Recall that the circumcenter is the intersection of perpendicular bisectors. Similarly, a Voronoi vertex is equidistant to three or more vertices in $$$S$$$, so it is their circumcenter. The way the regions were defined, we see that a segment contains the points equidistant to two points in $$$S$$$, so it is part of their perpendicular bisector. A vertex in this graph is where three or more segments meet (or a point at infinity), and an edge is a segment connecting two of these vertices. The Voronoi diagram can be represented by the planar graph of boundaries between regions. In other words, a point $$$q$$$ belongs to the region of its nearest neighbor. The region associated with a point $$$p\in S$$$ contains precisely the points $$$q$$$ such that $$$q$$$ is closer to $$$p$$$ than any other point in $$$S$$$. +1 because it's a unique tool with no equivalent, it's free and many people like it. My first frustration might come from the fact that I developed a tool (the websvg github voronoi) and know what are all the cool options a vornoi tool could have, but admittedly, I'm no competitor of this tool as I don't plan to challenge a Fusion360 plugin development, and I'd be happy if you take over any feature to this plugin.Given a set $$$S$$$ of $$$n$$$ points in the 2D plane, the Voronoi diagram is a partition of the plane into regions. Kellner, you're right, I edited my review, I put 4 stars, actually 3 +1. My first frustration might come from the fact that I developed a tool (the websvg github voronoi) and know what are all the cool options a vornoi tool could have, but admittedly, I'm no competitor of this tool as I don't plan to challenge a Fusion360 plugin development, and I'd be happy if you take over any feature to this plugin. +1 because it's a unique tool with no equivalent, it's free and many people like it. You're right, I edited my review, I put 4 stars, actually 3 +1. ![]()
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