Polygon - how to query a voronoi diagram?

Belle Windler asked a question: Polygon - how to query a voronoi diagram?
Asked By: Belle Windler
Date created: Wed, Feb 17, 2021 7:12 PM



Those who are looking for an answer to the question «Polygon - how to query a voronoi diagram?» often ask the following questions:

❓ C++ - how to construct a voronoi diagram inside a polygon?

Do a Delaunay triangulation of your polygon first: Calculate section points between segments; add these points, split the segments and rearrange the input so that "inside" is always on the same side of the edge when traversing the polygon's points. Trangulate all points in your polygon. Delete the triangles that lie outside your polygon.

❓ How to draw a voronoi diagram using a compass?

  • PRO-TIP: User nax left a great tip for finding the perpendicular bisectors using a compass. Basically, what you do is place the point of your compass on one of the the nodes, extend it roughly past the middle of the segment, and draw an arc above and below the line.

❓ How to make a voronoi diagram of oblong island?

  • This is a diagram of Oblong Island which, amazingly, is in the shape of a rectangle. Any location on the island can be defined by a set of coordinates. There are seven camping sites on the island labelled A to G. 1. What are the coordinates of site C ? [Type in your answer as two numbers separated by a comma] 2. What are the coordinates of site B ?

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I am using boost to compute the voronoi diagram of a set of points in 2-D, very straightforward; std::vector points; ... voronoi_diagram vd; construct_voronoi(points.begin(), points.end(), &vd); Is there an algorithm to process the resulting polygons such that the query "Which site does a given point belong to?"

Construction of the Voronoi Diagram At this point we are ready to construct the Voronoi diagram: voronoi_diagram vd; construct_voronoi(points.begin(), points.end(), segments.begin(), segments.end(), &vd); Traversing Voronoi Graph Voronoi graph traversal is the basic operation one would like to do once the Voronoi diagram is constructed.

Given a Voronoi diagram and a query point, how do we tell which cell a query falls into? (I.e., solve the 1-NN problem) We can project down to the x-axis every point in the Voronoi diagram –This gives us a bunch of “slabs” –We can find which slab our query is in by using binary search –Within a slab, we can find the Voronoi cell

While constructing the Voronoi diagram today is as simple as providing cell site points in a two dimensional form (e.g. a point is {x: 1, y: 2} in a given array of such objects, rendering heightmap does have added work. The problem here involves using such polygon to form the points in polygon which then can be used to render the heightmap.

Given Voronoi diagram, how to locate which polygon contains query point q? What if q is one of the Voronoi sites, would it make any difference? Do we walk through each polygon and check point-in-

Find the closest points on the buffer of concave hull nearest to points in step 2. These are show as green in the diagram below. Add these points to original data set. Calculate the Voronoi diagram of this combined data set. As can be seen in the 3rd diagram, the points on the hull now have enclosing polygons.

This produces the following set of polygons for the sample points included in the query . Query Explanation Step 1. Create the Delaunay Triangles from the input geometries. SELECT (gd).Path id, ST_ExteriorRing((gd).Geom) g -- ID and make triangle a linestring FROM (SELECT (ST_Dump(ST_DelaunayTriangles(geom))) gd FROM Sample) a -- Get Delaunay Triangles

The region defined by V(p i ) = {p|d (p,p i)≤d(p, p j)} for j≠i, j∈I n where, d (p, p i ) denotes the distance between p and p i, is called the Voronoi Polygon associated with p i and the set given by: VD(P) = {V(p 1),…, V(p n)} is called the Voronoi Diagram (also called the first order Voronoi diagram) generated by P and the boundaries of the polygons are called Voronoi edges.

The Voronoi diagram of a set of points is dual to its Delaunay triangulation. The Voronoi diagram is named after Georgy Voronoy, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons.

So use your own data structure (tree) alongside the diagram: as points sites are inserted into the voronoi diagram, insert the same points into the tree. Query the tree to find the (N) nearest voronoi site (s). Then it's up to you about how to map that site coordinate to a voronoi cell object. Share.

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