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Data dołączenia: 14 maj 2022
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Q: Plane-tiling algorithms and polygon splitting I was thinking of a method of tiling a plane using non-overlapping quadrilaterals. (I do not know of any tiling algorithm that can guarantee non-overlapping shapes. In fact, I'm surprised that tiling algorithms can be proven to exist). For this example, I'll just use quadrilaterals. I'm also interested in splitting polygons. A single vertex of a quadrilateral is called a quadrilateral divider. There are two options here: Split a single vertex of a polygon into two smaller polygons Split a polygon into two smaller polygons by cutting a polygon along a line that contains exactly one quadrilateral divider Here is a visualization: Q1: What is the best known algorithm for option 1? Q2: What is the best known algorithm for option 2? A: There is an algorithm to split a polygon into three smaller polygons. The question is where the three smaller polygons split the polygon, i.e. where the three edges are perpendicular to the diagonal line. This can be done by scanning the polygon from one edge to the other, and calculating the angle (or vector) for each edge. The edge that has the maximum angle is the one that will be split. There are some variations of this algorithm. The best known version is due to Patkotirakkawat and Ulrich. The paper is available on the Arxiv (note that I haven't looked into the algorithm for 30+ years). Edit: Using a similar algorithm, you can split a polygon into four smaller polygons. The edges that are perpendicular to the diagonal line should be points of inflection of the polygon. These points are typically the intersections of lines perpendicular to the diagonal line. Edit: There is a paper on this topic, where a search with the keywords'split polygon' was performed on PUBMED. Here is a link to the paper (doubt that I can post more than two links at once). A: I know that it's not a direct answer to the question, but I'd like to point out that the second picture is not a tiling. A tiling uses regular shapes, whereas the second one uses irregular shapes. A regular tiling of a square is

 

 

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