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Polygon triangulation. In computational geometry, polygon triangulation is the partition of a polygonal area (simple polygon) P into a set of triangles, [1] i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is P. Triangulations may be viewed as special cases of planar straight-line graphs.
For a hole-free polygon with vertices, a triangulation can be calculated in time (). For a polygon with holes , there is a lower bound of Ω ( n log n ) {\displaystyle \Omega (n\log n)} . A related problem is partitioning to triangles with a minimal total edge length, also called minimum-weight triangulation .
However, this polygon also has other ears that are not evident in this triangulation. In geometry , the two ears theorem states that every simple polygon with more than three vertices has at least two ears , vertices that can be removed from the polygon without introducing any crossings.
Cell is the intersection of all of these half-spaces, and hence it is a convex polygon. [6] When two cells in the Voronoi diagram share a boundary, it is a line segment , ray , or line, consisting of all the points in the plane that are equidistant to their two nearest sites.
Compute a center for the polygon face, e.g. the average of all its vertices. Connecting the vertices of a polygon face with its center gives a planar umbrella-shaped triangulation. Trivially, a set of tetrahedra is obtained by connecting triangles of the cell's hull with the cell's site.
The following pseudocode describes a basic implementation of the Bowyer-Watson algorithm. Its time complexity is ().Efficiency can be improved in a number of ways. For example, the triangle connectivity can be used to locate the triangles which contain the new point in their circumcircle, without having to check all of the triangles - by doing so we can decrease time complexity to ().
The triangulation number is T = b 2 + bc + c 2. This number times the number of original faces expresses how many triangles the new polyhedron will have. This number times the number of original faces expresses how many triangles the new polyhedron will have.
Only 4 of those 15 chords occur in the 16-cell, 8-cell and 24-cell. The four hypercubic chords √ 1, √ 2, √ 3 and √ 4 are sufficient to build the 24-cell and all its component parts. The 24-cell is the unique solution to the combination of these 4 chords and all the regular polytopes that can be built from them.