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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.
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.
Notable among the new parts found in the 120-cell which do not occur in the 600-cell are regular 5-cells and √ 5/2 chords. [ ba ] The relationships between the regular 5-cell (the simplex regular 4-polytope) and the other regular 4-polytopes are manifest directly only in the 120-cell.
(Think of geographical regions where each region is defined as all the points closest to a given city or post office.) [53] [54] The Voronoi cell for each defining point is a convex polygon. The Delaunay triangulation is a tessellation that is the dual graph of a Voronoi tessellation.
An ear of a polygon is defined as a triangle formed by three consecutive vertices ,, of the polygon, such that its edge lies entirely in the interior of the polygon. The two ears theorem states that every simple polygon that is not itself a triangle has at least two ears.
In geometry, a polygon with holes is an area-connected planar polygon with one external boundary and one or more interior boundaries (holes). [1] Polygons with holes can be dissected into multiple polygons by adding new edges, so they are not frequently needed. An ordinary polygon can be called simply-connected, while a polygon-with-holes is ...
Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F − E + V = 2).
It has 10 chapters, whose topics include the original art gallery theorem and Fisk's triangulation-based proof; rectilinear polygons; guards that can patrol a line segment rather than a single point; special classes of polygons including star-shaped polygons, spiral polygons, and monotone polygons; non-simple polygons; prison yard problems, in ...