In Euclidean plane geometry, a pseudotriangle (pseudo-triangle) is the simply connected subset of the plane that lies between any three mutually tangent convex sets. Thus, it is a shape bounded by three inward-curved sides, formed by the boundaries of these three convex sets. A pseudotriangulation (pseudo-triangulations) is a partition of a region of the plane into pseudotriangles, and a pointed pseudotriangulation is a pseudotriangulation in which at each vertex the incident edges span an angle of less than π. Although the words "pseudotriangle" and "pseudotriangulation" have been used with various meanings in mathematics for much longer, the terms as used here were introduced in 1993 by Michel Pocchiola and Gert Vegter in connection with the computation of visibility relations and bitangents among convex obstacles in the plane. Pointed pseudotriangulations were first considered by Ileana Streinu (2000, 2005) as part of her solution to the carpenter's ruler problem, a proof that any simple polygonal path in the plane can be straightened out by a sequence of continuous motions. Pseudotriangulations have also been used for collision detection among moving objects and for dynamic graph drawing and shape morphing. Pointed pseudotriangulations arise in rigidity theory as examples of minimally rigid planar graphs, and in methods for placing guards in connection with the art gallery theorem. The shelling antimatroid of a planar point set gives rise to pointed pseudotriangulations, although not all pointed pseudotriangulations can arise in this way. For a detailed survey of much of the material discussed here, see Rote, Santos, and Streinu (2008).
Pseudotriangles Pocchiola and Vegter (1996a, 1996b, 1996c) originally defined a pseudotriangle to be a simply-connected region of the plane bounded by three smooth convex curves that are tangent at their endpoints. However, subsequent work has settled on a broader definition that applies more generally to polygons as well as to regions bounded by smooth curves, and that allows nonzero angles at the three vertices. In this broader definition, a pseudotriangle is a simply-connected region of the plane, having three convex vertices. The three boundary curves connecting these three vertices must be convex, in the sense that any line segment connecting two points on the same boundary curve must lie entirely outside or on the boundary of the pseudotriangle. Thus, the pseudotriangle is the region between the convex hulls of these three curves, and more generally any three mutually tangent convex sets form a pseudotriangle that lies between them. For algorithmic applications it is of particular interest to characterize pseudotriangles that are polygons. In a polygon, a vertex is convex if it spans an interior angle of less than π, and concave otherwise (in particular, we consider an angle of exactly π to be concave). Any polygon must have at least three convex angles because the total exterior angle of a polygon is 2π, the convex angles contribute less than π each to this total, and the concave angles contribute zero or negative amounts. A polygonal pseudotriangle is a polygon that has exactly three convex vertices. In particular, any triangle, and any nonconvex quadrilateral, is a pseudotriangle. The convex hull of any pseudotriangle is a triangle. The curves along the pseudotriangle boundary between each pair of convex vertices either lie within the triangle or coincide with one of its edges.
Pseudotriangulations A pseudotriangulation is a partition of a region of the plane into pseudotriangles. Any triangulation of a region of the plane is a pseudotriangulation. While any two triangulations of the same region must have the same numbers of edges and triangles, the same is not true of pseudotriangulations; for instance, if the region is itself an n-vertex polygonal pseudotriangle, then a pseudotriangulation of it may have as few as one pseudotriangle and n edges, or as many as n − 2 pseudotriangles and 2n − 3 edges. A minimal pseudotriangulation is a pseudotriangulation T such that no subgraph of T is a pseudotriangulation covering the same convex region of the plane. A minimal pseudotriangulation with n vertices must have at least 2n − 3 edges; if it has exactly 2n − 3 edges, it must be a pointed pseudotriangulation, but there exist minimal pseudotriangulations with 3n − O(1) edges. Agarwal et al. (2002) describe data structures for maintaining pseudotriangulations of moving points or moving polygons. They show that using pseudotriangulations in place of triangulations allows their algorithms to maintain these structures with relatively few combinatorial changes as the inputs move, and they use these dynamic pseudotriangulations to perform collision detection among the moving objects. Gudmundsson et al. (2004) consider the problem of finding a pseudotriangulation of a point set or polygon with minimum total edge length, and provide approximation algorithms for this problem.
Pointed pseudotriangulations
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