Franz Aurenhammer

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Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason(More)
The power pow (x, s) of a point x with respect to a sphere s in Euclidean d-space E d is given by d2(x, z) 2, where d denotes the Euclidean distance function, and z and are the center and the radius of s. The power diagram of a finite set S of spheres in E d is a cell complex that associates each s S with the convex domain {x E d Ipow (x, s) < pow (x, t),(More)
A new internal structure for simple polygons, the straight skeleton, is introduced and discussed. It is composed of pieces of angular bisectores which partition the interior of a given n-gon P in a tree-like fashion into n monotone polygons. Its straight-line structure and its lower combinatorial complexity may make the straight skeleton preferable to the(More)
Dissecting Euclidean d -space with the power diagram of n weighted point sites partitions a given m -point set into clusters, one cluster for each region of the diagram. In this manner, an assignment of points to sites is induced. We show the equivalence of such assignments to constrained Euclidean least-squares assignments. As a corollary, there always(More)
-Let S denote a set ofn points in the plane such that each point p has assigned a positive weight w(p) which expresses its capability to influence its neighbourhood. In this sense, the weighted distance of an arbitrary point x from p is given by de(x, p)/w(p) where d e denotes the Euclidean distance function. The weighted Voronoi diagram for S is a(More)
The farthest line segment Voronoi diagram shows properties different from both the closest-segment Voronoi diagram and the farthest-point Voronoi diagram. Surprisingly, this structure did not receive attention in the computational geometry literature. We analyze its combinatorial and topological properties and outline an O(n log n) time construction(More)
We present a very simple algorithm for maintaining order-k Voronoi diagrams in the plane. By using a duatity transform that is of interest in its own right we show that the insertion or deletion of a site involves little more than the construction of a single convex hull in three-space. In particular, the order-k Voronoi dlagratn for n sites can(More)
A criterion is given that decides, for a convex tiling C of R d, whether C is the projection of the faces in the boundary of some convex polyhedron P in R d+l. Its applicability is restricted neither by the properties nor by the dimension of C. It turns out to be conceptually simpler than other criteria and allows the easy examination of various classes of(More)