Ansgar Grüne

Let K ⊂ R be a compact convex set in the plane. A halving chord of K is a line segment pp̂, p, p̂ ∈ ∂K, which divides the area of K into two equal parts. For every direction v there exists exactly one halving chord. Its length hA(v) is the corresponding (area) halving distance. In this article we give inequalities relating the minimum and maximum (area)(More)
• Comput. Geom.
• 2007
Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the ratio between the length of a shortest path connecting p and q in G and their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers-Baumann, Grüne and Klein have shown(More)
• ISAAC
• 2003
Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The maximum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph(More)
• Algorithmica
• 2005
Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph(More)
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• Algorithms
• 2009
We consider a pursuit-evasion problem where some lions have the task to clear a grid graph whose nodes are initially contaminated. The contamination spreads one step per time unit in each direction not blocked by a lion. A vertex is cleared from its contamination whenever a lion moves to it. Brass et al. [5] showed that n 2 lions are not enough to clear the(More)
Let G be a geometric graph in the plane whose edges may be curves. For two arbitrary points on its edges, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set S, we would like to know the smallest possible(More)
• Comput. Geom.
• 2008
Given S1, a finite set of points in the plane, we define a sequence of point sets Si as follows: With Si already determined, let Li be the set of all the line segments connecting pairs of points of ⋃i j=1 Sj , and let Si+1 be the set of intersection points of those line segments in Li, which cross but do not overlap. We show that with the exception of some(More)
A pride of lions are prowling among the vertices and edges of an n×n grid. If their paths are known in advance, is it possible to design a safe path for a man that avoids all lions, assuming that man and lion move at the same speed? In their recent paper [4], Dumitrescu et al. employed probabilistic arguments to show that O( √ n) lions can always be(More)
• EuroCG
• 2005
Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the length of a shortest path connecting p and q in G divided by their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). EbbersBaumann, Grüne and Klein have shown that every(More)
Let G be an embedded planar graph whose edges may be curves. The detour between two points, p and q (on edges or vertices) of G, is the ratio between the shortest path in G between p and q and their Euclidean distance. The supremum over all pairs of points of all these ratios is called the geometric dilation of G. Our research is motivated by the problem of(More)