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  • 1
    Electronic Resource
    Electronic Resource
    Lines in space: Combinatorics and algorithms (1996)
    Springer
    Algorithmica 15 (1996), S. 428-447 
    Details
    ISSN: 1432-0541
    Keywords: Computational geometry ; Lines in space ; Plücker coordinates ; ε-Nets
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Questions about lines in space arise frequently as subproblems in three-dimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements ofn lines in three-dimensional space. Our main results include: 1. A tight Θ(n 2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to then given lines. 2. A similar bound of Θ(n 3) for the complexity of the set of all lines passing above then given lines. 3. A preprocessing procedure usingO(n 2+ɛ) time and storage, for anyε〉0, that builds a structure supportingO(logn)-time queries for testing if a line lies above all the given lines. 4. An algorithm that tests the “towering property” inO(n 2+ɛ) time, for anyε〉0; don given red lines lie all aboven given blue lines? The tools used to obtain these and other results include Plücker coordinates for lines in space andε-nets for various geometric range spaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01955043
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Ray shooting in polygons using geodesic triangulations (1994)
    Springer
    Algorithmica 12 (1994), S. 54-68 
    Details
    ISSN: 1432-0541
    Keywords: Computational geometry ; Ray-shooting ; Triangulation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract LetP be a simple polygon withn vertices. We present a simple decomposition scheme that partitions the interior ofP intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles. This decomposition can be used to preprocessP in a very simple manner, so that any ray-shooting query can be answered in timeO(logn). The data structure requiresO(n) storage andO(n logn) preprocessing time. By using more sophisticated techniques, we can reduce the preprocessing time toO(n). We also extend our general technique to the case of ray shooting amidstk polygonal obstacles with a total ofn edges, so that a query can be answered inO(√ logn) time.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01377183
    Library Location Call Number Volume/Issue/Year Availability
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    Paper (German National Licenses)
    Fulltext
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