ISSN:
1432-0541
Keywords:
Polygonal approximation
;
Algorithmic paradigms
;
Shape approximation
;
Computational geometry
;
Implicit complexity parameters
;
Banach-Mazur metric
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
Abstract For compact Euclidean bodiesP, Q, we define λ(P, Q) to be the smallest ratior/s wherer 〉 0,s 〉 0 satisfy $$sQ' \subseteq P \subseteq rQ''$$ . HeresQ denotes a scaling ofQ by the factors, andQ′,Q″ are some translates ofQ. This function λ gives us a new distance function between bodies which, unlike previously studied measures, is invariant under affine transformations. If homothetic bodies are identified, the logarithm of this function is a metric. (Two bodies arehomothetic if one can be obtained from the other by scaling and translation.) For integerk ≥ 3, define λ(k) to be the minimum value such that for each convex polygonP there exists a convexk-gonQ with λ(P, Q) ≤ λ(k). Among other results, we prove that 2.118 ... 〈-λ(3) ≤ 2.25 and λ(k) = 1 + Θ(k −2). We give anO(n 2 log2 n)-time algorithm which, for any input convexn-gonP, finds a triangleT that minimizes λ(T, P) among triangles. However, in linear time we can find a trianglet with λ(t, P)〈-2.25. Our study is motivated by the attempt to reduce the complexity of the polygon containment problem, and also the motion-planning problem. In each case we describe algorithms which run faster when certain implicitslackness parameters of the input are bounded away from 1. These algorithms illustrate a new algorithmic paradigm in computational geometry for coping with complexity.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01758852
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