Abstract
For compact Euclidean bodies P, Q, we define λ(P, Q) to be the smallest ratio r/s where r > 0, s > 0 satisfy {Mathematical expression}. Here sQ denotes a scaling of Q by the factor s, and Q′, Q″ are some translates of Q. 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 are homothetic if one can be obtained from the other by scaling and translation.) For integer k ≥ 3, define λ(k) to be the minimum value such that for each convex polygon P there exists a convex k-gon Q 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 an O(n 2 log2 n)-time algorithm which, for any input convex n-gon P, finds a triangle T that minimizes λ(T, P) among triangles. However, in linear time we can find a triangle t 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 implicit slackness parameters of the input are bounded away from 1. These algorithms illustrate a new algorithmic paradigm in computational geometry for coping with complexity.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 365-389 |
| Number of pages | 25 |
| Journal | Algorithmica (New York) |
| Volume | 8 |
| Issue number | 1-6 |
| DOIs | |
| State | Published - Dec 1992 |
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Keywords
- Algorithmic paradigms
- Banach-Mazur metric
- Computational geometry
- Implicit complexity parameters
- Polygonal approximation
- Shape approximation
ASJC Scopus subject areas
- Applied Mathematics
- Safety, Risk, Reliability and Quality
- Software
- Computer Graphics and Computer-Aided Design
- Computer Science Applications
- Computer Science(all)
Cite this
Simultaneous inner and outer approximation of shapes. / Fleischer, Rudolf; Mehlhorn, Kurt; Rote, Günter; Welzl, Emo; Yap, Chee.
In: Algorithmica (New York), Vol. 8, No. 1-6, 12.1992, p. 365-389.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Simultaneous inner and outer approximation of shapes
AU - Fleischer, Rudolf
AU - Mehlhorn, Kurt
AU - Rote, Günter
AU - Welzl, Emo
AU - Yap, Chee
PY - 1992/12
Y1 - 1992/12
N2 - For compact Euclidean bodies P, Q, we define λ(P, Q) to be the smallest ratio r/s where r > 0, s > 0 satisfy {Mathematical expression}. Here sQ denotes a scaling of Q by the factor s, and Q′, Q″ are some translates of Q. 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 are homothetic if one can be obtained from the other by scaling and translation.) For integer k ≥ 3, define λ(k) to be the minimum value such that for each convex polygon P there exists a convex k-gon Q 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 an O(n 2 log2 n)-time algorithm which, for any input convex n-gon P, finds a triangle T that minimizes λ(T, P) among triangles. However, in linear time we can find a triangle t 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 implicit slackness parameters of the input are bounded away from 1. These algorithms illustrate a new algorithmic paradigm in computational geometry for coping with complexity.
AB - For compact Euclidean bodies P, Q, we define λ(P, Q) to be the smallest ratio r/s where r > 0, s > 0 satisfy {Mathematical expression}. Here sQ denotes a scaling of Q by the factor s, and Q′, Q″ are some translates of Q. 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 are homothetic if one can be obtained from the other by scaling and translation.) For integer k ≥ 3, define λ(k) to be the minimum value such that for each convex polygon P there exists a convex k-gon Q 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 an O(n 2 log2 n)-time algorithm which, for any input convex n-gon P, finds a triangle T that minimizes λ(T, P) among triangles. However, in linear time we can find a triangle t 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 implicit slackness parameters of the input are bounded away from 1. These algorithms illustrate a new algorithmic paradigm in computational geometry for coping with complexity.
KW - Algorithmic paradigms
KW - Banach-Mazur metric
KW - Computational geometry
KW - Implicit complexity parameters
KW - Polygonal approximation
KW - Shape approximation
UR - http://www.scopus.com/inward/record.url?scp=52449148577&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=52449148577&partnerID=8YFLogxK
U2 - 10.1007/BF01758852
DO - 10.1007/BF01758852
M3 - Article
VL - 8
SP - 365
EP - 389
JO - Algorithmica
JF - Algorithmica
SN - 0178-4617
IS - 1-6
ER -