### 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} log^{2} 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 |

### Fingerprint

### 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

*Algorithmica (New York)*,

*8*(1-6), 365-389. https://doi.org/10.1007/BF01758852

**Simultaneous inner and outer approximation of shapes.** / Fleischer, Rudolf; Mehlhorn, Kurt; Rote, Günter; Welzl, Emo; Yap, Chee.

Research output: Contribution to journal › Article

*Algorithmica (New York)*, vol. 8, no. 1-6, pp. 365-389. https://doi.org/10.1007/BF01758852

}

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

AN - SCOPUS:0000099132

VL - 8

SP - 365

EP - 389

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 1-6

ER -