### Abstract

Let S be a family of m convex polygons in the plane with a total number of n vertices and let each polygon have a positive weight associated with it. This paper presents algorithms to solve the weighted minmax approximation and the weighted minsum approximation problems. For the first problem, a line minimizing the maximum weighted orthogonal Euclidean distance to the polygons can be found in O(n^{2}logn) time and O(n^{2}) space. The time and space complexities can be reduced to O(n log n) and O(n), respectively, when the weights are equal. For the second problem, a line minimizing the sum of the weighted distances to the polygons can be found in O(nm log m) time and O(n) space. For both problems, we also consider constrained versions of these problems.

Original language | English (US) |
---|---|

Pages (from-to) | 27-52 |

Number of pages | 26 |

Journal | Computational Geometry: Theory and Applications |

Volume | 4 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1994 |

### Fingerprint

### Keywords

- Dual transform
- Line sweep algorithms
- Linear approximation

### ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Computational Geometry: Theory and Applications*,

*4*(1), 27-52. https://doi.org/10.1016/0925-7721(94)90015-9

**Linear approximation of simple objects.** / Robert, Jean Marc; Toussaint, Godfried.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 4, no. 1, pp. 27-52. https://doi.org/10.1016/0925-7721(94)90015-9

}

TY - JOUR

T1 - Linear approximation of simple objects

AU - Robert, Jean Marc

AU - Toussaint, Godfried

PY - 1994/1/1

Y1 - 1994/1/1

N2 - Let S be a family of m convex polygons in the plane with a total number of n vertices and let each polygon have a positive weight associated with it. This paper presents algorithms to solve the weighted minmax approximation and the weighted minsum approximation problems. For the first problem, a line minimizing the maximum weighted orthogonal Euclidean distance to the polygons can be found in O(n2logn) time and O(n2) space. The time and space complexities can be reduced to O(n log n) and O(n), respectively, when the weights are equal. For the second problem, a line minimizing the sum of the weighted distances to the polygons can be found in O(nm log m) time and O(n) space. For both problems, we also consider constrained versions of these problems.

AB - Let S be a family of m convex polygons in the plane with a total number of n vertices and let each polygon have a positive weight associated with it. This paper presents algorithms to solve the weighted minmax approximation and the weighted minsum approximation problems. For the first problem, a line minimizing the maximum weighted orthogonal Euclidean distance to the polygons can be found in O(n2logn) time and O(n2) space. The time and space complexities can be reduced to O(n log n) and O(n), respectively, when the weights are equal. For the second problem, a line minimizing the sum of the weighted distances to the polygons can be found in O(nm log m) time and O(n) space. For both problems, we also consider constrained versions of these problems.

KW - Dual transform

KW - Line sweep algorithms

KW - Linear approximation

UR - http://www.scopus.com/inward/record.url?scp=0041305423&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041305423&partnerID=8YFLogxK

U2 - 10.1016/0925-7721(94)90015-9

DO - 10.1016/0925-7721(94)90015-9

M3 - Article

VL - 4

SP - 27

EP - 52

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 1

ER -