### Abstract

A natural approach to defining continuous change of shape is in terms of a metric that measures the difference between two regions. We consider four such metrics over regions: the Hausdorff distance, the dual-Hausdorff distance, the area of the symmetric difference, and the optimal-homeomorphism metric (a generalization of the Fréchet distance). Each of these gives a different criterion for continuous change. We establish qualitative properties of all of these; in particular, the continuity of basic functions such as union, intersection, set difference, area, distance, and smoothed circumference and the transition graph between RCC-8 relations. We also show that the history-based definition of continuity proposed by Muller is equivalent to continuity with respect to the Hausdorff distance.

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

Pages (from-to) | 31-54 |

Number of pages | 24 |

Journal | Fundamenta Informaticae |

Volume | 46 |

Issue number | 1-2 |

State | Published - Apr 2001 |

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### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Safety, Risk, Reliability and Quality
- Applied Mathematics

### Cite this

*Fundamenta Informaticae*,

*46*(1-2), 31-54.

**Continuous shape transformation and metrics on regions.** / Davis, Ernest.

Research output: Contribution to journal › Article

*Fundamenta Informaticae*, vol. 46, no. 1-2, pp. 31-54.

}

TY - JOUR

T1 - Continuous shape transformation and metrics on regions

AU - Davis, Ernest

PY - 2001/4

Y1 - 2001/4

N2 - A natural approach to defining continuous change of shape is in terms of a metric that measures the difference between two regions. We consider four such metrics over regions: the Hausdorff distance, the dual-Hausdorff distance, the area of the symmetric difference, and the optimal-homeomorphism metric (a generalization of the Fréchet distance). Each of these gives a different criterion for continuous change. We establish qualitative properties of all of these; in particular, the continuity of basic functions such as union, intersection, set difference, area, distance, and smoothed circumference and the transition graph between RCC-8 relations. We also show that the history-based definition of continuity proposed by Muller is equivalent to continuity with respect to the Hausdorff distance.

AB - A natural approach to defining continuous change of shape is in terms of a metric that measures the difference between two regions. We consider four such metrics over regions: the Hausdorff distance, the dual-Hausdorff distance, the area of the symmetric difference, and the optimal-homeomorphism metric (a generalization of the Fréchet distance). Each of these gives a different criterion for continuous change. We establish qualitative properties of all of these; in particular, the continuity of basic functions such as union, intersection, set difference, area, distance, and smoothed circumference and the transition graph between RCC-8 relations. We also show that the history-based definition of continuity proposed by Muller is equivalent to continuity with respect to the Hausdorff distance.

UR - http://www.scopus.com/inward/record.url?scp=0035301827&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035301827&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0035301827

VL - 46

SP - 31

EP - 54

JO - Fundamenta Informaticae

JF - Fundamenta Informaticae

SN - 0169-2968

IS - 1-2

ER -