### Abstract

A space-filling curve is a way of mapping the multi-dimensional space into the 1-D space. It acts like a thread that passes through every cell element (or pixel) in the D-dimensional space so that every cell is visited exactly once. There are numerous kinds of space-filling curves. The difference between such curves is in their way of mapping to the 1-D space. Selecting the appropriate curve for any application requires knowledge of the mapping scheme provided by each space-filling curve. A space-filling curve consists of a set of segments. Each segment connects two consecutive multi-dimensional points. Five different types of segments are distinguished, namely, Jump, Contiguity, Reverse, Forward, and Still. A description vector V = (J,C,R,F,S), where J,C,R,F, and S are the percentages of Jump, Contiguity, Reverse, Forward, and Still segments in the space-filling curve, encapsulates all the properties of a space-filling curve. The knowledge of V facilitates the process of selecting the appropriate space-filling curve for different applications. Closed formulas are developed to compute the description vector V for any D-dimensional space and grid size N for different space-filling curves. A comparative study of different space-filling curves with respect to the description vector is conducted and results are presented and discussed.

Original language | English |
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Pages (from-to) | 179-209 |

Number of pages | 31 |

Journal | GeoInformatica |

Volume | 7 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Sep 2003 |

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### Keywords

- Fractals
- Locality-preserving mapping
- Performance analysis
- Space-filling curves

### ASJC Scopus subject areas

- Information Systems
- Geography, Planning and Development

### Cite this

*GeoInformatica*,

*7*(3), 179-209. https://doi.org/10.1023/A:1025196714293