### 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 |
---|---|

Pages (from-to) | 179-209 |

Number of pages | 31 |

Journal | GeoInformatica |

Volume | 7 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Sep 2003 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Geography, Planning and Development
- Information Systems

### Cite this

*GeoInformatica*,

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

**Analysis of multi-dimensional space-filling curves.** / Mokbel, Mohamed F.; Aref, Walid G.; Kamel, Ibrahim.

Research output: Contribution to journal › Article

*GeoInformatica*, vol. 7, no. 3, pp. 179-209. https://doi.org/10.1023/A:1025196714293

}

TY - JOUR

T1 - Analysis of multi-dimensional space-filling curves

AU - Mokbel, Mohamed F.

AU - Aref, Walid G.

AU - Kamel, Ibrahim

PY - 2003/9/1

Y1 - 2003/9/1

N2 - 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.

AB - 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.

KW - Fractals

KW - Locality-preserving mapping

KW - Performance analysis

KW - Space-filling curves

UR - http://www.scopus.com/inward/record.url?scp=0141682542&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141682542&partnerID=8YFLogxK

U2 - 10.1023/A:1025196714293

DO - 10.1023/A:1025196714293

M3 - Article

AN - SCOPUS:0141682542

VL - 7

SP - 179

EP - 209

JO - GeoInformatica

JF - GeoInformatica

SN - 1384-6175

IS - 3

ER -