Analysis of multi-dimensional space-filling curves

Mohamed F. Mokbel, Walid G. Aref, Ibrahim Kamel

Research output: Contribution to journalArticlepeer-review

75 Scopus citations


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 languageEnglish (US)
Pages (from-to)179-209
Number of pages31
Issue number3
StatePublished - Sep 2003


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


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