Space-Filling Curves

Mohamed F. Mokbel
Walid G. Aref

Encyclopedia of Database Systems, 2009, Part 19, 2674-2675


A space-filling curve (SFC) is a way of mapping the multi-dimensional space into the one-dimensional space. It acts like a thread that passes through every cell element (or pixel) in the multi-dimensional space so that every cell is visited exactly once. Thus, a space-filling curve imposes a linear order of points in the multi-dimensional space. A D-dimensional spacefilling curve in a space of N cells (pixels) of each dimension consists of ND  1 segments where each segment connects two consecutive D-dimensional points. There are numerous kinds of space-filling curves (e.g., Hilbert, Peano, and Gray). The difference between such curves is in their way of mapping to the one-dimensional space, i.e., the order that a certain space-filling curve traverses the multi-dimensional space. The quality of a space-filling curve is measured by its ability in preserving the locality (or relative distance) of multi-dimensional points in the mapped one-dimensional space. The main idea is that any two D-dimensional points that are close by in the 2674S Source D-dimensional space should be also close by in the one-dimensional space.