Space-filling curve

A space-filling curve ( English space-filling curve ) is a line in the analysis , the (or the regular two-dimensional surface, or a multi-dimensional space lattice , the / describes this / n surface area) completely ( surjektiv passes). Such a curve cannot be bijective and continuous at the same time , since otherwise the unit interval and the unit square would have the same dimension ( theorem of the invariance of dimension ).

The standard example: the Hilbert curve

The acronym FASS curve stands for "space- f illing, self- a voiding, s imple and self- s imilar" (space-filling, self-evasive, simple and self-similar ). FASS curves fill the room.

Space-filling curves do not have to be self-evasive, they can also cross over themselves; they do not have to be self-similar either, according to Jens-Michael Wierum's βΩ curve.

Examples of space-filling curves are:

Remarks

↑ It is a quadrant-based, "face-continuous" (cell-continuous), "facet-gated" (cell-connected), but not self-similar closed 2-dimensional space-filling curve (see H. Haverkort, 2016 ) .

Individual evidence

↑ J.-M. Why. Definition of a new circular space-filling curve: βΩ-indexing. Technical Report TR-001-02, Paderborn Center for Parallel Computing (PC ² ) (2002)

literature

Hans Sagan: Space-Filling Curves. Springer-Verlag 1994.
Michael Bader: Space-Filling Curves - An Introduction with Applications in Scientific Computing. Springer-Verlag 2012.
Herman Haverkort: How many three-dimensional Hilbert curves are there? , 2016. Retrieved July 28, 2018.

[2] It is a quadrant-based, "face-continuous" (cell-continuous), "facet-gated" (cell-connected), but not self-similar closed 2-dimensional space-filling curve (see H. Haverkort, 2016 ) .

[1] J.-M. Why. Definition of a new circular space-filling curve: βΩ-indexing. Technical Report TR-001-02, Paderborn Center for Parallel Computing (PC ² ) (2002)