Bounds on the number of discontinuities of Morton-type space-filling curves

TL;DR

This paper investigates the discontinuities of Morton-type space-filling curves, establishing bounds on the number of face-connected subdomains they produce, and provides theoretical and computational insights into their jump frequency.

Contribution

It proves bounds on the number of subdomains created by Morton-type curves in various dimensions and analyzes their jump behavior both theoretically and computationally.

Findings

For hypercube cases, the number of subdomains is bounded by two.

For simplicial cases in 2D and 3D, the bound grows with refinement depth.

The study includes both theoretical proofs and computational analysis of jump frequency.

Abstract

The Morton- or z-curve is one example for a space filling curve: Given a level of refinement L, it maps the interval [0, 2**dL) one-to-one to a set of d-dimensional cubes of edge length 2**-L that form a subdivision of the unit cube. Similar curves have been proposed for triangular and tetrahedral unit domains. In contrast to the Hilbert curve that is continuous, the Morton-type curves produce jumps. We prove that any contiguous subinterval of the curve divides the domain into a bounded number of face-connected subdomains. For the hypercube case and arbitrary dimension, the subdomains are star-shaped and the bound is indeed two. For the simplicial case in dimensions 2 and 3, the bound is proportional to the depth of refinement L. We supplement the paper with theoretical and computational studies on the frequency of jumps for a quantitative assessment.