Arithmetic properties of homogeneous Hilbert curves

TL;DR

This paper investigates the properties of homogeneous Hilbert curves, exploring their affine transformations, analytical representations, and recursive relations to deepen understanding of their mathematical structure.

Contribution

It provides new analytical representations and recursive relations for homogeneous Hilbert curves, expanding theoretical understanding of their properties.

Findings

Properties of homogeneous Hilbert curves are characterized.

Affine transformations involved are analyzed.

Recursive relations of Hilbert curves are established.

Abstract

Properties of the recently reported homogeneous Hilbert curves are deduced and reported. The nature of the affine transformations involved in the construction of the Hilbert curves is explored. The analytical representation of proper and improper Hilbert curves is obtained. The one-to-one mapping between two Hilbert curves is deduced. Recursive relation of Hilbert curves is reported.