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Arie Bos

Publications and source records attributed to Arie Bos.

4 recordsLinked to original sources
arXiv2022

Fractal Images as Number Sequences I An Introduction

In this article, we considered a fractal image as a fractal curve, that is, as a walk on a grid in Euclidean space $\R^d$. We placed integers on the generating vectors of a grid, such that opposite directions have opposite numbers. This numbering system converts a curve on that grid into a sequence of integers, corresponding with the curve's edges. The corresponding sequence contains the same fractal structure, i.e., an approximant of the curve corresponds to that of the sequence. We introduced a normalized sequence which is unique for a curve. The morphisms of the grid generators were translated into signed permutations on the alphabet of all the numbers used. By ordering the fractal sequences, we obtained an encyclopedia of fractals. A variety of examples and images enriched the text.

cs.CG↗
arXiv2015

Hyperorthogonal well-folded Hilbert curves

R-trees can be used to store and query sets of point data in two or more dimensions. An easy way to construct and maintain R-trees for two-dimensional points, due to Kamel and Faloutsos, is to keep the points in the order in which they appear along the Hilbert curve. The R-tree will then store bounding boxes of points along contiguous sections of the curve, and the efficiency of the R-tree depends on the size of the bounding boxes---smaller is better. Since there are many different ways to generalize the Hilbert curve to higher dimensions, this raises the question which generalization results in the smallest bounding boxes. Familiar methods, such as the one by Butz, can result in curve sections whose bounding boxes are a factor $Ω(2^{d/2})$ larger than the volume traversed by that section of the curve. Most of the volume bounded by such bounding boxes would not contain any data points. In this paper we present a new way of generalizing Hilbert's curve to higher dimensions, which results in much tighter bounding boxes: they have at most 4 times the volume of the part of the curve covered, independent of the number of dimensions. Moreover, we prove that a factor 4 is asymptotically optimal.

cs.CG↗