Calculation of dimensions of curves generated by subdivision schemes

ABSTRACT

Little attention has been paid to estimating dimensions of the curves generated by the subdivision algorithms. A unified method is proposed to estimate the dimension of curves generated by the arbitrary, stationary, linear subdivision schemes with given control points, based on a theorem about the Hausdorff dimension of iterated function systems. Several examples are given to demonstrate the implementation of the method, including the Koch curve, the uniform quadratic B-spline curve and the curves generated by the four-point binary and ternary interpolatory subdivision schemes with a free parameter. Compared with the method of the traditional iterated function system collage theorem, our algorithm overcomes the disadvantage of choosing points and collage, avoiding a large amount of calculation to find the contractive affine transformations and the contraction constants. Furthermore, we can calculate not only the dimension of the special curves with the geometric structure of self-similarity, but also the dimension of the curves generated by more general subdivision algorithms.

KEYWORDS:

• Dimension calculation
• subdivision schemes
• iterated function system
• subdivision curves

2010 MSC SUBJECT CLASSIFICATIONS:

• 14Q05
• 14Q15
• 51H05
• 51H99

Acknowledgments

We would like to thank the anonymous reviewers and the associate editor for many useful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Our work is supported by the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2016JM6056).