Visualization of constrained data by smooth rational fractal interpolation

Abstract

A family of smooth rational spline fractal interpolation function (SRFIF) is presented with the help of classical rational cubic spline. Each SRFIF of the family is identified uniquely by the values of vertical scaling factor $s_i$ and shape parameters ${\alpha }_i$ and ${\beta }_i$, and the shape of the fractal interpolating curves can be constrained and modified by selecting suitable parameters and vertical scaling factor. In order to meet the needs of practical design, a new method is developed to visualize constrained data in the view of constrained curves. In the special case, the methods of linear constraint and quadratic constraint are discussed. Also, convergence analysis shows that SRFIF gives a good approximation to the original function.

Keywords:

• fractals
• fractal rational splines
• curve modelling
• visualization of constrained data
• convergence analysis

2010 AMS Subject Classifications:

• 28A80
• 65D05
• 65D07
• 65D17

Acknowledgements

The authors are thankful to the anonymous referees for their helpful and valuable comments and suggestions in the development of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper is supported by the National Nature Science Foundation of China (No. 61070096) and the Nature Science Foundation of the Shandong Province of China (No. ZR2012FL05).