Abstract
In this paper, we construct the $$\mathscr {C}^1$$ -rational cubic fractal interpolation function (RCFIF) and its application in preserving the constrained nature of a given data set. The $$\mathscr {C}^1$$ -RCFIF is the fractal design of the traditional rational cubic interpolant of the form $$\frac{p_i(\theta )}{q_i(\theta )}$$ , where $$p_i(\theta )$$ and $$q_i(\theta )$$ are the cubic polynomials with three tension parameters. We derive the uniform error bound between the RCFIF with the original function in $$\mathscr {C}^3[x_1,x_n]$$ . When the data set is constrained between two piecewise straight lines, we deduce the sufficient conditions on the parameters of the RCFIF so that it lies between those two lines. Numerical examples are given to support that our method is interactive and smooth.
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