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Fractal interpolation of polygons

Fractal interpolation of polygons is a local procedure which applies to polyline model of curve. This model represents non-local features of curve, or shape. The curve created by fractal interpolation has fine structure (texture) which is fully matched with it's shape. For flat curve fractal interpolation can be easily described using complex coordinates. Let z_n be vertex of polygon. Fractal interpolation insert two point z_n+1/3 and z_n+2/3 between z_n and z_n+1, such that

${\begin{aligned}{\frac {Z_{n+{\frac {1}{3}}}-Z_{n}}{Z_{n+1}-Z_{n}}}&={\frac {Z_{n}-Z_{n-1}}{Z_{n+2}-Z_{n-1}}}\\{\frac {Z_{n+{\frac {2}{3}}}-Z_{n}}{Z_{n+1}-Z_{n}}}&={\frac {Z_{n+1}-Z_{n-1}}{Z_{n+2}-Z_{n-1}}}\end{aligned}}$

Example: fractal "Tree of Life"

More details, as well as 3D curve interpolation algorithm, can be found in original paper:

Tikhomirova T.A., Fedorenko G.T., Kirillova L.N. "Fractal interpolation of random curves", Journal of Computer and Information Technology, № 3, 2014