Art doesn’t have to be confined to the artistic. There can be method to art. Did you know that mathematics plays an important role in computer-aided graphic design?
Finding use in computer graphics and animation are a set of polynomials called Bernstein basis polynomials that form the basis for the space of polynomials of a given degree. Polynomials include mathematical elements such as variables, coefficients, exponents etc. joined together by an addition sign, a minus sign, or a multiplication sign. Examples of polynomials include: 3x +2, 4x2 +5x + 1, and even integers like the number 5.
In the late 1960s, Pierre Bézier revolutionized the field of design with his innovative development of the Bézier curve. Bézier utilized Bernstein basis polynomials to create this mathematical breakthrough. This discovery significantly advanced computer-aided design.
Bernstein basis polynomials can be generalized by what are known as fractal interpolants/approximants. Fractals are complex geometrical shapes that exhibit self-similarity. A fractal interpolant is a special type of interpolating (interjecting) function that constructs a self-similar, fractal-like curve passing through a given set of data points.
Fractal interpolants exhibit a range of smoothness, from infinitely differentiable functions to nowhere differentiable functions. They find use in a variety of fields such as science and engineering, computer graphics, geometric modelling, optimization, robotics, etc.
A concept known as the iterated function system (IFS) was utilized to develop the theory of fractal interpolation functions. IFS is a mathematical framework used to generate self-similar fractals, using a set of recursive transformations. To create non-smooth designs, fractal theory can be used.
For a given dataset of ‘N’ points and a fixed set of scalings, we only get one interpolant through IFS. However, with zipper IFS, it is possible to generate 2N – 1 different interpolants (referred to as zipper fractal interpolants), including the interpolant obtained through IFS, by simply varying a binary vector known as the signature. Zipper fractals can create intricate branching, twisting, or interlocking structures.
In this study, the authors Dr. Vijay and Prof. M. Guru Prem Prasad from the Department of Mathematics, Indian Institute of Technology Guwahati (IITG), Guwahati, India, and Prof. Gurunathan Saravana Kumar from the Department of Engineering Design, Indian Institute of Technology Madras (IITM), Chennai, India, have used a method involving the fractal perturbation of Bézier basic functions to generate fractal Bézier curves.
Using this novel class of zipper fractal Bézier curves, some appealing, mesmerizing, and visually pleasing designs have been generated. The visual representations are created by constructing the zipper fractal Bézier curves with appropriate colour maps in the MATLAB (2024a) environment.
The approximants constructed using Bernstein basis polynomials or Bézier curves exhibit a smooth nature. In contrast, signals can be highly irregular and non-smooth, making them more complex to analyze due to their diverse range of frequencies and amplitudes. Future research could be directed by applying these results to electrocardiogram (ECG) waveforms, as these demand high accuracy and regular calibration and precision.
This article has introduced a novel generalization of Bézier curves that can be non-smooth while achieving endpoint interpolation, symmetry, the convex-hull property, and geometric invariance. One can generate an endless variety of distinct graphical objects.
Given below are a few flower designs by zipper fractal Bézier curves:
Fig: Flower designs by zipper fractal Bézier curves.
Prof. A. V. Tetenov from the Department of Mathematics, Sobolev Institute of Mathematics, Novosibirsk, Russia, praised the work done by the authors with the following comments: “The authors of the paper “A novel class of zipper fractal Bézier curves and its graphics applications” explain the construction of a basis of fractal Bernstein polynomials that are used to create fractalized Bezier curves. In my opinion, it is a remarkable achievement of Professor G. Saravana Kumar and his enthusiastic students. In their search, they reached a place where sophisticated numerical analysis and refined technical design can get a revitalizing portion of fractal perturbations to create state-of-art masterpieces. I believe the results contained in the paper will find their application in all these areas.”
Article by Akshay Anantharaman
Click here for the original link to the paper
