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Bernstein Polynomials: Overview

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Bernstein Polynomials as Approximations of Functions

Bernstein polynomials, named after Russian mathematician, Sergei Natanovich Bernstein, area a way to approximate the core ingredients of functions.

These polynomials converge (settle on) a particular function of choice, giving good approximations with very few combined Bernstein polynomials. Another advantage is that by using a set of polynomials, rather than a function, points can be calculated and stored efficiently [1]. One disadvantage is that the classic Bernstein polynomials tend to converge slowly [2], a fact that caused them to “languish in obscurity” until the advent of the modern computer [3].

Bernstein polynomials

Bernstein polynomial approximating a curve.


In many situations, it’s better to use Bernstein polynomials rather than an explicit function of the form y = f(x), because of the limitations of function notation. These include the fact that a vertical line (i.e. a vertical asymptote) x = c cannot be described using function notation.

Formula

A Bernstein polynomial of degree n is a linear combination of the (n + 1) Bernstein basis polynomials of degree n. The n + 1 Bernstein basis polynomials of degree n are defined as:
bernstein basis polynomials

Where is a binomial coefficient.
Every Bernstein basis polynomial is nonnegative on [0,1], and zero only at the endpoints [3].


Practical Uses of Bernstein Polynomials

Bernstein polynomials are used in statistics for smoothing and as a basis for Bezier elements used in isogeometric analysis [4]. They form the mathematical foundation for computer-aided geometric design (CAGD); Adobe’s Illustrator and Flash, as well as font imaging systems like Postscript; all use Bernstein polynomials to create Bezier curves— a type of parametric curve.

bezier curve

Bernstein polynomials are behind the create of Bezier curves.

References

[1] Timmerman, G. (2014). Approximating Continuous Functions and Curves&hellip Retrieved July 15, 2021 from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.754.2049&rep=rep1&type=pdfhttps://sites.math.washington.edu/~morrow/336_14/papers/grant.pdf
[2] Guan, Z. Iterated Bernstein polynomial approximations. Retrieved July 15, 2021 from:
[3] BERNSTEIN_POLYNOMIAL. Retrieved July 15, 2021 from: https://people.sc.fsu.edu/~jburkardt/f_src/bernstein_polynomial/bernstein_polynomial.html
[4] Farouki, R. The Bernstein polynomial basis: A centennial retrospective. Computer Aided Geometric Design. Volume 29, Issue 6, August 2012, Pages 379-419. Retrieved July 15, 2021 from: https://www.sciencedirect.com/science/article/abs/pii/S0167839612000192

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Stephanie Glen. "Bernstein Polynomials: Overview" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/bernstein-polynomials-overview/
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