Bernstein Polynomial
What you are looking at below is vector art, it is different from normal artwork and images. If you download vector art its format would be .svg which stands for Scalable Vector Graphics. It doesn't store the image in form of pixels rather it stores it as mathematical vectors!
But when we hear the word “Vector” we imagine a straight line with direction, then how are we able to construct such beautiful curves?
The answer is Bezier curves!
I have been doing graphic designing and CGI a lot recently and that’s how I stumbled across Bezier curves which are the essential part without which vector graphics that we see everywhere wouldn’t have been possible. It is such a common occurrence that I think any graphic designer irrespective of the software must have used it.
But do you know what is even more interesting? It’s the math behind it… And to understand and admire the beauty of Bezier curves one needs to understand Bernstein polynomials. This is named after Sergei Natanovich Bernstein. Bernstein’s form of polynomials was primarily used to prove the Weierstrass approximation theorem but with the rise of computer graphics now it forms the backbone of Bezier curves.
This article explains briefly the basic math behind Bernstein polynomials which is just a prelude to Bezier curves. I tried to simplify things …. hope u find it informative.
Mathematical definition of n+1 Bernstein basis polynomials of degree n:-
Bernstein basis polynomial ex:-
Using the Bernstein basis polynomials we can approximate any polynomial just like how we use Taylor expansion to represent a function as an infinite series.
So a Bernstein polynomial is a polynomial that can be represented as the linear combination of the Bernstein basis polynomials in Vector space ∏_n :-
Some important properties of Bernstein polynomials:-
The next property is an interesting one as it shows that the sum of the basis polynomials is 1. This result can also be represented through the Binomial Theorem. If you notice closely then the ith term of n-order Bernstein polynomial is just the ith term of binomial expansion of x and (1-x) with power n.
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2.
3. Mean
4. Variance
5. The maximum of every basis polynomial occurs at x=k/n. This property is an interesting one and it also has a beautiful geometrical interpretation which will be more evident when we will consider Bezier curves. This property kind of determines the “influence” of individual basis polynomial on the polynomial. As you can see in the image ….
Now I will prove properties 3 and 4 as they are important. In order to prove them we will use a formula that will make it easy :
Now for property 4, we will proceed in a similar way:-
So this is how we can prove these 2 properties in a quite simple way.
Now, these properties will come in handy while trying to prove the Weierstrass Approximation Theorem using the Bernstein polynomials. However, I am not going to go into that part as it's different. My goal was to provide a basic idea behind Bernstein polynomials as the Bezier curves are based upon these.
