Polynomials are incredibly useful mathematical tools as they are simply defined, can be calculated quickly on computer systems and represent a tremendous variety of functions. They can be differentiated and integrated easily, and can be pieced together to form spline curves that can approximate any function to any accuracy desired. Most students are introducted to polynomials at a very early stage in their studies of mathematics, and would probably recall them in the form below:
In general, any polynomial function that has degree less than or equal to , can be written in this way, and the reasons are simply
In these notes we discuss another of the commonly used bases for the space of polynomials, the Bernstein basis, and discuss its many useful properties.
For a pdf version of these notes look here.
The Bernstein polynomials of degree are defined by
These polynomials are quite easy to write down: the coefficients can be obtained from Pascal's triangle; the exponents on the term increase by one as increases; and the exponents on the term decrease by one as increases. In the simple cases, we obtain
A Recursive Definition of the Bernstein Polynomials
The Bernstein polynomials of degree can be defined by blending together two Bernstein polynomials of degree . That is, the th th-degree Bernstein polynomial can be written as
The Bernstein Polynomials are All Non-Negative
A function is non-negative over an interval if for . In the case of the Bernstein polynomials of degree , each is non-negative over the interval . To show this we use the recursive definition property above and mathematical induction.
It is easily seen that the functions and are both non-negative for . If we assume that all Bernstein polynomials of degree less than are non-negative, then by using the recursive definition of the Bernstein polynomial, we can write
In this process, we have also shown that each of the Bernstein polynomials is positive when .
The Bernstein Polynomials form a Partition of Unity
A set of functions is said to partition unity if they sum to one for all values of . The Bernstein polynomials of degree form a partition of unity in that they all sum to one.
To show that this is true, it is easiest to first show a slightly different fact: for each , the sum of the Bernstein polynomials of degree is equal to the sum of the Bernstein polynomials of degree . That is,
Once we have established this equality, it is simple to write
The partition of unity is a very important property when utilizing Bernstein polynomials in geometric modeling and computer graphics. In particular, for any set of points , , , , in three-dimensional space, and for any , the expression
Any of the lower-degree Bernstein polynomials (degree ) can be expressed as a linear combination of Bernstein polynomials of degree . In particular, any Bernstein polynomial of degree can be written as a linear combination of Bernstein polynomials of degree . We first note that
Converting from the Bernstein Basis to the Power Basis
Since the power basis forms a basis for the space of polynomials of degree less than or equal to , any Bernstein polynomial of degree can be written in terms of the power basis. This can be directly calculated using the definition of the Bernstein polynomials and the binomial theorem, as follows:
To show that each power basis element can be written as a linear combination of Bernstein Polynomials, we use the degree elevation formulas and induction to calculate:
Derivatives of the th degree Bernstein polynomials are polynomials of degree . Using the definition of the Bernstein polynomial we can show that this derivative can be written as a linear combination of Bernstein polynomials. In particular
The Bernstein Polynomials as a Basis
Why do the Bernstein polynomials of order form a basis for the space of polynomials of degree less than or equal to ?
This is easily seen if one realizes that The power basis spans the space of polynomials and any member of the power basis can be written as a linear combination of Bernstein polynomials.
If this were true, then we could write
A Matrix Representation for Bernstein Polynomials
In many applications, a matrix formulation for the Bernstein polynomials is useful. These are straightforward to develop if one only looks at a linear combination in terms of dot products.
Given a polynomial written as a linear combination of the Bernstein basis functions
In the quadratic case (), the matrix representation is