I know what Bernstein polynomials themselves are, and am intimately familiar with one of their usage cases - Bezier curves.
However, I recently came across someone mentioning of "Not Normalized Bernstein Basis". What exactly does that mean and how does it differ from a normalized Bernstein Basis?
From (eg. quadratic) Bezier curves I know that if you have:
$y=(1-x)^2+2*(1-x)x+x^2$
That function is the same as:
$y=1$
Would that mean it is normalized?
Would removing the 2 from the middle term (aka removing the binomial co-efficients) cause it to be non normalized? If so, is this just one possible non normalized form, or is it the only one?
Thanks!
It's difficult to know what the term means without more context. But here's one thing that it might mean ...
In simple discussions, Bernstein polynomials are typically defined on the interval $[0,1]$. So, the $i$-th Bernstein polynomial of degree $m$ is given by $$ \phi^m_i(u) = {m \choose i}u^i (1-u)^{m-i} $$ In slightly more advanced work, when you start talking about subdivision, b-splines, and especially blossoming, it's useful to have Bernstein polynomials (and Bézier curves) that are defined on an arbitrary interval $[a,b]$ rather then on $[0,1]$. This just requires a simple change of variable effected by shifting/scaling: $$ u = \frac{t-a}{b-a} $$ So, the more general form of Bernstein polynomial is $$ \psi^m_i(t) = {m \choose i}\frac{(t-a)^i (b-t)^{m-i}}{(b-a)^m} $$ If my guess is correct, $\phi^m_i$ would be called a normalized Bernstein polynomial, and $\psi^m_i$ would be called a non-normalized one.
Another possibility ...
Sometimes the Bezier curve $$ \mathbf{C}(u) = \sum_{i=0}^m \phi^m_i(u) \mathbf{P}_i \quad (0 \le u \le 1) $$ is written in the form $$ \mathbf{C}(u) = \sum_{i=0}^m \theta^m_i(u) \mathbf{Q}_i \quad (0 \le u \le 1) $$ where $$ \theta^m_i(u) = u^i (1-u)^{m-i} \;\; ; \;\; \mathbf{Q}_i = {m \choose i}\mathbf{P}_i $$ So, the binomial coefficients have been absorbed into the control points, which makes computations more efficient, sometimes. The functions $\theta^m_i$ are then called "non-normalized" Bernstein polynomials.