Normalized vs Non Normalized Bernstein 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!
 A: 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.
