In an answer to the question Modifying and Generalizing the De Casteljau Algorithm, I found out that you can convert a polynomial into a polar form, aka make it into a symmetric multiaffine function.
For example, if $f(t)=(1−t)^3$, then $F(u,v,w)=(1−u)(1−v)(1−w)$.
I've been reading the sources listed and I can see how this conversion of polynomials does make it easier to label and understand control points in the de Casteljau and de Boor algorithms.
What i'm wondering about though is what are other benefits of converting a polynomial to this form? Not restricting to the curve/spline usage case.
I understand that in my example, when $u$,$v$ and $w$ are the same value, it acts like the original cubic equation. What happens though when they aren't equal? Is there anything meaningful to be said about that, or do you just simply end up with something equivalent to a different polynomial when doing this?
Do these symmetric multiaffine functions have any other meaningful properties besides convenient labeling? The papers say there are, but I so far haven't been able to find or understand what they are so am a bit confused about that.
Thanks!
