NURBS, parametrized curves and manifolds

Question

Let's start with the definitions:

A parametrized curve is a map $γ : (α,β) → R^n$ , for some $α,β$ with $−∞ ≤ α < β ≤ ∞$.

A NURBS curve is defined by $C(u)=\sum_{i=1}^n R_{i,p}(u)\mathbf{P_i}$ as a rational function from the domain $\Omega=[0,1]$ to $R^n$.

A parametrized manifold in $R^n$ is a smooth map $σ:U → R^n$ , where $U ⊂ R^m$ is a non-empty open set. It is called regular at $x ∈ U$ if the $n × m$ Jacobi matrix $Dσ(x)$ has rank $m$ (that is, it has linearly independent columns), and it is called regular if this is the case at all $x ∈ U$.

Now, I might be misunderstanding some things but I have a couple of questions:

i) We can say that a NURBS curve is defined is parametric form. However it doesn't fit in the definition of a parametrized curve because the interval $[0,1]$ is not open. Why is that and is this important?

ii) We can use NURBS to exactly represent conics. For example the circle can be exactly represented. If we consider the circle as a manifold, can we also consider the NURBS mapping that defines the geometry of the circle as a chart, or is the open set (interval) condition again a problem?

I guess there are two notions here that have me confused. One the use of open intervals vs closed intervals and second, what's the relationship between parametrizations, manifolds and charts. Are all parametric curves (or surfaces) manifolds and vice versa?

Answer 1

To answer your question in brief, All curves and surfaces are manifolds, parametrized or not. Manifolds are abstractions of surfaces or curves.Just look up the definition of a chart and atlas on Wikipedia. In a way , a parametrization is a kind of chart. A precise answer might be possible if you add a bit more to your question or your background.

As for the other question, open intervals and closed intervals are both used for defining curves. Take a look at this related question:

open interval in definition of curve

Hope this helped.

Answer 2

Regarding your first point, a parameterized curve is sometimes defined to be open because we usually want it to be smooth in some sense. Leaving its domain open allows us to be lazy, because we don't have to deal with the cases where $t=a$ and $t=b$. If the question of differentiability at the terminal points is a non-issue, then nothing is lost by closing the domain.

Regarding your second question, I don't know specifically. But typically, leaving it open allows us to use some key theorems (e.g. the open mapping theorem). I'm not entirely sure if there's anything lost here by closing the parametric domain. Take a look at some of the theorems and definitions and see if something breaks!