Curve scaled by vector. Given a closed curve $C$ which is defined as a set of vectors pointing to each point $V_a$. Let the curve's "range" be  $0 \leq a \leq 1$. Let the normal vector pointing outwardly tangent of infinitely small magnitude at $V_a$ to be $\lambda_a$. Now, define a "vectoral scale" as a transformation $T: V, \lambda  \to V$ such that $T(V, \lambda ) = \{V_a+\lambda_a\}$. If $\underset{h \to 0}{\lim}\lambda_{a+h}-\lambda_a = 0$, define them as the same point.
Because $T(V,\lambda)$ only extends the curve an infinitely small length by the unit normal vector, define $T^n$ as $T(V, \lambda)$ occuring $n$ times.
Now, I need to prove or disprove that $\underset{\beta \to \infty}{\lim}T^\beta(V,\lambda)$ eventually converges or "rounds" to a curve which is a circle of any radius.
Here is a (rather amateur) visualization of my question.

As you can see, the curve's depression where $V_a$ is starts to close up and become more rounded as $T^x$ is applied. When the curve become infinitely rounded, it will become a circle.
 A: There are a few things in the problem description which are not entirely clear, such as what happens when the curve is not smooth, e.g. makes an angle.
One way to reformulate the problem, as I suspect it is intended, is as follows:
Let $C$ be a closed curve enclosing a compact region $U$: i.e., $C$ is the boundary of $U$. For any $r\ge0$, let $U_r$ consist of all points within distance $r$ of $U$,
$U_r=\{x \mid d(x,U)\le r\}$, and define $C_r$ to be the boundary of $U_r$.
As $r$ increases, wherever the boundary of $U_r$ is smooth, $U_r$ will grow in the direction perpendicular to the boundary $C_r$ with the same speed as $r$ increases, just as you describe.
If there is a point $O\in U$ so that $U$ is within a ball of radius $R$ centered at $O$, then $C_r$ with lie between circles of radius $r$ and $r+R$. In that respect, $C_r$ becomes more and more like a large circle. However, unless you started with a circle, you will never get a perfect circle.
To given an example, for some $a>0$, let $U$ be the union of balls of radius $a$ around points $(a,0)$ and $(-a,0)$. Then $U_r$ will be the union of balls of radius $a+r$ around the same two points. This will never be convex, but always have an indentation and for an angle where the two circles meet. So in that respect, it will never become a circle.
