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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.