Formal definition of concave side of a plane curve

Question

Let $\alpha:(a,b)\subseteq \mathbb{R} \rightarrow \mathbb{R}^{2}$ be a plane regular curve which is parametrized by arc length with non-vanishing curvature. Why the principal unit normal vector $N = \frac{1}{||\alpha^{''}||}\alpha^{''}$points toward the concave side of $\alpha$? My college textbooks mention this as something obvious, without further discussion. What is the formal definition of "concave side of a plane curve"?.

Answer 1

I have not a textbook reference in mind but I write my simple intuitions about your question. I hope it would be useful.

Intuitively, at one point with non-vanishing curvature you can approximate (locally) the curve by a circle of radius $1/k$ (osculating circle), being $k$ the value of the curvature at that point. This can be done only in one side of the curve, this one is the side in which $N$ is pointing. The formal definition of this "side" is the following: As we now, $N$ is orthogonal to $T$ (the tangent vector of the curve) and there are 2 possible choices of orthogonal vectors to $T$. How to choose properly that one that we will call $N$?

It is easy. Look how $T$ is changing: the direction of $dT/ds$ is the direction of $N$, therefore we have the definition of "concave side of a curve" at one point: is the side signaled by the vector $dT/ds$.

If you want a concise definition of "concave side" define it as the side in which is located the center of the osculating circle at the point of interest, following the precise method to find its center, given, e.g., in https://mathworld.wolfram.com/OsculatingCircle.html.

I find the term "concave side" misleading because in fact $N$ must be "drawn" inside something that seems a "convex" figure...