I'm interested in using curvilinear coordinates to represent points of a subregion $R \subset \mathbb R^2$, and I'm trying to determine conditions on $\gamma$ and $R$ to ensure that the representation is unique. Use case: represent the position of a robot on a track defined by a reference curve and positive and negative bounds on lateral dimensions.

Mathematically, $\gamma(s)$ is a $C^2$ curve with $s$ a curvilinear abscissa with arbitrary origin and $\gamma$ is injective. At point $s$, we have the unit tangent vector $\mathbf T = \gamma'(s)$ and we let $\mathbf N$ the unit normal vector such that $(\mathbf T, \mathbf N)$ is positively oriented. For a point $X \in R$, I want to define the curvilinear coordinates of $X$ as $(s,r)$ such that $X = X_\gamma + r \mathbf N$ with $X_\gamma = \gamma(s)$ the point of $\gamma$ closest to $X$.

However $X_\gamma$ may not be unique, for instance if $\gamma$ is a circular arc with center $X$. Requiring that \begin{equation}R \subset \left\{ \gamma(s) + r\mathbf N \ :\ |r| \leq \frac{1}{|k(s)|} \right\}\end{equation} where $k$ is the local curvature seems to avoid the issue "locally".

My problem is finding a condition to avoid the non-local issue: for instance, if $\gamma$ is shaped like a $\cup$ (ie, with parallel lines), there is an infinity of points for which $X_\gamma$ is not unique. However, also requiring that $r < \frac d 2$ where $d$ is the width of the $\cup$ gets rid of the issue.

I'm pretty much convinced that such a $d > 0$ can be found for any curve with finite length. However, I'm having trouble formalizing a demonstration. Any thoughts?

Precision: I'm only interested in finding a good enough sufficient condition on $R$ and $\gamma$ for $X_\gamma$ to be unique, I don't need it to be necessary.

Thanks for any help


Thanks to the article pointed out by Moishe Cohen, I was able to (learn a lot about the reach of a manifold and) confirm my initial thought.

From Proposition 14 of http://www.utgjiu.ro/math/sma/v03/p07.pdf:

Compact $C^2$-smooth submanifolds of $\mathbb R^n$ have positive reach

The reach of a manifold (in $\mathbb R^2$) is exactly $\frac d 2$ in the original question.


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