Consider a simple, closed curve $\left(x(s),y(s)\right)$, being $s$ the arc-length.
The quantity
\begin{align} \frac{d x(s)}{d s}\frac{d y(s)}{d s} \end{align}
is found in some problems concerning Chebyshev nets, so I wonder if we can give to it some meaning or ascribe a property. In terms of the unit tangent $\mathbf{T}$ it can be written as $\left(\mathbf{T}\cdot \mathbf{\hat{x}}\right)\left(\mathbf{T}\cdot \mathbf{\hat{y}}\right)=\mathbf{\hat{x}}\cdot \left(\mathbf{T}\mathbf{T} \right)\cdot \mathbf{\hat{y}}$, but it apparently doesn't hint to something useful.