What is Curvilinear abscissa; Frenet basis?

Along a trajectory, the curvilinear abscissa $s(t)$ is the arc length from a reference point, counted algebraically; $v = \dot s$. At $M$, the unit tangent $\vect T = \dd\vect{OM}/\dd s$ points along increasing $s$; the radius of curvature $\rho > 0$ and the unit normal $\vect N$, pointing toward the concave side (the center of curvature), are defined by

$(\vect T, \vect N)$ is the Frenet basis; $\rho = \infty$ on a straight line, $\rho = R$ on a circle of radius $R$.