What is Curvilinear abscissa; Frenet basis? — Physics

Definition 11.12 University Physics — Year 1 · Chapter 11 — Kinematics of a Point

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

\frac{\dd\vect T}{\dd s} = \frac{1}{\rho}\,\vect N .

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

The Frenet basis at a point of a curve: T along the motion, N toward the center of curvature C. The acceleration splits into a tangential part v T (speeding up or braking) and a normal part (v2/ ) N (turning). — The Frenet basis at a point of a curve: $\vect T$ along the motion, $\vect N$ toward the center of curvature $C$ . The acceleration splits into a tangential part $\dot v\vect T$ (speeding up or braking) and a normal part $(v^2/\rho)\vect N$ (turning).

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