Acceleration of Point in Plane in Intrinsic Coordinates

Theorem
Let $\mathbf a$ be the acceleration of a particle $P$ in space.

Let $P$ be moving in a plane $\PP$.

Then the motion of $P$ can be expressed in intrinsic coordinates as:
 * $\mathbf a = \dfrac {\d \map v t} {\d t} \mathbf s + \dfrac {\paren {\map v t}^2} \rho \bspsi$

where:
 * $\mathbf s$ denotes the unit vector along the tangential direction of $P$
 * $\bspsi$ denotes the unit vector toward the center of curvature of the motion of $P$
 * $\map v t$ is the speed of $P$ at the time $t$
 * $\rho$ is the radius of curvature of the motion of $P$ at time $t$.