# Curvilinear abscissa confusion

How's exactly defined the curvilinear abscissa in kinematics? Surfing on the net I found different definitions:

a) Fixed a point $O$ and a direction, the curvilinear abscissa $s$ at a point $P$ is the lenght of the arc of curve $OP,$ positive in the direction of the curve.

b) Fixed a point $O$ and nothing else, the curvilinear abscissa is the lenght of the arc of curve $OP$.

In calculus, I've seen the abscissa (referred to a curve $\phi$) defined as:

$$s:=\int _0 ^t ||\phi '(\tau) || d \tau$$

That is (b). However seems to me that (a) is the tendency. Thank you in advance.

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