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In 3D, the trajectory, which is a curve, represents all the points that an object occupies during its motion. Given a certain basis (Cartesian, cylindrical, spherical, etc.), the instantaneous position of the moving object, relative to the origin, along its trajectory can always be identified by three functions that are the components of the position vector ##P(t)##.

For example, in Cartesian coordinates, the objects' position is given instant by instant by the three parametric functions $$P(t) = x(t) \hat{\textbf x} +y(t) \hat{\textbf y} + z(t) \hat{\textbf z}$$ or $$P(t)=r(t) \hat{\textbf r}+\theta(t) \hat{\theta}+ z(t) \hat{\textbf z}$$ in cylindrical coordinates, etc.

But the scalar parameter can be anything, correct? The same curve/trajectory can be parametrized with the parameter being time ##t##, distance travelled ##s## or any other constructed parameter. Is that correct? Are there other specific parameters that are commonly used besides time and distance travelled ##s##?

When intrinsic coordinates ##\textbf (T, N, B) ## are used, it is common to use the distance travelled as the scalar parameter ##s## instead of the time parameter ##t##. Why? The vector ##T## is the tangential unit vector tangent to the trajectory at a specific point, ##N## is the unit vector perpendicular to the trajectory and ##B## is normal to both ##T## and ##N##.

How is the position of an object along its trajectory represented using the unit basis vectors ##\textbf (T, N, B) ## and their components ##(a,b,c)##? For example, how would $$P(s)= a(s) \textbf T +b(s) \textbf N +c(s) \textbf B $$ ? I am not sure how the vector ##P(t)## expressed this way would identify the object in space along its trajectory...