Defining a generalized coordinate system

[dipole]

(Note that the title of this thread might be incorrect - I'm just drawing on the vocabulary people use when discussing Lagrangian Mechanics...)

Hi, I'm trying to set up a coordinate system to represent points in space where one of the coordinates is the distance along a parametric curve, one is the shortest distance from a point to the curve, and one is an angle defined relative to some direction.

I have some curve in space defined by:

$x = x(t)$
$y = y(t)$
$z = z(t)$

and I want to define a coordinate system $(s, \rho, \omega)$ relative to this curve.

$\rho$ is defined by the shortest distance from a point $p' = (x',y',z')$ (I will use primes to denote points not on the curve) in space to the curve (assume this is unique).

$s$ is defined by the distance along the curve, starting from some initial point $\left (x(t_0), y(t_0), z(t_0) \right)$ to the point $\left (x(t), y(t), z(t) \right)$ such that

$$d \left (x', y', z', x(t), y(t), z(t) \right ) = \rho$$

where $d$ is the euclidean distance.

In other words, say I have some point in space $p' = (x',y',z')$, then $\rho$ is the length of the smallest line segment between $p'$ and some point on the curve $p(t) = (x(t), y(t), z(t))$. I want to represent the vector from $p(t)$ to $p'$ by an orthogonal coordinate system that is attached to the curve.

Let $\hat{s}(t)$ be the unit tangent vector to the curve at the point $p(t)$. This defines a plane where the point $p'$ lies on a circle of radius $\rho$ in the plane. The only other thing I need to uniquely define the point $p'$ in this plane is some reference direction to measure the angle at which the point $p'$ lies on the circle. Call this direction $\hat{\omega}(t)$.

Here's the kicker though - I want $\hat{\omega}(t)$ to be defined in such a way that when the curve is a straight line, and in the z-direction, that the coordinate system becomes plain old cylindiral coordinates, and $\hat{\omega}(t) = \hat{x}$ . However, when the curve is not a straight line, then $\hat{\omega}$ should rotate with the curve in such a way that it is always orthogonal to $\hat{s}(t)$, and locally if $\hat{s}(t) \simeq \hat{z}$ then $\hat{\omega}(t) \simeq \hat{x}$.

My question is how to define $\hat{\omega}(t)$ in such a way that satisfies these constraints.

Does this make sense? I hope at least the idea of what I want to do is clear. I further hope this problem is not ill-posed. Any help would be appreciated and please let me know if something is unclear.

 

[Ssnow]

Hi, this remember me the Frenet system associated to a curve:

https://en.wikipedia.org/wiki/Frenet–Serret_formulas

 

[dipole]

Hi, yes I remembered that as I was writing this, but I'm still not sure how to enforce that the Frenet-Serret frame becomes normal cylindrical coordinates in the limit that the curvature goes to zero (assuming that any straight line will be in the z-direction).