Frenet frame

We want our way of specifying curves to be agnostic to coordinate system. What are the coordinates we should give our curve in? How about a coordinate frame defined by the curve itself? So, at every point on the curve, we define tangent, normal, and binormal directions. Then, the derivatives of these quantities can be written in terms of the quantities themselves. This gives rise to a system of differential equations, which has a unique solution: the curve.