Smooth curve

A smooth curve is a smooth map from an interval into a manifold or Euclidean space.

A curve may be described by a parametric equation; smoothness requires its coordinate functions to be differentiable to all orders.

A curve parameter labels the points visited by a parametric curve $\gamma (t)$. Choosing time as the parameter determines its velocity; an arc-length parametrization instead measures accumulated arc length. A regular monotone reparametrization changes the labels and speed while preserving the route and its arc length.

A reparametrization composes a parametrized curve with a smooth change of parameter. An orientation-preserving reparametrization has positive derivative.

The operation changes the parametric equation while preserving the image of the curve.

A smooth parametrized curve is regular when its velocity never vanishes.

Regularity is an additional differential condition on a smooth curve, not a synonym for an arbitrary curve.

An embedded curve is a smooth embedding of a one-dimensional manifold. Its image has no self-intersections and carries the subspace topology.

For a regular curve, $s(t)={\int }_{t_0}^t|{\alpha }^{\prime }(r)|dr$ has positive derivative and hence a smooth inverse. Reparametrization by $s$ gives unit speed.

For a regular parametric curve $F(u)$ traversed in increasing $u$ with positive speed $v(F(u))$, its travel time isInvert this strictly increasing function to sample at equal time intervals, or integrate $\dot{u}=v(F(u))/\Vert F^{\prime }(u)\Vert$. A zero of the speed requires separate analysis: whether travel time diverges depends on the integrability of this reciprocal-speed expression, and the scalar speed alone does not determine the direction after a turning event.