Smooth curve

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

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

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

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.