212 Smooth Curve
A curve whose derivative is continuous and never the zero vector that is used to define a unique tangent direction along the path.
definition [d] (Smooth Curve = Regular Curve) A curve with a parametrization \(\mathbf{r}(t)\) on an interval \(I\) such that
- \(\dfrac{d\mathbf{r}}{dt}\) is continuous on \(I\) .
- \(\dfrac{d\mathbf{r}}{dt}(t) \neq \mathbf{0}\) for all \(t \in I\) except possibly at the endpoints.
where
- \(\mathbf{r}(t)\) is the position vector of the curve.
- \(\dfrac{d\mathbf{r}}{dt}(t)\) is the tangent vector.
- \(t\) is the parameter.
- \(I\) is the parameter interval.
- \(\mathbf{0}\) is the zero vector.
Note:
- \(\dfrac{d\mathbf{r}}{dt}(t)\) is also called the velocity vector.
- geometrically there are no sharp corners.
- geometrically there are no cusps.
- the tangent turns continuously.