Smooth Curves

Smooth Curve

A curve $C$ parametrized by $\vec{r}(t)=⟨x(t),y(t),z(t)⟩$ for $\alpha \le t\le \beta$, is said to be a smooth curve if:

• The derivatives $x^{\prime }(t),y^{\prime }(t),z^{\prime }(t)$ are all continuous for all $\alpha \le t\le \beta$
• The derivatives $x^{\prime }(t),y^{\prime }(t),z^{\prime }(t)$ do not simultaneously vanish for $\alpha \le t\le \beta$

Piecewise-Smooth Curve

A curve $C$ is a piecewise-smooth curve if it is continuous and can be divided into a finite number of smooth subcurves.