Classification of Curves

Before we look more in-depth into curves, we should first look at some different types of curves. Intuitively, these definitions should make sense and will be important to acknowledge when we look at computing arc lengths of curves.

Definition: If $\vec{r}(t)$ traces a curve $C$ for $a ≤ t ≤ b$, then $C$ is said to be a Closed Curve if $\vec{r}(a) = \vec{r}(b)$ and an Open Curve otherwise.

Definition: If $\vec{r}(t)$ traces a curve $C$ for $a ≤ t ≤ b$, then $C$ is said to be Self-Intersecting if for $t_0, t_1 \in (a, b)$ and $t_0 \neq t_1$ we have that $\vec{r}(t_0) = \vec{r}(t_1)$. $C$ is said to be Non-Self-Intersecting if for $t_0, t_1 \in (a, b)$ and $t_0 \neq t_1$ then we have that $\vec{r}(t_0) \neq \vec{r}(t_1)$.

Note that by our definition above, a closed curve is non-self-intersecting. Curves that are closed and is non-self-intersecting are called Simple Closed Curves.