📂Geometry

Definition of a Simple Curve

Definition ¹

A regular curve $\beta (t)$ is said to be simple if $\beta$ is an injective function or it is a closed curve with period $a > 0$ that satisfies the following for some integer $n \in \mathbb{Z}$: $$\beta \left( t_{1} \right) = \beta \left( t_{2} \right) \iff t_{1} - t_{2} = na$$

Example

Cases like the above, which cannot be represented as an injective function but are considered simple curves,

are closed curves without any twisted parts. If there are twisted parts, it cannot satisfy the mathematical conditions at that point.