logo

Definition of a Simple Curve 📂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

Figure 1

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

Figure 2

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

  1. Millman. (1977). Elements of Differential Geometry: p54. ↩︎