Curve

• Let $\gamma :I\to {\mathbb{R}}^n$ be a continuous function, where $I\subseteq \mathbb{R}$ is an interval.

  • The curve defined by $\gamma$ is the set $\gamma (I)=\{\gamma (t)\mid t\in I\}$.

  • The function $\gamma$ is called the parameterization of the curve.

  • $\gamma (t)=(x_1(t),x_2(t),\dots ,x_n(t))$ for $x_i:I\to \mathbb{R}$.

  • The variable $t$ is called the parameter of the curve.

• (Remark: A curve can have multiple parameterizations.)
• A curve is simple if $\gamma$ is injective.
• A curve is closed if $I=[a,b]$ and $\gamma (a)=\gamma (b)$.
• A curve is regular if $\forall t\in I,{\gamma }^{\prime }(t)\ne 0$.