Orientation of Parametric Curves

Definition: Orientation

Let $\gamma :[a;b]\to {\mathbb{R}}^n$ and $\phi :[c;d]\to {\mathbb{R}}^n$ be equivalent parametric curves.

If $\gamma (a)=\phi (c)$ and $\gamma (b)=\phi (d)$, we say that $\gamma$ and $\phi$ have the same orientation or that $\gamma$ (or $\phi$) is orientation-preserving.

If $\gamma (a)=\phi (d)$ and $\gamma (b)=\phi (c)$, we say that $\gamma$ and $\phi$ have opposite orientations or that $\gamma$ (or $\phi$) is orientation-reversing.