# Definitions > [!NOTE] Definition (Regular Parametrisation) > Let $\underline{r}:I\to \mathbb{R}^{n}$ where $I$ is a [[Real intervals|real interval]] be [[Vector-Valued Function of Real variable|vector-valued function]]. Then $\underline{r}(t)$ is regular parametrisation iff $\underline{r}(t)$ is [[Fréchet Differentiation|differentiable]] and for all $t\in I,$ $\underline{r}'(t)\neq \underline{0}.$ # Properties The vector $\underline{r}'(c)$ is a ***tangent vector*** to the curve at $t=c$. A regular curve has [[Arc length of regular parametrized curve|arc length]]. If the derivative of a parametrisation is $1$ then we say the curve has [[Unit speed parametrisation|unit speed]]. # Examples > [!Example] > Show that the ellipse with equation $\left( \frac{x}{a} \right)^{2}+\left( \frac{y}{b} \right)^{2}=1$, with $a,b \neq 0$ is a regular curve > > **Soltuion** > The curve is parametrised by $\underline{r}(t)=(a\cos t,b\sin t), t\in[0,2\pi]$. > Its derivative is given by $\underline{r}'(t)=(-a\sin t,b\cos t)\neq \underline{0}$ ($\cos$ and $\sin$ can't be zero at the same time). > > [!Example] > Find a regular and a non-regular parametrisation for the line $y = x$. > > **Solution** > Regular: $\underline{r}(t)=(t,t)$, $t\in \mathbb{R}$. $\underline{r}'(t)=(1,1)\neq(0,0)$. > Irregular: $\underline{r}(t)=(t^{3,}t^{3})$, $t\in \mathbb{R}$. $\underline{r}'(t)=(3t^{2},3t^{2})=(0,0)$ when $t=0$. > [!Example] > Find the unit tangent to the helix parametrised by $\underline{r}(t)=(\cos t,\sin t,t), t\in[0,2\pi]$ at $t=0$. > > **Solution** > $\underline{r}'(t)=(-\sin t,\cos t, 1)$ > $\underline{r}'(0)=(0,1,1)$ > $\underline{T}(0)=\left( 0, \frac{1}{\sqrt{ 2 }}, \frac{1}{\sqrt{ 2 }} \right)$ > > The standard symbol for the unit tangent is $\underline{T}(t)$. > [!Example] > Find the derivative of $f(t)=||\underline{r}(t)||$ (assuming that $f(t)\neq 0$). > > **Solution** > Note that $f^{2}(t)=\underline{r}\cdot \underline{r}$. Differentiating w.r.t $t$ gives $2\underline{f}(t)\underline{f}'(t)=2\underline{r}'(t)\cdot \underline{r}(t)\implies f'(t)=\frac{2\underline{r}'(t)\cdot\underline{r}(t)}{2f(t)}=\frac{2\underline{r}'(t)\cdot\underline{r}(t)}{2||\underline{r}(t)||}$ > >[!Example] >Let $\underline{r}(t)$ be a function such that $||\underline{r}(t)||= \text{constant}$. Then $\underline{r}(t)$ and $\underline{r}'(t)$ are [[Angle Between Nonzero Real Vectors#^3616e1|orthogonal]]. > >**Proof** >Note that $||\underline{r}||^{2}=\text{constant}\implies \underline{r}\cdot \underline{r}=\text{constant}$. >Differentiating w.r.t $t$ gives $2\underline{r}\cdot \underline{r}'=0 \implies \underline{r}\cdot \underline{r}'=0$ as required.