---- > [!definition] Definition. ([[vertex of a (regular) parameterized curve]]) > A **vertex** of a (sufficiently differentiable) [[regular curve]] $\alpha:[a,b] \to \mathbb{R}^{n}$ is a point $t \in [a,b]$ where $\kappa'(t)=0$. Here, $\kappa'(t)$ denotes the [[derivative]] of the [[curvature of parameterized curve|curvature]] of $\alpha$. ^64f9df > [!basicexample] > Consider the planar [[ellipse]] $\alpha(t)=\big(x(t),y(t)\big)$, where $x=a \cos t, \ \ y=b \sin t, \ \ t \in [0, 2\pi], a \neq b.$ We claim that the [[curvature of parameterized curve|curvature]] of $\alpha$ is $\kappa(t)=\frac{ab}{\|\alpha''(t)\|^{3}}$ and the [[ellipse]] has vertices at the points $(a,0),(-a,0),(0,b),(0,-b)$. > $\alpha'(t)= (-a\sin t, b \cos t)$ and $\alpha''(t)=(-a \cos t, -b \sin t)$, $\alpha'''(t)=(a\sin t, -b \cos t)$. Since $\alpha$ is a plane curve, we know $\kappa(t)=\frac{\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} x'(t) \\ y'(t) \end{bmatrix} \cdot \begin{bmatrix} x''(t) \\ y''(t) \end{bmatrix}}{\|\alpha''(t)\|^{3}}=\frac{x'(t)y''(t) - x''(t)y'(t)}{\|\alpha''(t)\|^{3}}$ and substitution yields $\begin{align} \kappa(t) = & \frac{(-a \sin t) (-b \sin t) - (-a \cos t)(b \cos t)}{\|\alpha''(t)\|^{3}} \\ = & \frac{ab }{\|\alpha''(t)\|^{3}} \end{align}.$ It follows that $\begin{align} \kappa'(t) = & -\frac{ab \|D(\|\alpha''(t)\|^{3})}{\|\alpha''(t)\|^{6}} \end{align}$ and since ![[IMG_579332DC0B99-1.jpeg]] we conclude $\kappa'(t)=\frac{3ab (b^{2} - a^{2})\sin t \cos t}{\|\alpha''(t)\|^{5}}.$ $\kappa'(t)$ has zeros at the zeros of $\sin t$ and the zeros of $\cos t$: these are $t \in \left\{ 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \right\}$. Evaluating $\alpha$ at such $t$, we obtain $\begin{align} \alpha(0) = & (a,0) \\ \alpha\left( \frac{\pi}{2} \right) = & (0, b) \\ \alpha(\pi) = & (-a,0) \\ \alpha\left( \frac{3\pi}{2} \right) = & (0, -b), \end{align}$ which is what we had wanted to show. ^19afe2 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```