# Math Insight

### Applet: Parametrized ellipse

The vector-valued function $\adllp(t)=(3\cos \frac{t^2}{2\pi})\vc{i} + (2 \sin \frac{t^2}{2\pi}) \vc{j}$ parametrizes an ellipse, shown in cyan. This ellipse is the image of the interval $[0,2\pi]$ (shown in red) under the mapping of $\adllp$. For each value of $t$, the blue vector is $\adllp(t)$. As you change $t$ by moving the blue point along the interval $[0,2\pi]$, the head of the arrow traces out the ellipse.

Can you show mathematically that $\adllp(t)$ traces out an ellipse? To do so, let $(x,y)$ be the point defined by $(x,y)=\adllp(t)$ for some value of $t$. Since $\adllp(t) = (3\cos \frac{t^2}{2\pi}, 2 \sin \frac{t^2}{2\pi})$, we conclude that $x = 3\cos \frac{t^2}{2\pi}$ and $y=2\sin \frac{t^2}{2\pi}$. Can you see that $x$ and $y$ satisfy \begin{align*} \frac{x^2}{3^2} + \frac{y^2}{2^2} = 1, \end{align*} which is the equation for the above ellipse? (You remember that $\cos^2s + \sin^2s = 1$ for any $s$. You can calculate expressions for $x^2/3^2$ and $y^2/2^2$, and add them together.)

Applet file: parametrized_ellipse_2.ggb

#### Applet links

This applet is found in the pages

List of all applets

#### General information about Geogebra Web applets

This applet was created using Geogebra. In most Geogebra applets, you can move objects by dragging them with the mouse. In some, you can enter values with the keyboard. To reset the applet to its original view, click the icon in the upper right hand corner.

You can download the applet onto your own computer so you can use it outside this web page or even modify it to improve it. You simply need to download the above applet file and download the Geogebra program onto your own computer.