# 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