3.9 The ellipse E

We can take the standard equation for an ellipse to be

where $a,b>0$ are constants.

The centre of the ellipse is $(0,0)$ and the axes are the horizontal line segment: $y=0,-a\leq x\leq a$ and the vertical line segment: $x=0$, $-b\leq y\leq b$.

We can parametrize this ellipse by setting $x=a\cos t$, $y=b\sin t$, where $0\leq t<2\pi$.

In the special case $a=b$, we obtain a circle of radius $a$, with centre $(0,0)$.