TITLE: What shape is traced out by this animation?
QUESTION [5 upvotes]: Found this animation circulating online, and was wondering what shape the rod's end traces out. It seems to be an ellipse, but can that be proved somehow?
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REPLY [6 votes]: Call the two pivots moving on a cross $a$ and $b$. $a$ moves on $\{(0,t)|t\in[0,1]\}$, and $b$ on $\{(s,0)|t\in[0,1]\}$. Their position must furthermore satisfy
$$s^2+t^2=1$$
The drawing point is positioned at
$$a(t) + (1+l)(b(s)-a(t)) =$$
$$= ((1+l)\sqrt{1-t^2},t+(1+l)t)$$
By parametrizing $t = \cos(\phi)$ we obtain the position
$$((1+l)\sin(\phi),(2+l)\cos(\phi))$$
which is indeed an ellipse when we let $\phi$ run from $0$ to $2\pi$.
Sorry if my solution is a bit messy, but I worked it up while writing it.