TITLE: Tracking the movement of the watch hand
QUESTION [9 upvotes]: I love watches, and I had an idea for a weird kind of watch movement (all of the stuff that moves the hands). It is made up of a a central wheel, with one of the hands connected to it (in this case, it will be the hour hand). This hand goes through a pivot, and then displays the time. I attached a video of a 3d mock up here, because it is kinda hard to explain. My question is, is there any functions that would be able to graph the movement of the end of the hand? I don't want to make the real prototype just yet.

REPLY [5 votes]: Denote with $l$ the length of the hand and with $R$ the radius of the circle. 
Parametric coordinates of point $M$ (as a function of $\alpha\in[0,2\pi)$) are: 
$$x_M=(l-2R\cos\frac\alpha2)\cos\frac\alpha2=l\cos\frac\alpha2-R(1+\cos\alpha)$$
$$y_M=-(l-2R\cos\frac\alpha2)\sin\frac\alpha2=-l\sin\frac\alpha2+R\sin\alpha$$
As an exercise you can eliminate angle $\alpha$ and obtain an implicit relation between coordinates of point $M$, but there is not much that you can do with it. It is better to work with parametric equations. Select $l,R$ and calculate coordinates for a range of $\alpha$ angles.