Have I forgotten all my secondary-school geometry?
(That's not actually my question.)
Suppose $R>r>0$ and consider this circle
(later edit: I think $R>0$, $r>0$ is enough; we don't really need $R>r.$ end of later edit) $ (x-R)^2 + z^2 = r^2 $ in the $xz$-plane. For any point $p$ on the circle, draw the line through $p$ and the origin $(x,z)=(0,0)$, intersecting the circle at a second point $q$. Thus $p\mapsto q$ is a sort of projection, with the center of projection at the origin. Let $dp/dq$ denote (OK, slightly odd notation here since $p$ and $q$ are not scalars) the ratio of rates of motion of $p$ and $q$ as $p$, and hence also $q$, move along their circular arcs.
If I'm not mistaken, the following differential equation is satisfied: $ \frac{dp}{dq} = \frac{x\text{-coordinate of }p}{x\text{-coordinate of }q}. $ Is this
- a known result in the sense of being in all the books, for suitable values of "all"; or
- a known result in the sense that any fool would "recall" it if the occasion arose (e.g. like $19\times23=437$); or
- other (specify).
(A bit of a vague question maybe, but some smart people here can sometimes survive that.)
(BTW, if you like circles, check this out: http://en.wikipedia.org/wiki/List_of_circle_topics)