# Focus of a rolling parabola traces a catenary - geometric explanation

It is known that the focus of a rolling parabola along the x-axis traces a catenary. I'm interested in a geometric explanation.

But I don't get why $\cos \angle PFK = \frac{dx}{ds}$. Can someone explain this equation?

At this point my view is as follow: we have the focus $F = (x,y)$ and denote the unit tangent of catenary at $F$ (or in other words the unit tangent to the curve traced out by $F$) with $T = (\frac{dx}{ds}, \frac{dy}{ds})$. Now assume that $\angle PFK$ is the angle between $T$ and $x$-axis. Then, $\cos \angle PFK = <T,(1,0)> = \frac{dx}{ds}$. At this point we are finished. (By the way, our $x$-axis is the line $PK$).

But why should $\angle PFK$ denote the angle between $T$ and $x$-axis? Why should we know that? I think we know nothing about $T$ at the moment.

Best regards!

Well it is well known that $$x = \int \cos∠PFK ds$$. Does this help? I can explain more but it should be clear now.