i was reading through some work on curves of pursuit and the author stated a certain equivalence without any comment as to where it came from.
Say i have a differentiable curve in the plane. Given a point let the distance of the tangent line from the origin at be , the angle the tangent line makes with the x-axis be , and the length of the curve be . Then we have the relation:
this sort of thing looks like it should be intuitive but i can't see why it is at all, or at least like it should follow from some other nice results. i've been able to show it by letting :
and then working out all the derivatives manually but that doesn't seem to help me understand it either.
i hope that someone recognizes it and can point me in the direction for a nicer explanation of it.