By constructing differential triangles in polar coordinates in the plane show that 

$$ \dfrac{dp}{ds}= \dfrac{t}{R} = t \,\kappa$$

[ pedal length $p$ , tangent projection $t$, instant radius $R$ , curvature $\kappa$, arc $s$].

EDIT 1:

In place of $(12), (13)$ I obtained by scalar arc differentiation the RHS as $ (t \kappa, 1- \kappa \,p )$ respectively in which sign of $\kappa $ is the opposite.

I reasoned for the new *latter Equation* (13) taking the example of an involute of a circle.... since the magnitude of normal is constant (= radius of circle from which a taut string is unwound), it should vanish, so that

$$ \kappa p = +1,\, p = +R $$

But Robert Lewis and by implication and also Ted Shifrin put the sign as 

$$ \kappa p = -1,\, p = -R $$

which should be reconciled.

[![Either pdash at Inflection](https://i.stack.imgur.com/7GPYS.png)](https://i.stack.imgur.com/7GPYS.png)