# 19.3: The Intrinsic Equation to the Cycloid

An element \(ds\) of arc length, in terms of \(dx\) and \(dy\), is given by the theorem of Pythagoras: \( ds = ((dx)^2 + (dy)^2))^{1/2} \) or, since \(x\) and \(y\) are given by the parametric Equations 19.1.1 and 19.1.2, by And of course we have just shown that the intrinsic coordinate \( \psi \) (i.e. the angle that the tangent to the cycloid makes with the horizontal) is equal to \( \theta \).

Exercise \(\PageIndex{1}\)

Integrate \(ds\) (with initial condition \(s\) = 0, \( \theta \) = 0) to show that the intrinsic equation to the cycloid is

\[ s = 4 a \sin \psi \label{19.3.1}\tag{19.3.1}\]

Also, eliminate \( \psi \) (or \( \theta \)) from Equations \( \ref{19.3.1}\) and 19.1.2 to show that the following relation holds between arc length and height on the cycloid:

\[ s^2 = 4 ay. \label{19.3.2}\tag{19.3.2}\]