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:

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