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# 19.3: The Intrinsic Equation to the Cycloid

[ "article:topic", "authorname:tatumj" ]

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:

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