Consider one arch of the cycloid as shown in the figure. Let s() be the arc length

Consider one arch of the cycloid

as shown in the figure. Let s(θ) be the arc length from the
highest point on the arch to the point (x(θ), y(θ)), and let
p(θ) = 1/K be the radius of curvature at the point (x(θ), y(θ)).
Show that s and p are related by the equation s² + p² = 16.
(This equation is called a natural equation for the curve.)

Transcribed Image Text:

r(0) (0 sin )i + (1 − cos 0)j, 0 ≤ 0 ≤ 2π = -