Parametric Curves: Example 7: The Cycloid: Proof Part 1

In this video I go over the cycloid curve and derive the parametric equations for the case in which the angle inside the circle is between 0 and π/2. The cycloid is formed by tracing a point on the circumference of a circle as it rotates along a straight line. The resulting parametric equations are shown below:

x = r(θ – sinθ)
y = r(1 – cosθ)

Although I derived these equations for the case where θ is between 0 and π/2, these equations do in fact work for all values of θ. I will go over the proof for other values of θ in a later video so stay tuned for that!

Also, these parametric equations can be rearranged to eliminate θ and to write x as a function of y, but the resulting Cartesian equation is much more complicated. I will nonetheless derive the Cartesian form of the cycloid in a later video, as well as the very interesting history of the cycloid, so stay tuned!

Download the notes in my video:!As32ynv0LoaIht9rJUo4SMEVW-eiJg

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