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: https://1drv.ms/b/s!As32ynv0LoaIht9rJUo4SMEVW-eiJg

View Video Notes on Steemit: https://steemit.com/steemstem/@mes/parametric-curves-example-7-the-cycloid-proof-part-1

Related Videos:

Parametric Curves: Example 6: Graphing Devices: https://youtu.be/cwpEasc9mhQ
Parametric Curves: Example 5: Lissajous Figure: https://youtu.be/0OT1OnnAQ9I
Parametric Curves: Example 4: https://youtu.be/WZYIS1My-g4
Parametric Curves: Example 3: https://youtu.be/VpiJdEviBbk
Parametric Curves: Example 2: https://youtu.be/SBkzZDlqTTE
Parametric Curves: Example 1: https://youtu.be/0k7QyWa9-HA
Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Trigonometry: Sine, Cosine and Tan Functions: http://youtu.be/WKTIlF2oWw8
Angles – Degrees vs Radians: What are Radians??: http://youtu.be/bOdrVvQWRdE .


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