In this video I go over Part 5 of the Laboratory Project titled Running Circles Around Circles and this time derive the parametric equations for an Epicycloid. An epicycloid is a curve formed by the path a point on a circle takes as it rotates around the OUTSIDE of another fixed circle. This is similar to a hypocycloid, which I covered in Parts 1 to 5, but for a hypocycloid the circle rotates around the INSIDE of the fixed circle. The derivation for the parametric equations for the epicycloid is also very similar to that for the hypocycloid. In the derivation I set up two right angle triangles in order to apply basic trigonometry to obtain the coordinates of the point on the outer circle.
The derivation also involves using the definition of arc length, as well as simplifying the final equations by first deriving the trig identities for sin(π-x) and cos(π-x). Although the derivation gets a bit messy because of the many angles and trigonometric ratios, it is definitely worth following the video very closely to see how distances can be derived using trigonometry. So make sure to watch this video, and watch it slowly!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhuIFzu-V4_zu15qzyQ
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/running-circles-around-circles-part-5-epicycloid-proof
Related Videos:
Laboratory Project: Running Circles Around Circles: Part 4:
Laboratory Project: Running Circles Around Circles: Part 3:
Laboratory Project: Running Circles Around Circles: Part 2:
Laboratory Project: Running Circles Around Circles: Part 1:
Parametric Curves: Superellipses:
Parametric Curves: Example 11: Conchoids of Nicomedes:
Parametric Curves: Example 7: The Cycloid: Proof Part 1:
Parametric Curves: Example 6: Graphing Devices:
Parametric Curves: Example 5: Lissajous Figure:
Parametric Equations and Curves:
Parametric Equations and Polar Coordinates: .
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I don't always prove parametric equations for an epicycloid but when I do it's usually pretty epic ;)
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/running-circles-around-circles-part-5-epicycloid-proof
Very good article friend, these issues very little are in the community because they stop another language, from college just remember that the first derivative gives me the speed while the second provides me the acceleration. If I've been wrong already is a long time. I support you with such interesting articles
Its quite interesting to know about all ths circle things. Didnt even think that curves can produce illusive movements too. A really informative post.
Cheers man.