Exercise 4: Parametric Equations of a Trochoid

In this video I derive the parametric equations of a trochoid, which is the shape obtained by tracing a point as a circle rotates. If the point traced is on the circle itself, then the shape is the famous cycloid. When the point is inside the circle, the shape is a curtate trochoid, and when the point is outside the circle, the shape is a prolate trochoid, which has the characteristic loop. I derive the equations by using basic trigonometry and the fact that the distance traveled by the circle is equal to its arc length because there is no slippage as it rotates. I graph out the various types of trochoids using Desmos 2D graphing calculator, including an amazing animated graph someone made on Desmos!

#math #trochoid #cycloid #desmos #calculus

Timestamps

• Exercise 4: Trochoid: 0:00
• Solution: Recap on the cycloid: 2:56
• Sketching a curtate trochoid where d is less than r: 4:04
• Distance travel is the arc length: 7:08
• Zooming into our circle diagram: 8:20
• Coordinates of Q and P: 11:36
• Recall trigonometric identities for subtracting by pi: 11:36
• Simplifying our coordinates of Q and P: 16:16
• Coordinates of P are the parametric equations of our Trochoid: 18:15
• When d = r, we get a cycloid: 18:50
• Prolate trochoid has d greater than r: 20:41
• Graphing the trochoid with Desmos 2D graphing calculator: https://www.desmos.com/calculator/zrw9kc2tmb 20:58
• Amazing Trochoid animator on Desmos: https://www.desmos.com/calculator/1orvf2yhrk 23:00

Notes and playlists

• Summary: @mes/re-leothreads-2y6of6khw
• Sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GwdMY-E9_4LsmYmO-IxcoX
• Notes: @mes/omhbcxkw
• Vector Functions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HQl7xTQBS_O8Te8PcpNm4U .

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