Prove parametric equations trochoid


enter image description here take a hard look at picture take some fixed point anywhere on the circle and when the circle is rolled around the curve drawn by the the fixed point is the TROCHOID

in the diagram i have drawn i labelled it as point D some where on circle also i have taken the angle to be "a"

from the diagram you can see that point D is online CP and P started initially from origin and it has travelled by an angle of "a" so PR=OR= ra ( PCR is a sector with angle a and arc length ra)

our job is to find the x and y co-ordinates of D which is making the curve

from triangle CDQ

CD=d which is given in our question

CQ= d $cosa$ and DQ= d $sina$

now x-coordinate of point D= OR- DQ

SO x= r*a-d $sina$

and y- coordinate of point D = CR- QR

y=r- d $cosa$

hence the parametric representation of the curve is

x= r*a-d $sina$

y=r- d $cosa$


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Sam Creamer
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Sam Creamer

Updated on August 01, 2022


  • Sam Creamer
    Sam Creamer 1 day

    I have to show that the parametric equations of a trochoid are:

    $x = r\theta - d\sin\theta$ and $y=r-d\cos\theta$

    where r is radius and d is the distance between center of the circle and a point P.

    Can someone please explain this to me? I'm in my second week of advanced Calculus, thanks

    • Spine Feast
      Spine Feast almost 9 years has a reasonable derivation, have you studied it?
    • Sam Creamer
      Sam Creamer almost 9 years
      not really... it was just thrown at me and I had never seen it before and I don't think I fully understand it
  • Mr. Math
    Mr. Math almost 9 years
    my pleasure hope it helps