[Math] Prove parametric equations trochoid

Question

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

Answer 1

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 $cosa$