Describe the cycloid described by a wheel rolling inside a circle

I'm trying to describe the cycloid described by a wheel rolling inside a circle:
The (rotor)vector attached to the center of the rolling wheel is described by this equation:
r = r cos(w t)e(x') + r sin(w t)e(y')

w is the angular velocity of the wheel, e(x') and e(y') are the unit vectors and t is the time

This is the relative position vector of the point on the wheel that touches the big circle as the wheel rolls.
The origin of the absolute coordinate system x y is the center of the big circle, the origin (the center of the wheel) of the other coordinate system x' y' (where x' always stays parallel to x and y' to y) moves in a smaller circle with radius R - r, R is the radius of the big circle, r is the radius of the wheel. So the position vector of the origin in the absolute coordinate system is given by:

p = (R - r) cos( (r/R) w t)e(x) - (R - r) sin( (r/R) w t)e(y)

The angular velocity W of the smaller circle is given by (r/R)*w, since r*w = R*W (the wheel doesn't slip). The minus sign is because the wheel rolls clockwise.

r + p should describe the curve, now, when R = 4r I should plot an astroid, which is not the case, where have I gone wrong?


Science Advisor
Homework Helper
Gold Member
Re: Astroid

You have taken into account that it doesn't slip, but you haven't taken into account that the little wheel is additionally moving in a circle itself. So the amount the little wheel turns if the big wheel turns wt is (R/r)wt + wt. Try that.

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