I'm trying to describe the cycloid described by a wheel rolling inside a circle:(adsbygoogle = window.adsbygoogle || []).push({});

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

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# Describe the cycloid described by a wheel rolling inside a circle

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