Suppose we have a particle of charge Q, mass m at rest at the origin at t=0, with an electric field E=Ek and a magnetic field B=Bi with i,j and k the cartesian unit vectors. Then the solution for the trajectory of this particle is x=0, y=E(wt-sinwt)/wB, z=E(1-coswt)/wB, where w=QB/m. These parametric desribe the motion of the particle, but they can be recast into the form (y-Rwt)2+(z-R)2=R2 where R=E/wB which also describes the particle motion. This form is that of a circle, radius R, with centre (0,Rwt,R) that travels in along the y-axis with speed Rw. Now my book states 'The particle moves as though it were a spot on the rim of a wheel, rolling down the y axis at speed Rw. The curve generated in this way is called a cycloid. However I've always had trouble being able to properly see this from the equations, but haven't worried about it too much. However I think not understanding it is stopping me from solving a similar problem with a slightly more complicated motion. So my problem is this. The equation (y-Rwt)2+(z-R)2=R2 describes the particle trajectory. All I can see from this is that at time t, the particle must be on a circle of radius R, centre (0,Rwt,R), which tells me the circle is moving along the y-axis. How can I see that the particle moves around this circle (and so appears to be a point on a rolling wheel)? And in order to appear like a rolling wheel, the particle would need a certain speed around the circumference of the moving circle (I believe it would have to move around the circumference at speed Rw, whilst the circle itself moves with this speed along the y-axis) - how do I see this is satisfied? Thanks.