Modified Cycloid motion of charged particle

[peasqueeze]

I need a charged particle trajectory in crossed E and B that is a cycloid in the x-y plane. This part is simple enough, but I want the trajectory to be parabolic in the x-z plane. The images will help explain what I need.

The left image shows the shape I want to achieve, and the right image shows my current attempt. The fields producing the image on the right are...
$$
\vec{B} = B_{0}\hat{z}
$$
$$
\vec{E}=\gamma E_{y}\hat{x}+E_{y}\hat{y}+E_{z}\hat{z} ; E_{z}<<1
$$

So- how can I modify the above electric and magnetic fields to produce this 'parabolic cycloid'
Note: I know the image on the right doesn't look like a cycloid exactly, but, for my model, I needed really tight loops. So don't get hung up on that detail.

 

[gtoguy97]

The key to achieving the desired trajectory is to modify the electric field so that it has an additional component in the x-direction. This component should be of the same magnitude as the y-component, but opposite in sign. In other words, you should have:$$\vec{E}= \gamma E_{y}\hat{x} - E_{y}\hat{y} + E_{z}\hat{z} ; E_{z} << 1$$This will produce a cycloid in the x-y plane and a parabolic trajectory in the x-z plane.