Hmm, now that I investigate further my post is making less and less sense. First, you are absolutely right when say a helix isn't in any one plane, so it certainly is NOT a circle projected 45 degrees back. I tried recreating the revolved helix in Solidworks, and it wouldn't let me, since it's a 3-dimensional curve... any ideas there (AutoCAD seemed to do it fine...)? Anyways, I can't confirm what I posted earlier.
A projected circle would be an ellipse with a=r and b=r*sqrt(2). The problem with using this ellipse as a profile is that you are rotating around an axis 45 degrees offset from the wheel. Looking at the profile of the wheel (along its axis of rotation) part of the elliptical profile will be revolved to be
in front of the profile, so the profile is not always necessarily the point of contact or the farthest point from the center of the wheel. See what I'm saying? This inspired to do some insane algebra to find a curve that, when revolved, would satisfy this. Lots of circle to ellipse tangency, differentiating, and disgusting algebra that wolfram alpha couldn't solve, so that got me nowhere as well.
There seems to no easy solution to this problem, but that doesn't I'm going to stop trying!
