If a basic quadratic, $y=x^2 $ for example, represented on a 2-d sheet, is wrapped around a sphere, will the resulting curve intersect itself at a given point, more than once?
closed as unclear what you're asking by Stefan Mesken, David Quinn, Jesko Hüttenhain, Daniel W. Farlow, zz20s Apr 7 '16 at 14:27
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(This is really a long comment.)
I think I understand what you are asking --- and I think it is a nice question!
I am imagining "wrapping" as a function $w: \mathbf R^2 \rightarrow S^2$, where $S^2$ is the sphere, that tells us where each point in the plane (your "2-d sheet") should go to on the sphere. You also require that $w$ maps the $x$-axis in $\mathbf R^2$ to a circumference of the sphere.
You are then asking about the image $w(P)$, where $P$ is the parabola $y=x^2$.
However, there is a basic issue with this set-up, namely curvature. Have you ever tried to wrap a sheet of paper around a ball? You find that it is impossible to do so without crumpling the paper in some way. This is precisely because the sheet of paper is flat, but the surface of the ball is curved. (Fun question: why doesn't this happen when you wrap the toilet paper around the cardboard tube?) In fact, it is impossible to map even a tiny piece of the sheet onto the sphere in a way that preserves distances (i.e without stretching or shrinking) and does not "crumple" the sheet.
On the other hand, if you are willing to shrink as you wrap, then there is a very elegant wrapping function called stereographic projection. Rather than try to describe it in words, let me just reproduce the nice illustration from the Wikipedia article (created by User:Mark.Howison):
I will leave it as an exercise to imagine how a parabola in the plane will map to the surface of the sphere under this function.