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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    $\begingroup$ It's unclear what you are asking. There is more than one way to 'wrap a parabola around a circle. Depending on the curvature of the parabola (relative to that of the circle) there is either exactly one intersection at the vertex of the parabola or two intersections at equal distance from the vertex. (Assuming that the line connecting the midpoint of the circle and the vertex of the parabola is on the axis of symmetry of both of them and that there is a point of intersection.) $\endgroup$ – Stefan Mesken Apr 7 '16 at 12:03
  • $\begingroup$ If the curve y=x^2 is in the plane, wrap the plane about a sphere such that the x axis of the plane corresponds to any circumference of the sphere. $\endgroup$ – cjg123 Apr 7 '16 at 12:14
  • $\begingroup$ Sorry, I still don't get it. $\endgroup$ – Stefan Mesken Apr 7 '16 at 12:23
  • $\begingroup$ @cjg123 For "sphere", do you mean circle? Like laying down the parabola onto the circle? But there could be many circles that you can choose from. Mind explaining? $\endgroup$ – lEm Apr 7 '16 at 13:17
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    $\begingroup$ Sure - No, I mean a 3d sphere, like a ball. So take y=x^2 in the plane and wrap it around a ball (lets say of unit radius). So imagine an infinitely large piece of paper with the function y=x^2 printed on it, and it being wrapped around a ball of unit radius. It should be wrapped in such a way that the flat sheet is transformed into a surface of nonzero curvature (acting as the new surface of the ball), and will be continually wrapped around the ball an infinite number of times. Now, project the resulting curve created onto the original surface of the ball. $\endgroup$ – cjg123 Apr 7 '16 at 13:30

(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):enter image description here

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.

  • $\begingroup$ Excellent, great response! Your interpretation of my question was spot on. Yes, I realized that the issue you pointed out would arise, but I thought it might be handled by continually folding the edges until only an infinitesimal gap remained, and thus the "sheet" approaches the full covering of the surface of the ball. Regarding the toilet paper tube question, could it be that the tube is topologically isomorphic to the paper itself, and so you can wrap it without the issues experienced with the sphere? $\endgroup$ – cjg123 Apr 7 '16 at 14:10
  • $\begingroup$ The stereographic projection bit is interesting. In this case, would it sort of be like a reverse stereographic projection, since the mapping is from 2-d to the surface of a 3-d figure, instead of the opposite? $\endgroup$ – cjg123 Apr 7 '16 at 14:13
  • $\begingroup$ Yes, I guess "stereographic projection" usually means mapping the sphere to the plane, so I am talking about the inverse map. $\endgroup$ – Nefertiti Apr 7 '16 at 14:14
  • $\begingroup$ About your other points: I guess I don't know what "continually folding the edges" means --- this stuff gets hard to describe in words! Anyway, all I was trying to get at in my response was that once this kind of crumpling or folding starts to happen, it is (to me at least) difficult to understand what is going on, whereas if we allow some shrinking instead then we get a very nice picture. $\endgroup$ – Nefertiti Apr 7 '16 at 14:55
  • $\begingroup$ About the toilet roll question: it is not just a question of topology, which doesn't care about stretching or shrinking, but really of geometry: wrapping without crumpling or shrinking is an isometry. For more details, you can try this Wikipedia article: en.wikipedia.org/wiki/Gaussian_curvature $\endgroup$ – Nefertiti Apr 7 '16 at 14:56

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