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Question: Can a disc drawn in the Euclidean plane be mapped to the surface of a hemisphere in Euclidean space ?


If $U$ is the unit disc drawn in the Euclidean plane is there a map, $\pi$, which sends the points of $U$ to the surface of a hemisphere, $H,$ in Euclidean space ?

Background and Motivation:

If $U$ is the unit disc centered at the origin consider $n$ chords drawn through the interior of $U$ such that no two chords are parallel and no three chords intersect at the same point. The arrangement graph $G$ induced by the discs and the chords has a vertex for each intersection point in the interior of $U$ and $2$ vertices for each chord incident to the boundary of $U.$ Naturally $G$ has an edge for each arc directly connecting two intersection points. $G$ is planar and $3-$connected. I know that $G$ has $n(n+3)/2, n(n+2)$ and $(n^2+n+4)/2$ vertices, edges and faces respectively. That $G$ is Hamiltonian-connected follows from R. Thomas and his work on Plummer's conjecture. I have conjectured that $G$ is $3-$colourable and $3-$choosable.

Independently Felsner, Hurtado, Noy and Streinu :Hamiltonicity and Colorings of Arrangement Graphs, ask if the arrangement graph of great circles on the sphere is $3-$colourable. In addition they conjecture such an arrangement is $3-$choosable.

Now I began to think the following

  1. Show my graph $G$ is $3-$colourable
  2. $\pi:G \to H$
  3. Glue $H$ to a copy of itself at the equator

then I could solve the conjecture by Felsner and his colleagues. Moreover if the map $\pi$ is bijective then any solution to Felsner's conjecture will also solve mine. The map $\pi$ does not need to preserve angles or surface areas. $\pi$ necessarily will have to map chords to great circles. See @JohnHughes excellent answer concerning the map $\pi$ and why vertical projection will not work.

enter image description here

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If by a "circle" you mean the set of all points inside a circle (e.g., points whose distance from some center $C$ is less than or equal to 1), then the answer is "yes" and one solution is called "stereographic projection;" another is "vertical projection".

If you have a point $(x, y)$ in the unit disk (the "filled in circle"), the corresponding point, using vertical projection, is $(x, y, \sqrt{1 - x^2 - y^2})$.

For stereographic projection, you send the point $P = (x, y)$ to a new point $Q$: \begin{align} h &= x^2 + y^2 \\ Q &= (\frac{x}{h+1}, \frac{y}{h+1}, \frac{h-1}{h+1}) \end{align}

The latter has the charm that it takes chords in your disk to great-circle arcs in the hemisphere, and preserves angles of intersection, although not lengths.

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  • $\begingroup$ I thought Stereographic Projection required the "whole" sphere. $\endgroup$ – Antonio Hernandez Maquivar Apr 5 '16 at 14:50
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    $\begingroup$ Stereographic projection from the north pole $(0,0,1)$ of a unit sphere whose equator is on the $xy$-plane will take points in the lower hemisphere to those within the unit disk in the plane, and points in the upper hemisphere to those outside the unit disk. So the map I've given you will send points of your disk (for which $h < 1$) to points in the southern hemisphere (for which $z < 0$), as you can see by noting that $h + 1$ is positive, but $h-1$ is negative. $\endgroup$ – John Hughes Apr 5 '16 at 15:02
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    $\begingroup$ If by "blow it up" you mean "project vertically," then you'll end up with a circle on the sphere if you do what you've described. But it won't in general be a great circle (unless your initial line segment is a diameter of the disk), and the angles between segments will not be preserved as they are when you use stereographic projection. $\endgroup$ – John Hughes Apr 5 '16 at 17:56
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    $\begingroup$ OK...then any bijection at all will do. But for others who might care about that, I'm gonna leave it in my answer. :) $\endgroup$ – John Hughes Apr 5 '16 at 22:20
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    $\begingroup$ If two arcs (not necessarily circular -- any nice image of a closed interval will do) on the hemisphere intersect, then their images, under any bijection with the disk, intersect in the disk, and vice versa. Indeed $f(A \cap B) = f(A) \cap f(B)$ for any bijection $f$ and any two subsets of $f$'s domain. $\endgroup$ – John Hughes Apr 6 '16 at 14:59
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I think you mean to say disk instead of circle.

If you mean disk then you can do it.

Just imagine of punching the disk from center, it will give you a hemisphere.

And if you want precisely the map then you can get it from Stereographic projection.

EDIT

It will look something like this when you punch a disk.

enter image description here

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Projecting the hemisphere onto the circle bounded by the equator from a point infinitely far above the North pole maps the hemisphere bijectively to the circle. Take the inverse of this map.

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