Can a Center of a Sphere be conceived at infinity

[neelakash]

Hello everyone...I was wondering if it is possible to conceive a 2nd center of a finite sphere in infinity...(I am not a mathematician and therefore, my words might look ridiculous)

apparently, it looks like every point on the sphere is at the same distance from infinity...Anyway, this statement is incorrect.What I should say is that say P is a point at $$\ r\rightarrow\infty$$ from the center O of the sphere. In this limit the distance PS where S is any point on the sphere is the same...That is PS1=PS2=...=PSn

One of my fiends told that the answer depends on the definition of the center of the sphere: like if one defines the center as the point of intersection of two diameters etc...In that case, I would say it's a chicken and egg problem for there is scope to ask to define diameter.

The problem actually appears for in the context of image problems. One can show that if a charge is placed outside a sphere and is interested in the same region,an image charge may be conceived inside the sphere. This is done in all undergrad books.However,one can also show that if you place the real charge inside the sphere and try to calculate the potential inside,you get the image charge outside the sphere. In fact, it is just the inverse problem. The value and positions of the image charge are inverted with respect to the sphere.

Once you are inside the sphere,the outside world is outside the spherical surface (that looks concave to you). However, it is the same surface by which you are separated from an observer outside the sphere. If the definition of convexity or concavity is a matter of terminology to both of you you should see the outside world confined within a sphere of radius infinity...

What do you think about this?

 

[arkajad]

Hello everyone...I was wondering if it is possible to conceive a 2nd center of a finite sphere in infinity..

Of course it is possible. That is what the http://en.wikipedia.org/wiki/Inversive_geometry" does: interchanges the origin inside with the point at infinity outside.

It is much like the equator circle may be thought of as having two centers - one on the North pole and another one on the South pole - if you are restricted only to the two dimensional surface of the Earth. Similarly you can realize 2d sphere of a 3d space as an "equator" of a 3d sphere in 4d space.

 

[neelakash]

@arkajad: Thank you very much for your reply. It is interesting to note that the radius vectors (from these two centers on the sphere) are not collinear in general.

 

[arkajad]

@arkajad: Thank you very much for your reply. It is interesting to note that the radius vectors (from these two centers on the sphere) are not collinear in general.

As long as you stay on the surface of the sphere - they are. They are on the same great circle. And great circles are "straight lines" on the surface of a sphere.

 

[neelakash]

I agree...

 

[neelakash]

@arkajad: I understood what you suggested. But how do I know if the inverse point is actually a center? May be because it is difficult to visualize, I am having some problem...Is it something like this:

From inside of the sphere,the outside looks as a sphere (let us call this as inverse/ complementary sphere) bounded by the same spherical surface (as far as the problem is concerned). From a point on this surface, the normal to this outside sphere passes through its center. How would I know this normal will correspond to a line parallel to the axis out to infinity (it is when we are looking from outside)...

 

[arkajad]

Get rid of one dimension and think of the two-sphere and its equator. Note also that the conformal inversion preserves the angles but not distances. So the concept of two different lines being parallel to each other is not well defined.