Projecting 3-Sphere onto a Plane | Stereographic Projection

[Carol_m]

Hello,

Is it possible to project a hypersphere (a 3-sphere) onto a plane? is this possible using stereographic projection?

Please, if this is possible I would appreciate a nice explain me about how to do it.

Thank you!

Carol

 

[xts]

Of course, not!
You may stereographically project the 2-dimensional sphere (bended in 3 dims) on 2-dimensional flat plane. Similiraly, you could develop some projections of 3-D hypershere, bended in 4th dim onto flat 3-D space. But not on 2-D one.

You may think about projecting it first on flat 3D Euclidean space, and then use some common method of visualisation to present it as a 2-D picture.

 

[Carol_m]

Hello xts,

Thank you for your reply and sorry if I am being too naive but I do not follow your answer. Could you please explain me a bit more? (sorry but this is not my background)

From what I have read in textbooks I know that you can project the surface of a sphere in a 2-D plane, my problem is that I want to project every point in the sphere (not only the surface) on a 2D plane, that's why my question was about hypersphere projection (but maybe I am also confuse about this??).

If you can help me about this I would really appreciate it and sorry again if I am asking the obvious.

Thank you!

 

[xts]

I want to project every point in the sphere (not only the surface) on a 2D plane

Then you must project infinite number of points (some line segment) of your sphere into a single point on your plane.

 

[pmacias]

,

Yes, it is possible to project a 3-sphere onto a plane using stereographic projection. This is a commonly used technique in mathematics and geometry to visualize higher dimensional objects in lower dimensions.

To understand how this projection works, let's first review what a 3-sphere is. A 3-sphere, also known as a hypersphere, is a four-dimensional object that is analogous to a three-dimensional sphere. It is defined as the set of points in four-dimensional space that are equidistant from a fixed point, just like how a sphere is the set of points in three-dimensional space equidistant from a fixed point.

Now, imagine placing a plane tangent to the 3-sphere at a single point. This plane is known as the stereographic plane. In stereographic projection, we draw a line from the center of the 3-sphere to any point on its surface and extend it until it intersects the stereographic plane. The point of intersection is the projected point of the original point on the 3-sphere. This process is repeated for every point on the 3-sphere, resulting in a projection of the 3-sphere onto the plane.

The stereographic projection preserves certain properties of the 3-sphere, such as distance and angles, making it a useful tool for visualizing this higher dimensional object. It is also a conformal projection, meaning that it preserves the local shape of the 3-sphere at each point.

I hope this explanation helps you understand how to project a 3-sphere onto a plane using stereographic projection. If you have any further questions, please let me know.