What maps to a plane in a stereographic projection of a 3-sphere to $\mathbb{R}^3$?

Question

Any straight line in a stereographic projecton from a 2-sphere to $\mathbb{R}^2$ maps back to a circle passing through the projection point. That I can see why: the rays between the projection point and the straight line defines a half-plane that intersects with the 2-sphere and passes through the projection point. Since the intersection between a sphere and a plane is a circle, a circle it is.

What about a plane in a stereographic projecton from a 3-sphere to $\mathbb{R}^3$ ?

I assume it maps back to a 2-sphere passing through the point of projection in the 3-sphere, but I am just extrapolating from the lower dimension case.

EDIT: Backgroung: in my game, I use a stereographic projection to map the surface of a 3-sphere (the game world, where points are represented by 4-D unit vectors), into a "ordinary" 3-D flat space, in order to be rendered. I want to know what the far clipping plane in my view frustum maps back into 4-D coordinates. I hope it is a sphere. Is it?

Answer 1

It is a 2-sphere embedded in the 3-sphere.

Reasoning as you did in the lower-dimensional case, you could say that the set of rays from the projection point in the 3-sphere to the plane in $\mathbb{R}^3$ form a half-hyperplane intersecting the 3-sphere at the projection point. The intersection between the 3-sphere and a hyperplane is a 2-sphere.

But let's check that using coordinates. Stereographic projection of the sphere $x_1^2+x_2^2+x_3^2+x_4^2=1$ from $(0,0,0,1)$ to $\mathbb{R}^3$ has the coordinate expression \begin{equation} (x_1,x_2,x_3,x_4) \mapsto \left( \frac{x_1}{1-x_4} , \frac{x_2}{1-x_4} , \frac{x_3}{1-x_4} \right) \end{equation} A point $(x_1,x_2,x_3,x_4)$ in the 3-sphere maps to the plane \begin{equation} Ay_1 + By_2 + Cy_3 + D = 0 \end{equation} in $\mathbb{R}^2$ if and only if \begin{equation} Ax_1+Bx_2+Cx_3-Dx_4 + D=0 \end{equation} This is the equation of a hyperplane through $(0,0,0,1)$.

That a hyperplane does in fact intersect a 3-sphere in a 2-sphere can be verified either synthetically or analytically, using the lower-dimensional case as a guide.