- Supposing a four-dimensional euclidian space, with a right-handed basis of (i, j, k, l).
- Supposing a 2-sphere S with radius r such as:
$$x^2 + y^2 + z^2 = r^2, w = 0$$
What is the shape of the projection of S along an arbitrary axis into a three-dimensional space orthogonal to said axis?
These specific cases seem straightforward:
- Projections along basis vectors i, j, k yield a circle or radius r.
- Projection along basis vector l yields a sphere of radius r.
First hypothesis: By analogy with a circle projected from 3D space onto a plane, I tend to believe that the general case yields an oblate spheroid whose major axis has a length of r and whose minor axis is in [0,r]. My intuition is reinforced by the fact this is the simplest way to cover both specific cases.
In case the my first hypothesis is right, what is the shape of obtained by projecting a spherinder (a 4D cylinder with equal-radius 2-spheres at each end) in the same way? (In case the first hypothesis is wrong, I guess I'll need to open a second topic.)
Second hypothesis: We get an oblate spheroid (as in hypothesis 1) swept along a segment which corresponds to the projection of the spherinder's height.
How can I verify each hypothesis? What would be broad steps to proofs? (I don't need detailed proofs, just the steps.)
Your first hypothesis is correct, but it can be made more explicit:
If the angle from your "axis" to the $\ell$-direction is $\theta$, then the minor axis will have length $r \cos \theta$.
Your second hypothesis is also correct, but again can be made more explicit: the length of the swept segment (assuming the original segment has length $k$ and is aligned with the $\ell$ axis!) will be $k \cos(\theta)$.