parametric curve between points $x, y \in S^{n}$ on $n$-sphere

Question

This is somewhat related to my question here.

I'd like to take any arbitrary points $x$ and $y$ on the $n$-sphere and connected them with a curve $\gamma : [0,1] \to S^{n}$ such that $\gamma(0) =x$ and $\gamma(1) = y$.

I believe that this problem reduces to finding a such a curve between the points $(1,0,...,0)$ and arbitrary $x$ on the $n$-sphere. How can this be done?

I'd like to get something in the spirit of: $$ \gamma(t) = \mathrm{something} \cdot \cos(2 \pi t) + \mathrm{something} \cdot \sin(2 \pi t) $$

I know that the vector $\tfrac{1}{\sqrt{n-1}}\left( 0, 1, \ldots, 1 \right)$ is a a unit vector that is normal to the point $(1,0,...,0)$, so I'm thinking I would need to use this in the construction at some point.

EDIT: I'm not looking for a proof that $S^{n}$ is path-connected - I'd really just like to see how such a path would look like - explicitly.

Answer 1

as long as they are not identical or antipodal, they define a 2-plane. One of them can be a basis vector, the other (orthogonal) basis vector is found using the second vector and Gram-Schmidt. Apparently you want $0,1,$ Then, you get $v \cos (\pi t/2) + w \sin (\pi t/2).$ Now that I thinkof it, this gets us to $w$ at $t=1.$