Suppose I have two 4D points, $\mathbf{a}=(a_1,a_2,a_3,a_4)$ and $\mathbf{b}=(b_1,b_2,b_3,b_4)$, that both lie on a unit 3-sphere (i.e. unit distance from origin). In addition, I have a 2-D plane that is formed by the intersection of $\mathbf{p}\cdot\mathbf{x}=0$ ($\mathbf{p}$ is known) and the 3-sphere. I am trying to find the intersection of this plane and the unique "line" that passes through $\mathbf{a}$ and $\mathbf{b}$ within the 3-sphere (analogous to a great circle in spherical geometry). What system of equations would I solve to find this intersection?
2026-03-26 12:41:22.1774528882
intersection of a line and plane on a 3-sphere
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