# How to find the intersection between the great circle and a hyperplane?

Let $s = (\frac{1}{\sqrt{d}}, \ldots, \frac{1}{\sqrt{d}})$ and $u \in \mathbb{R}^d$ be two distinct unit norm vectors in the first orthant. Consider moving along the great circle defined by $s$ and $u$ (in the direction from $s$ to $u$) until the first intersection of the great circle and one of the following hyperplanes $\{x_1 = 0, x_2 = 0, \ldots, x_d = 0\}$ is reached. Let $v$ denote that intersection.

Is it possible to explicitly give an equation for the point $v$ or at least the angle between $u$ and $v$?

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$s$ and $u$ span a 2-dimensional plane, and the great circle that contains them is precisely this plane intersected with the unit sphere. Let $p_i$ be the point of intersection of the great circle and the hyperplane $\{x_i = 0\}$. Then you know $p_i$ is of the form $as + bu$, and you can solve the equations
$as_i + bu_i = 0$
$\lVert as + bu \rVert = 1$
where, $s_i$ and $u_i$ indicate the $i^{th}$ components of $s$ and $u$, respectively. This gives you the (typically) two points where the the great circle crosses the hyperplane, and by finding the $i$ whose solution has all nonnegative coordinates, you should get your answer.