# Coordinates of octahedron's vertices and checking if a point is inside it.

Given that I have the distance between the center of an octahedron and any of its faces (regular octahedron, so all the distances are equal), how can I calculate the coordinates of its vertices, considering that the octahedron may have rotation in any of the 3 Euler angles? I can't find this formula anywhere and I'm having serious trouble calculating it. Also, after calculating the coordinates of the vertices, I need to check if a point is inside or outside the octahedron. I am thinking of doing something like:

$\alpha_1 = dR *\frac{L_1}{L_1^2}$

$\alpha_2 = dR * \frac{L_2}{L_2^2}$

$\alpha_3 = dR * \frac{L_3}{L_3^2}$

And dR is the vector from the point being checked to the center of the octahedron. $L_1$, $L_2$ and $L_3$ are the vectors from the center of the octahedron to the nearest vertices to the point. And then if:

$\alpha_1 + \alpha_2 + \alpha_3 \leq 1$

The point is inside the octahedron. Is my calculation correct? Sorry if this question is confusing, I've never posted a question in a math forum and I'm not a native English speaker.

Wikipedia says that the radius of an inscribed sphere (tangent to each of the octahedron's faces) is $r=L/\sqrt 6$, where $L$ is the length of the edges of the octahedron's. The distance you know is this radius. The coordinates of the vertices are $(\pm R,0,0)$, $(0,\pm R,0)$, $(0,0,\pm R)$, where $R$ is the radius of a circumscribed sphere and which is given by $R=L/\sqrt 2=r \sqrt 3$.