Suppose I am given an arbitrary 3-dimensional *convex polyhedron* $P\subset\Bbb R^3$ that contains the origin.
I can "blow it up" to a *spherical polyhedron* by projecting all edges and vertices (away from the origin) to the unit sphere (centered at the origin):

![](https://i.stack.imgur.com/t6Q9e.png)
What about the other direction?

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> **Question:** Given a spherical polyhedron, is there a "convex polyhedron" whose projection is *exactly* the given spherical polyhedron? And how to construct it explicitly?
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For me, a spherical polyedron is a tiling of the 2-sphere where the edges are great circle arcs.
And I know that there is always a convex polyhedron with the same combinatorics as the given spherical polyhedron, but I ask specifically about a convex polyhedron that *projects* to the given spherical polyhedron.