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I am reading a paper by Christine Bernardi, available here http://epubs.siam.org/doi/pdf/10.1137/0726068, my question relates to page 1237, which I shall elaborate on:

In this part we have the unit sphere (in $\Bbb R^3$), and a regular tetrahedron $T$ inscribed in the sphere. The paper states that if we let $\omega_j$ be the faces of $T$, then the six planes containing an edge of $T$ and the origin divide the unit sphere F into four equal parts, $\Gamma_j$.

The paper then reads that each $\Gamma_j$ is the stereographic projection of $\omega_j$, but my question is how one defines the stereographic projection of a face (i.e. a plane) onto a sphere, since to my knowledge, one would project a sphere onto a plane.

Thanks in advance for any help!

As an addendum I believe I have figured out what is meant by this type of projection though I welcome any comments or criticisms, I believe you consider the face $\omega_j$, and at each point $x\in\omega_j$ you consider the inscribed sphere with its (relative) north pole at $x$ and south pole lying on the unit sphere, you then project the point onto the plane tangent to the south pole of this sphere (which lies on the original unit sphere), taking all projections of all points $x\in\omega_j$ we get $\Gamma_j$.

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For what it's worth at this late date (and not having read the paper): I'd guess the author means "radial projection from the origin", which sends each point $x$ on the tetrahedron to its normalization $x/\|x\|$:

Faces of a tetrahedron projected onto a sphere

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