Determining a rotation of a tetrahedron from a projection

Abstract

The following problem, arising from medical imaging, is addressed: Suppose that $T$ is a known tetrahedron in ${\mathbb{R}}^3$ with centroid at the origin. Also known is the orthogonal projection $U$ of the vertices of the image $\varphi T$ of $T$ under an unknown rotation $\varphi$ about the origin. Under what circumstances can $\varphi$ be determined from $T$ and $U$?

16 pages, 3 figures; minor revision based on reviewers' comments