Determining a rotation of a tetrahedron from a projection

TL;DR

This paper investigates the problem of identifying an unknown rotation of a tetrahedron in three-dimensional space based on its orthogonal projection, with applications in medical imaging.

Contribution

It provides conditions under which the rotation of a tetrahedron can be uniquely determined from its projected vertices.

Findings

Rotation can be uniquely determined under certain geometric conditions.

The problem is formulated in the context of medical imaging applications.

Conditions for reconstructing the rotation are explicitly characterized.

Abstract

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