A fast algorithm to compute the $\pi$-line through points inside a helix cylinder

In the context of helical cone-beam CT, Danielsson et al. discovered that for each point interior to the cylindrical surface containing a given helix, there is exactly one line segment passing through the point which intersects two points less than one turn apart on the helix. This segment is called a $\pi$-line. A new constructive algebraic proof of this result is presented along with a fast algorithm to compute the endpoints of the $\pi$-line through an arbitrary point in the interior of the helix cylinder. This proof exposes the geometry of the decomposition of a cylinder interior as a disjoint union of $\pi$-lines.