A Fast Algorithm to Compute the Π-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 π-line. A new constructive algebraic proof of this result is presented along with a fast algorithm to compute the endpoints of the π-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 π-lines.