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

Author: Steven H. Izen
Journal: Proc. Amer. Math. Soc. 135 (2007), 269-276
MSC (2000): Primary 65H05; Secondary 51N05, 65R10
Published electronically: July 28, 2006
MathSciNet review: 2280195
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Abstract: 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.

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Additional Information

Steven H. Izen
Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106

Received by editor(s): February 26, 2004
Received by editor(s) in revised form: July 25, 2005
Published electronically: July 28, 2006
Communicated by: M. Gregory Forest
Article copyright: © Copyright 2006 American Mathematical Society