Orthogonal Zigzag Operation in An Absolute Plane and Criteria for Quasi-Parallelism

In works by Karzel, Pianta, Rostamzadeh, and Taherian, quasi-parallel lines — pairs of lines without intersection and common perpendicular— were introduced to classify absolute planes. However, the properties and characterization of this geometric relationship remain unexplored. In this paper, we answer the fundamental question, "When are two distinct lines in an absolute plane quasi-parallel?" Even more generally, "How to recognize two lines from their geometric relationship?"
To answer these questions, we introduce a novel geometric operation called the orthogonal zigzag operation, which provides new criteria for quasi-parallelism and other line relationships. Additionally, it offers a new perspective on the classification of absolute planes.
For distinct lines A and B and a point x∈A, the orthogonal projection π_A maps x to its foot on B, and π_B maps this foot back to a point x^''∈A. The composition π_AB=π_B∘π_A: A→A which maps x to x'' is called the orthogonal zigzag operation. We show this operation characterizes all possible line relationships between lines: π_AB is identity if and only if A and B are coperpendicular; constant if and only if A and B are orthogonal; has a contraction to a fixed point if and only if A and B intersect but are not perpendicular; contracts to a fixed point (extends from a fixed point) if and only if A and B have a coperpendicular and the plane has hyperbolic (elliptic) congruence, respectively; and finally, it is a fixed-point-free isotone permutation if and only if A and B are quasi-parallel.