Feet in Orthogonal-Buekenhout-Metz Unitals

TL;DR

This paper investigates the geometric structure of the set of feet of a point outside an Orthogonal-Buekenhout-Metz unital in projective plane, revealing they form either a line or two arcs, with additional geometric insights.

Contribution

It provides a geometric characterization of the feet sets in Orthogonal-Buekenhout-Metz unitals, distinguishing between collinear points and arc partitions.

Findings

Feet sets are either $q+1$ collinear points or partitioned into two arcs.

Geometric properties of feet sets are characterized.

Additional geometric properties of these sets are explored.

Abstract

Given an Orthogonal-Buekenhout-Metz unital $U_{\alpha,\beta}$, embedded in $PG(2,q^2)$, and a point $P\notin U_{\alpha,\beta}$, we study the set of feet, $\tau_{P}(U_{\alpha,\beta})$, of $P$ in $U_{\alpha,\beta}$. We characterize geometrically each of these sets as either $q+1$ collinear points or as $q+1$ points partitioned into two arcs. Other results about the geometry of these sets are also given.