Images of conics under derivation

Abstract The Hall plane of order q 2 is constructed from the Desarguesian plane of order q 2 by the process of derivation with respect to a fixed derivation set on a fixed ideal line. The affine points of a nondashdegenerate conic in the Desarguesian plane of order q 2 can be regarded as a set of points in the corresponding Hall plane. We give a classification of the structure of sets of points in Hall planes arising from conics with two points in the derivation set. Further, if q is even, we classify the sets of points in Hall planes arising from conics which admit the ideal line as a tangent and are such that either the point of contact or the nucleus of the conic (or both) is contained in the derivation set.