Parabelscharen und grenzkreise in der isotropen ebene

Abstract

It is known by H. Sachs [5] that the classical curve theorem of ABRAMESCU also holds in isotropic geometry. Generalising an idea due to O. Röschel [2] we regard all inscribed parabolas ∏(s, t) of a triangle Δ(t). This triangle is formed by the tangents of three neighbouring points of a C^ω -curve k(t) in an isotropic plane. Let U(Δ(t)) be the circumcircle of Δ(t) and I(δ(t)) the incircle of the triangle δ(t) whose midpoints of the sides are the vertices of Δ(t). The circle U(Δ(t)) is the locus of the isotropic focal points of ∏(s, t) and the incircle I(δ(T)) the envelope of the isotropic axes of ∏(s, t). We prove that the ABRAMESU-circle — lim U(Δ(t)) — is identical with the locus of the focal points of lim ∏(s, t) and the circle lim I(δ(t)) with the envelope of the axes of lim ∏(s, t). The characteristic points, different from k(t), of the circles lim U(Δ(t)) and lim I(δ(t)) determine the direction of the affine-normal of k(t).