Parabolic sheaves and limit circles in the isotropic plane (Q1919317)

Let \(\Delta = \Delta(t)\) be a triangle formed by three neighbouring tangents of a \(C^\infty\)-curve \(k(t)\) in an isotropic plane. Generalising an idea due to O. Röschel (1974) the author regards all inscribed parabolas \(\pi(s,t)\) of \(\Delta(t)\). Let \(U(\Delta)\) be the circumcircle of \(\Delta\) and \(I(\delta)\) the incircle of the triangle \(\delta = \delta(t)\) whose midpoints of the sides are the vertices of \(\Delta(t)\); \(U(\Delta(t))\) is the locus of the isotropic focal points of \(\pi(s,t)\), and \(I(\delta(t))\) is the envelope of the isotropic axes of \(\pi(s,t)\). The author proves that the Abramescu-circle \(\lim U(\Delta(t))\) (cf. H. Sachs, 1972) is identical with the locus of the focal points of \(\lim \pi(s,t)\), and that the circle \(\lim I(\delta(t))\) is the envelope of the isotropic axes of the parabola \(\lim \pi(s,t)\). The characteristic points, different from \(k(t)\), of the circles \(\lim U(\Delta(t))\) and \(\lim I(\delta(t))\) determine the direction of the affine normal of \(k(t)\).