Modeling of curls of Agnesi in non-Euclidean planes

Abstract

We investigate analogs of an Agnesi’s curl in non-Euclidean planes. We have proved that the Agnesi’s curl of a co-Euclidean plane E2* is a cubic curve with a node at the absolute point of the plane E2*. The strip width between the tangents of the Agnesi’s curl of the plane E2* in the node does not depend on the base co-circle of the curl and is equal to the bigger acute angle of the Egyptian triangle. The type of the Agnesi’s curl of the elliptic plane depends on the radius of its base circle ω. If the radius r of ω is equal to a quarter of an elliptic line, that is, r = πρ/4, then the curl of Agnesi degenerates to the tangent of the circle ω. If r < πρ/4 or r > πρ/4, then the curl of Agnesi is a cubic curve with acnode or node respectively.