New Straightedge-and-Compass Constructions to Determine the Curvature of Ellipses, Hyperbolas and Parabolas at any Point

摘要

Recent results in the geometry of conics, apart from their remarkable constructive and engineering applications, also give rise to certain novel geometric constructions to obtain the radius of curvature of conics at any point with compass and straightedge alone. This kind of constructions not only have an ancient mathematical tradition, but also are of genuine mathematical interest. Here, versions of the method for the three kinds of conies-ellipses, hyperbolas and parabolas- are presented with figures, descriptions of constructive steps, and justifications of their exactitude to solve the problem stated. Additionally, an improved, optimized version for the elliptical case is also provided.