Degenerated algebraic curves

When there is an unique tangent line at a point on a curve, this point is called regular point. Almost all points on an algebraic curve are regular.

Beside regular points, there are points on a curve that can have more than one tangent lines and these points are called singular points.

Example of singular points are double points - points in which the curve intersects itself. There is an upper bound on the number of  double points.

  • Plane algebraic curve of order n cannot have more than  (n1)(n2)2 double points.

  • If a plane algebraic curve of order  n has more double point than this bound, it is degenerated into curves of lower order and the sum of orders of that curves equals n.

    Lower figure shown one non-degenerated curve of order 4 with 3 double points and four possible degenerated curves of order 4.

  • Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja - 3DGeomTeh - Developing project of the University of Zagreb