Еңіс жоқ қисықтардағы ерекше нүктелер

Point on a curve not given by a smooth embedding of a parameter
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.

Algebraic curves in the plane may be defined as the set of points (x, y) satisfying an equation of the form where f is a polynomial function f: \R^2 \to \R. If f is expanded as

If the origin (0, 0) is on the curve then If then the implicit function theorem guarantees there is a smooth function h so that the curve has the form 1=y = h(x) near the origin. Similarly, if then there is a smooth function k so that the curve has the form 1=x = k(y) near the origin. In either case, there is a smooth map from \R to the plane which defines the curve in the neighborhood of the origin. Note that at the origin

so the curve is non singular or regular at the origin if at least one of the partial derivatives of f is non zero. The singular points are those points on the curve where both partial derivatives vanish,

If has two real solutions for m, that is if then the origin is called a crunode. The curve in this case crosses itself at the origin and has two distinct tangents corresponding to the two solutions of The function f has a saddle point at the origin in this case.

If has no real solutions for m, that is if then the origin is called an acnode. In the real plane the origin is an isolated point on the curve; however when considered as a complex curve the origin is not isolated and has two imaginary tangents corresponding to the two complex solutions of The function f has a local extremum at the origin in this case.

If has a single solution of multiplicity 2 for m, that is if then the origin is called a cusp. The curve in this case changes direction at the origin creating a sharp point. The curve has a single tangent at the origin which may be considered as two coincident tangents.

The term node is used to indicate either a crunode or an acnode, in other words a double point which is not a cusp. The number of nodes and the number of cusps on a curve are two of the invariants used in the Plücker formulas. If one of the solutions of is also a solution of then the corresponding branch of the curve has a point of inflection at the origin. In this case the origin is called a flecnode. If both tangents have this property, so is a factor of then the origin is called a biflecnode.

In general, if all the terms of degree less than k are 0, and at least one term of degree k is not 0 in f, then curve is said to have a multiple point of order k or a k ple point. The curve will have, in general, k tangents at the origin though some of these tangents may be imaginary.