Singular point of a curve

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

Similarly, if b0 ≠ 0 then there is a smooth function k so that the curve has the form x = k(y) near the origin.

⁠ to the plane which defines the curve in the neighborhood of 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.

Assume the curve passes through the origin and write

is not 0 then f = 0 has a solution of multiplicity 1 at x = 0 and the origin is a point of single contact with line

then f = 0 has a solution of multiplicity 2 or higher and the line

is 0 but the coefficient of x3 is not then the origin is a point of inflection of the curve.

If the coefficients of x2 and x3 are both 0 then the origin is called point of undulation of the curve.

[1] If b0 and b1 are both 0 in the above expansion, but at least one of c0, c1, c2 is not 0 then the origin is called a double point of the curve.

Double points can be classified according to the solutions of

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.

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.

The curve in this case changes direction at the origin creating a sharp point.

The term node is used to indicate either a crunode or an acnode, in other words a double point which is not a cusp.

then the corresponding branch of the curve has a point of inflection at the origin.

In this case the origin is called a flecnode.

[2] 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.

Many curves can be defined in either fashion, but the two definitions may not agree.

Both definitions give a singular point at the origin.

⁠ never vanishes, and hence the node is not a singularity of the parameterized curve as defined above.

For instance the straight line y = 0 can be parameterised by

Hence, it is technically more correct to discuss singular points of a smooth mapping here rather than a singular point of a curve.

The above definitions can be extended to cover implicit curves which are defined as the zero set ⁠

⁠ of a smooth function, and it is not necessary just to consider algebraic varieties.

The definitions can be extended to cover curves in higher dimensions.

Any parameterized curve can also be defined as an implicit curve, and the classification of singular points of curves can be studied as a classification of singular points of an algebraic variety.