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Concept
Singular point of a curve
Summary
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 a_0 = 0. If b_1 ≠ 0 then the implicit function theorem guarantees there is a smooth function h so that the curve has the form y = h(x) near the origin. Similarly, if b_0 ≠ 0 then there is a smooth function k so that the curve has the form 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,
Assume the curve passes through the origin and write Then f can be written
If 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 If then f = 0 has a solution of multiplicity 2 or higher and the line or is tangent to the curve. In this case, if is not 0 then the curve has a point of double contact with If the coefficient of x^2, is 0 but the coefficient of x^3 is not then the origin is a point of inflection of the curve. If the coefficients of x^2 and x^3 are both 0 then the origin is called point of undulation of the curve. This analysis can be applied to any point on the curve by translating the coordinate axes so that the origin is at the given point.
If b_0 and b_1 are both 0 in the above expansion, but at least one of c_0, c_1, c_2 is not 0 then the origin is called a double point of the curve. Again putting f can be written
Double points can be classified according to the solutions of
Crunode
If has two real solutions for m, that is if then the origin is called a crunode.
Official source
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