Singular point of a curve

Assume the curve passes through the origin and write ${\displaystyle y=mx.}$ Then f can be written

$${\displaystyle f=(b_{0}+mb_{1})x+(c_{0}+2mc_{1}+c_{2}m^{2})x^{2}+\cdots .}$$

If ${\displaystyle b_{0}+mb_{1}}$ 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 ${\displaystyle y=mx.}$ If ${\displaystyle b_{0}+mb_{1}=0}$ then f = 0 has a solution of multiplicity 2 or higher and the line ${\displaystyle y=mx,}$ or ${\displaystyle b_{0}x+b_{1}y=0,}$ is tangent to the curve. In this case, if ${\displaystyle c_{0}+2mc_{1}+c_{2}m^{2}}$ is not 0 then the curve has a point of double contact with ${\displaystyle y=mx.}$ If the coefficient of x², ${\displaystyle c_{0}+2mc_{1}+c_{2}m^{2},}$ 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. 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.[1]

Three limaçons illustrating the types of double point. When converted to Cartesian coordinates as ${\displaystyle (x^{2}+y^{2}-x)^{2}=(1.5)^{2}(x^{2}+y^{2}),}$ the left curve acquires an acnode at the origin, which is an isolated point in the plane. The central curve, the cardioid, has a cusp at the origin. The right curve has a crunode at the origin and the curve crosses itself to form a loop.

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. Again putting ${\displaystyle y=mx,}$ f can be written

$${\displaystyle f=(c_{0}+2mc_{1}+c_{2}m^{2})x^{2}+(d_{0}+3md_{1}+3m^{2}d_{2}+d_{3}m^{3})x^{3}+\cdots .}$$

Double points can be classified according to the solutions of ${\displaystyle c_{0}+2mc_{1}+m^{2}c_{2}=0.}$

If ${\displaystyle c_{0}+2mc_{1}+m^{2}c_{2}=0}$ has two real solutions for m, that is if ${\displaystyle c_{0}c_{2}-c_{1}^{2}<0,}$ 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 ${\displaystyle c_{0}+2mc_{1}+m^{2}c_{2}=0.}$ The function f has a saddle point at the origin in this case.

If ${\displaystyle c_{0}+2mc_{1}+m^{2}c_{2}=0}$ has no real solutions for m, that is if ${\displaystyle c_{0}c_{2}-c_{1}^{2}>0,}$ 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 ${\displaystyle c_{0}+2mc_{1}+m^{2}c_{2}=0.}$ The function f has a local extremum at the origin in this case.

If ${\displaystyle c_{0}+2mc_{1}+m^{2}c_{2}=0}$ has a single solution of multiplicity 2 for m, that is if ${\displaystyle c_{0}c_{2}-c_{1}^{2}=0,}$ 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 ${\displaystyle c_{0}+2mc_{1}+m^{2}c_{2}=0}$ is also a solution of ${\displaystyle d_{0}+3md_{1}+3m^{2}d_{2}+m^{3}d_{3}=0,}$ 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 ${\displaystyle c_{0}+2mc_{1}+m^{2}c_{2}}$ is a factor of ${\displaystyle d_{0}+3md_{1}+3m^{2}d_{2}+m^{3}d_{3},}$ then the origin is called a biflecnode.[2]

A curve with a triple point at the origin: x(t) = sin(2t) + cos(t), y(t) = sin(t) + cos(2t)

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.[3]