Singular point curve

A singular point of the curve is a point in the vicinity of which there is no smooth parameterization. The exact definition depends on the type of curve being studied.

Algebraic Curves on a Plane

An algebraic curve in the plane can be defined as a set of points $\left(x,y\right)$ ${\ displaystyle \ left (x, y \ right)}$ $\left(x,y\right)$ satisfying an equation of the form $f\left(x,y\right)=0$ ${\ displaystyle f \ left (x, y \ right) = 0}$ $f\left(x,y\right)=0$ where $f\left(x,y\right)$ ${\ displaystyle f \ left (x, y \ right)}$ $f\left(x,y\right)$ - polynomial function $f\colon \mathbb {R} ^{2}\rightarrow \mathbb {R}$ ${\ displaystyle f \ colon \ mathbb {R} ^ {2} \ rightarrow \ mathbb {R}}$ $f\colon \mathbb {R} ^{2}\rightarrow \mathbb {R}$ :

f=a+b_{1}x+b_{2}y+c_{1}x^{2}+2c_{2}xy+c_{3}y^{2}+\dots

{\ displaystyle f = a + b_ {1} x + b_ {2} y + c_ {1} x ^ {2} + 2c_ {2} xy + c_ {3} y ^ {2} + \ dots}

f=a+b_{1}x+b_{2}y+c_{1}x^{2}+2c_{2}xy+c_{3}y^{2}+\dots

.

If the origin $\left(0,0\right)$ ${\ displaystyle \ left (0,0 \ right)}$ $\left(0,0\right)$ belongs to the curve then $a=0$ ${\ displaystyle a = 0}$ $a=0$ . If a $b_{2}\not =0$ ${\ displaystyle b_ {2} \ not = 0}$ $b_{2}\not =0$ , then the implicit function theorem guarantees the existence of a smooth function $g$ ${\ displaystyle g}$ $g$ such that the curve takes the form $y=g\left(x\right)$ ${\ displaystyle y = g \ left (x \ right)}$ $y=g\left(x\right)$ in the vicinity of the origin. Similarly, if $b_{1}\not =0$ ${\ displaystyle b_ {1} \ not = 0}$ $b_{1}\not =0$ , then there is such a function $h$ ${\ displaystyle h}$ $h$ that the curve satisfies the equation $x=h\left(y\right)$ ${\ displaystyle x = h \ left (y \ right)}$ $x=h\left(y\right)$ in the vicinity of the origin. In both cases, there is a smooth mapping $\mathbb {R} \rightarrow \mathbb {R}$ ${\ displaystyle \ mathbb {R} \ rightarrow \ mathbb {R}}$ $\mathbb {R} \rightarrow \mathbb {R}$ , which defines a curve in the vicinity of the origin. Note that in the vicinity of the origin

b_{1}={\partial f \over \partial x},\,b_{2}={\partial f \over \partial y}.

{\ displaystyle b_ {1} = {\ partial f \ over \ partial x}, \, b_ {2} = {\ partial f \ over \ partial y}.}

b_{1}={\partial f \over \partial x},\,b_{2}={\partial f \over \partial y}.

The singular points of the curve are those points of the curve at which both derivatives vanish:

f(x,y)={\partial f \over \partial x}={\partial f \over \partial y}=0.

{\ displaystyle f (x, y) = {\ partial f \ over \ partial x} = {\ partial f \ over \ partial y} = 0.}

f(x,y)={\partial f \over \partial x}={\partial f \over \partial y}=0.

Regular Points

Let the curve go through the origin. Putting $y=mx$ ${\ displaystyle y = mx}$ $y=mx$ can imagine $f$ ${\ displaystyle f}$ $f$ as

f=(b_{1}+b_{2}m)x+(c_{1}+2c_{2}m+c_{3}m^{2})x^{2}+\dots

{\ displaystyle f = (b_ {1} + b_ {2} m) x + (c_ {1} + 2c_ {2} m + c_ {3} m ^ {2}) x ^ {2} + \ dots}

f=(b_{1}+b_{2}m)x+(c_{1}+2c_{2}m+c_{3}m^{2})x^{2}+\dots

.

If a $b_{1}+b_{2}m\not =0$ ${\ displaystyle b_ {1} + b_ {2} m \ not = 0}$ $b_{1}+b_{2}m\not =0$ then the equation $f=0$ ${\ displaystyle f = 0}$ $f=0$ has a solution of multiplicity 1 at the point $x=0$ ${\ displaystyle x = 0}$ $x=0$ and the origin is the point of single contact of the curve with the line $y=mx$ ${\ displaystyle y = mx}$ $y=mx$ . If a $b_{1}+b_{2}m=0$ ${\ displaystyle b_ {1} + b_ {2} m = 0}$ $b_{1}+b_{2}m=0$ then $f=0$ ${\ displaystyle f = 0}$ $f=0$ has a point $x=0$ ${\ displaystyle x = 0}$ $x=0$ a solution of multiplicity 2 or higher and direct $y=mx$ ${\ displaystyle y = mx}$ $y=mx$ is tangent to the curve. In this case, if $c_{1}+2c_{2}m+c_{3}m^{2}\not =0$ ${\ displaystyle c_ {1} + 2c_ {2} m + c_ {3} m ^ {2} \ not = 0}$ $c_{1}+2c_{2}m+c_{3}m^{2}\not =0$ , the curve has double contact with direct $y=mx$ ${\ displaystyle y = mx}$ $y=mx$ . If a $c_{1}+2c_{2}m+c_{3}m^{2}=0$ ${\ displaystyle c_ {1} + 2c_ {2} m + c_ {3} m ^ {2} = 0}$ $c_{1}+2c_{2}m+c_{3}m^{2}=0$ , and the coefficient at $x^{3}$ ${\ displaystyle x ^ {3}}$ $x^{3}$ not equal to zero, then the origin is the inflection point of the curve. This reasoning can be applied to any point in the curve by transferring the origin to a given point. ^[one]

Double dots

The three snails of Pascal illustrate the types of double points. The left curve has an isolated point at the origin. The central curve, the cardioid , has a cusp at the origin. The right curve has a self-intersection point at the origin, forming a loop.

If in the above equation $b_{1}=0$ ${\ displaystyle b_ {1} = 0}$ and $b_{2}=0$ ${\ displaystyle b_ {2} = 0}$ but at least one of the quantities $c_{1}$ ${\ displaystyle c_ {1}}$ , $c_{2}$ ${\ displaystyle c_ {2}}$ or $c_{3}$ ${\ displaystyle c_ {3}}$ not equal to zero, then the origin is called the double point of the curve. Put again $y=mx$ ${\ displaystyle y = mx}$ then $f$ ${\ displaystyle f}$ will take the form

f=(c_{1}+2c_{2}m+c_{3}m^{2})x^{2}+(d_{1}+3d_{2}m+3d_{3}m^{2}+d_{4}m^{3})x^{3}+\dots .

{\ displaystyle f = (c_ {1} + 2c_ {2} m + c_ {3} m ^ {2}) x ^ {2} + (d_ {1} + 3d_ {2} m + 3d_ {3} m ^ {2} + d_ {4} m ^ {3}) x ^ {3} + \ dots.}

Double points can be classified by the roots of the equation $c_{1}+2c_{2}m+c_{3}m^{2}=0$ ${\ displaystyle c_ {1} + 2c_ {2} m + c_ {3} m ^ {2} = 0}$ .

Self-intersection points

If the equation $c_{1}+2c_{2}m+c_{3}m^{2}=0$ ${\ displaystyle c_ {1} + 2c_ {2} m + c_ {3} m ^ {2} = 0}$ has two real solutions for $m$ ${\ displaystyle m}$ that is, if $c_{2}^{2}-c_{1}c_{3}>0$ ${\ displaystyle c_ {2} ^ {2} -c_ {1} c_ {3}> 0}$ , then the origin is called . The curve in this case has two different tangents corresponding to two solutions of the equation $c_{1}+2c_{2}m+c_{3}m^{2}=0$ ${\ displaystyle c_ {1} + 2c_ {2} m + c_ {3} m ^ {2} = 0}$ . Function $f$ ${\ displaystyle f}$ in this case has a saddle point at the origin.

Isolated

If the equation $c_{1}+2c_{2}m+c_{3}m^{2}=0$ ${\ displaystyle c_ {1} + 2c_ {2} m + c_ {3} m ^ {2} = 0}$ has no material decisions on $m$ ${\ displaystyle m}$ that is, if $c_{2}^{2}-c_{1}c_{3}<0$ ${\ displaystyle c_ {2} ^ {2} -c_ {1} c_ {3} <0}$ , then the origin is called an isolated point . On the real plane, the origin will be isolated from the curve, but on the complex plane the origin will not be isolated and will have two imaginary tangents corresponding to two imaginary solutions of the equation $c_{1}+2c_{2}m+c_{3}m^{2}=0$ ${\ displaystyle c_ {1} + 2c_ {2} m + c_ {3} m ^ {2} = 0}$ . Function $f$ ${\ displaystyle f}$ in this case has a local extremum at the origin.

Caspians

If the equation $c_{1}+2c_{2}m+c_{3}m^{2}=0$ ${\ displaystyle c_ {1} + 2c_ {2} m + c_ {3} m ^ {2} = 0}$ has one real solution for $m$ ${\ displaystyle m}$ multiplicity 2, that is, if $c_{2}^{2}-c_{1}c_{3}=0$ ${\ displaystyle c_ {2} ^ {2} -c_ {1} c_ {3} = 0}$ , then the origin is called the cusp , or return point . The curve in this case at a singular point changes direction, forming a point. The curve at the origin has a single tangent, which can be interpreted as two coinciding tangents.

Further Classification

The term node is used as a generic name for isolated points and self-intersection points. The number of nodes and the number of cusps of the curve are two invariants used in the Plücker formulas .

If one of the solutions to the equation $c_{1}+2c_{2}m+c_{3}m^{2}=0$ ${\ displaystyle c_ {1} + 2c_ {2} m + c_ {3} m ^ {2} = 0}$ is also a solution to the equation $d_{1}+3d_{2}m+3d_{3}m^{2}+d_{4}m^{3}=0$ ${\ displaystyle d_ {1} + 3d_ {2} m + 3d_ {3} m ^ {2} + d_ {4} m ^ {3} = 0}$ , then the corresponding branch of the curve has an inflection at the origin. In this case, the origin is called the point of contact . If both branches have this property, then $c_{1}+2c_{2}m+c_{3}m^{2}$ ${\ displaystyle c_ {1} + 2c_ {2} m + c_ {3} m ^ {2}}$ is a divider $d_{1}+3d_{2}m+3d_{3}m^{2}+d_{4}m^{3}$ ${\ displaystyle d_ {1} + 3d_ {2} m + 3d_ {3} m ^ {2} + d_ {4} m ^ {3}}$ , and the origin is called a bifflectoidal point (double contact point). ^[2]

Multiple Points

A curve with a triple point at the origin.

In the general case, when all terms with degree less than $k$ ${\ displaystyle k}$ , and provided that at least one member with degree $k$ ${\ displaystyle k}$ is not equal to zero, they say that the curve has a multiple point of order k . In this case, the curve has $k$ ${\ displaystyle k}$ tangents at the origin, but some of them may be imaginary or coincide. ^[3]

Parametric Curves

The parametric curve in R ² is defined as the image of the function g : R → R ² , g ( t ) = ( g ₁ ( t ), g ₂ ( t )). The singular points of such a curve are the points at which

{dg_{1} \over dt}={dg_{2} \over dt}=0.

{\ displaystyle {dg_ {1} \ over dt} = {dg_ {2} \ over dt} = 0.}

Casp

Many curves can be set in both forms, but these two tasks are not always consistent. For example, cusp can be found both in the algebraic curve x ³ - y ² = 0 and in the parametric curve g ( t ) = ( t ² , t ³ ). Both curve assignments give a singular point at the origin. However, curve y ² - x ³ - x ² = 0 at the origin is special for an algebraic curve, but with the parametric specification g ( t ) = ( t ² −1, t ( t ² −1)) derivatives g ′ ( t ) never vanishes, and therefore the point is not singular in the above sense.

Caution should be exercised when choosing a parameterization. For example, the line y = 0 can be set parametrically as g ( t ) = ( t ³ , 0) and it will have a singular point at the origin. If, however, it is parameterized as g ( t ) = ( t , 0), it will not have singular points. Thus, it is technically more correct to speak of singular points of a smooth map, rather than singular points of a curve.

The above definitions can be extended to implicit curves , which can be defined as the set of zeros f ⁻¹ (0) of an arbitrary smooth function . Definitions can also be extended to curves in spaces of higher dimensions.

According to the Hassler-Whitney theorem, ^[4] ^[5] any closed set in R ⁿ is the set of solutions f ⁻¹ (0) for some smooth function f : R ⁿ → R. Therefore, any parametric curve can be defined as an implicit curve.

Feature Point Types

Examples of singular points of various types:

Isolated point : x ² + y ² = 0,
: x ² - y ² = 0,
Casp ( return point ): x ³ - y ² = 0,
Bill-shaped cusp: x ⁵ - y ² = 0.

Notes

↑ Hilton Chapter II § 1
↑ Hilton Chapter II § 2
↑ Hilton Chapter II § 3
↑ Brooker and Larden. Differential Germs and Catastrophes. - London Mathematical Society. Lecture Notes 17. Cambridge. - 1975.
↑ Bruce and Giblin, Curves and singularities , (1984, 1992) ISBN 0-521-41985-9 , ISBN 0-521-42999-4 (paperback)

Literature

Harold Hilton. Plane Algebraic Curves . - Oxford, 1920.

[1] Hilton Chapter II § 1

[2] Hilton Chapter II § 2

[3] Hilton Chapter II § 3

[4] Brooker and Larden. Differential Germs and Catastrophes. - London Mathematical Society. Lecture Notes 17. Cambridge. - 1975.

[5] Bruce and Giblin, Curves and singularities , (1984, 1992) ISBN 0-521-41985-9 , ISBN 0-521-42999-4 (paperback)