In
geometry, a singular point on a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
is one where the curve is not given by a
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.
Algebraic curves in the plane
Algebraic curves in the plane may be defined as the set of points satisfying an equation of the form
where is a
polynomial function If is expanded as
If the origin is on the curve then . If then the
implicit function theorem guarantees there is a smooth function so that the curve has the form near the origin. Similarly, if then there is a smooth function so that the curve has the form near the origin. In either case, there is a smooth map from 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 is non-zero. The singular points are those points on the curve where both partial derivatives vanish,
Regular points
Assume the curve passes through the origin and write
Then can be written
If
is not 0 then has a solution of multiplicity 1 at and the origin is a point of single contact with line
If
then 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 ,
is 0 but the coefficient of is not then the origin is a
point of inflection
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...
of the curve. If the coefficients of and 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.
Double points
If and are both in the above expansion, but at least one of , , is not 0 then the origin is called a double point of the curve. Again putting
can be written
Double points can be classified according to the solutions of
Crunodes
If
has two real solutions for , 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 has a
saddle point at the origin in this case.
Acnodes
If
has no real solutions for , that is if
then the origin is called an ''
acnode
An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are " isolated point or hermit point".
For example the equation
:f(x,y)=y^2+x^2-x^3=0
has an acnode at the origin, because it is ...
''. 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 has a
local extremum
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administra ...
at the origin in this case.
Cusps
If
has a single solution of multiplicity 2 for , 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.
Further classification
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''.
Multiple points
In general, if all the terms of degree less than are 0, and at least one term of degree is not 0 in , then curve is said to have a ''multiple point'' of order or a ''k-ple point''. The curve will have, in general, tangents at the origin though some of these tangents may be imaginary.
Parametric curves
A
parameterized curve in is defined as the image of a function
The singular points are those points where
Many curves can be defined in either fashion, but the two definitions may not agree. For example, the
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
can be defined on an algebraic curve,
or on a parametrised curve,
Both definitions give a singular point at the origin. However, a
node
In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex).
Node may refer to:
In mathematics
*Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two or more curves, lines, ...
such as that of
at the origin is a singularity of the curve considered as an algebraic curve, but if we parameterize it as
then never vanishes, and hence the node is ''not'' a singularity of the parameterized curve as defined above.
Care needs to be taken when choosing a parameterization. For instance the straight line can be parameterised by
which has a singularity at the origin. When parametrised by
it is nonsingular. Hence, it is technically more correct to discuss
singular points of a smooth mapping
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, s ...
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.
A theorem of
Hassler Whitney[Bruce and Giblin, ''Curves and singularities'', (1984, 1992) , (paperback)] states
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 point of an algebraic variety.
Types of singular points
Some of the possible singularities are:
*An isolated point:
an
acnode
An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are " isolated point or hermit point".
For example the equation
:f(x,y)=y^2+x^2-x^3=0
has an acnode at the origin, because it is ...
*Two lines crossing:
a
crunode
*A
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
:
also called a ''spinode''
*A
tacnode:
*A
rhamphoid cusp:
See also
*
Singular point of an algebraic variety
*
Singularity theory
*
Morse theory
References
*
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