Cusp (singularity)
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a cusp, sometimes called spinode in old texts, is a 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 ...
where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of
singular point of a curve 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 Algebraic cur ...
. For a
plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
defined by an analytic, parametric equation :\begin x &= f(t)\\ y &= g(t), \end a cusp is a point where both
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of and are zero, and the
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
, in the direction of the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
, changes sign (the direction of the tangent is the direction of the slope \lim (g'(t)/f'(t))). Cusps are ''local singularities'' in the sense that they involve only one value of the parameter , in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point. For a curve defined by an
implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...
:F(x,y) = 0, which is
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 algebrai ...
, cusps are points where the terms of lowest degree of the
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
of are a power of a
linear polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
; however, not all singular points that have this property are cusps. The theory of
Puiseux series In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + ...
implies that, if is an
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
(for example a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
), a linear change of coordinates allows the curve to be parametrized, in a neighborhood of the cusp, as :\begin x &= at^m\\ y &= S(t), \end where is a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, is a positive
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
, and is a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
of order (degree of the nonzero term of the lowest degree) larger than . The number is sometimes called the ''order'' or the ''multiplicity'' of the cusp, and is equal to the degree of the nonzero part of lowest degree of . In some contexts, the definition of a cusp is restricted to the case of cusps of order two—that is, the case where . The definitions for plane curves and implicitly-defined curves have been generalized by René Thom and
Vladimir Arnold Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...
to curves defined by
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
s: a curve has a cusp at a point if there is a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
of a neighborhood of the point in the ambient space, which maps the curve onto one of the above-defined cusps.


Classification in differential geometry

Consider 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 algebrai ...
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real f ...
of two variables, say where and are
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. So is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
from the plane to the line. The space of all such smooth functions is
acted Agency for Technical Cooperation and Development, commonly known as ACTED, is a French humanitarian non-governmental organisation. It is a non-governmental, non-political and non-profit organisatio. ACTED works in 37 countries responding to eme ...
upon by the
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of
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
s of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
in both the
source Source may refer to: Research * Historical document * Historical source * Source (intelligence) or sub source, typically a confidential provider of non open-source intelligence * Source (journalism), a person, publication, publishing institute o ...
and the
target Target may refer to: Physical items * Shooting target, used in marksmanship training and various shooting sports ** Bullseye (target), the goal one for which one aims in many of these sports ** Aiming point, in field artillery, f ...
. This action splits the whole function space up into equivalence classes, i.e.
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
s of the
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
. One such family of equivalence classes is denoted by where is a non-negative integer. A function is said to be of type if it lies in the orbit of x^2 \pm y^, i.e. there exists a diffeomorphic change of coordinate in source and target which takes into one of these forms. These simple forms x^2 \pm y^ are said to give normal forms for the type -singularities. Notice that the are the same as the since the diffeomorphic change of coordinate in the source takes x^2 + y^ to x^2 - y^. So we can drop the ± from notation. The cusps are then given by the zero-level-sets of the representatives of the equivalence classes, where is an integer.


Examples

* An ordinary cusp is given by x^2-y^3=0, i.e. the zero-level-set of a type -singularity. Let be a smooth function of and and assume, for simplicity, that . Then a type -singularity of at can be characterised by: # Having a degenerate quadratic part, i.e. the quadratic terms in the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of form a perfect square, say , where is linear in and , ''and'' # does not divide the cubic terms in the Taylor series of . * A rhamphoid cusp () denoted originally a cusp such that both branches are on the same side of the tangent, such as for the curve of equation x^2-x^4-y^5=0. As such a singularity is in the same differential class as the cusp of equation x^2-y^5=0, which is a singularity of type , the term has been extended to all such singularities. These cusps are non-generic as caustics and
wave front In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same '' phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal frequ ...
s. The rhamphoid cusp and the ordinary cusp are non-diffeomorphic. A parametric form is x = t^2,\, y = a x^4 + x^5. For a type -singularity we need to have a degenerate quadratic part (this gives type ), that ''does'' divide the cubic terms (this gives type ), another divisibility condition (giving type ), and a final non-divisibility condition (giving type exactly ). To see where these extra divisibility conditions come from, assume that has a degenerate quadratic part and that divides the cubic terms. It follows that the third order taylor series of is given by L^2 \pm LQ, where is quadratic in and . We can complete the square to show that L^2 \pm LQ = (L \pm Q/2)^2 - Q^4/4. We can now make a diffeomorphic change of variable (in this case we simply substitute polynomials with
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
linear parts) so that (L \pm Q/2)^2 - Q^4/4 \to x_1^2 + P_1 where is quartic (order four) in and . The divisibility condition for type is that divides . If does not divide then we have type exactly (the zero-level-set here is a
tacnode In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp). is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to the curve at that point are tangent. This m ...
). If divides we complete the square on x_1^2 + P_1 and change coordinates so that we have x_2^2 + P_2 where is
quintic In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
(order five) in and . If does not divide then we have exactly type , i.e. the zero-level-set will be a rhamphoid cusp.


Applications

Cusps appear naturally when projecting into a plane a
smooth 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 ...
in three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. In general, such a projection is a curve whose singularities are self-crossing points and ordinary cusps. Self-crossing points appear when two different points of the curves have the same projection. Ordinary cusps appear when the tangent to the curve is parallel to the direction of projection (that is when the tangent projects on a single point). More complicated singularities occur when several phenomena occur simultaneously. For example, rhamphoid cusps occur for
inflection point 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 ...
s (and for
undulation point 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 ...
s) for which the tangent is parallel to the direction of projection. In many cases, and typically in computer vision and
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, the curve that is projected is the curve of the critical points of the restriction to a (smooth) spatial object of the projection. A cusp appears thus as a singularity of the contour of the image of the object (vision) or of its shadow (computer graphics). Caustics and
wave front In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same '' phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal frequ ...
s are other examples of curves having cusps that are visible in the real world.


See also

*
Cusp catastrophe In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation theory studies and classifies phenomena ch ...
*
Cardioid In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal ...


References

* *


External links


Physicists See The Cosmos In A Coffee Cup
{{DEFAULTSORT:Cusp (Singularity) Algebraic curves Singularity theory