In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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.
For a
plane curve defined by an
analytic,
parametric equation
In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters.
In the case ...
:
a cusp is a point where both
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s of and are zero, and the
directional derivative
In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point.
The directional derivative of a multivariable differentiable (scalar) function along a given vect ...
, in the direction of the
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
, changes sign (the direction of the tangent is the direction of the slope
). 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 ...
:
which is
smooth, cusps are points where the terms of lowest degree of the
Taylor expansion of are a power of a
linear polynomial; however, not all singular points that have this property are cusps. The theory of
Puiseux series 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 a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
), a linear change of coordinates allows the curve to be
parametrized, in a
neighborhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
of the cusp, as
:
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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
, is a positive
even integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
, and is a
power series 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 to curves defined by
differentiable functions: a curve has a cusp at a point if there is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ...
of a
neighborhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
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 real-valued function 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. So is a
function from the plane to the line. The space of all such smooth functions is
acted upon by the
group of
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ...
s of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of
coordinate in both the
source and the
target. This action splits the whole
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
up into
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es, i.e.
orbit
In celestial mechanics, an orbit (also known as orbital revolution) 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 ...
s of the
group action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.
Many sets of transformations form a group under ...
.
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
i.e. there exists a diffeomorphic change of coordinate in source and target which takes into one of these forms. These simple forms
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
to
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
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
As such a singularity is in the same differential class as the cusp of equation
which is a singularity of type , the term has been extended to all such singularities. These cusps are non-generic as
caustics and
wave fronts. The rhamphoid cusp and the ordinary cusp are non-diffeomorphic. A parametric form is
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
where is quadratic in and . We can
complete the square to show that
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 exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concep ...
linear parts) so that
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). If divides we complete the square on
and change coordinates so that we have
where is
quintic (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 three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. 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 points (and for
undulation points) for which the tangent is parallel to the direction of projection.
In many cases, and typically in
computer vision
Computer vision tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical ...
and
computer graphics
Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...
, 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 fronts are other examples of curves having cusps that are visible in the real world.
See also
*
Cusp catastrophe
*
Cardioid
References
*
*
External links
Physicists See The Cosmos In A Coffee Cup
{{DEFAULTSORT:Cusp (Singularity)
Algebraic curves
Singularity theory