In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Whitney umbrella (or Whitney's umbrella) is a self-intersecting surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 placed in three dimensions

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

. It is the union of all straight lines that pass through points of a fixed parabola

Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

 and are perpendicular to a fixed straight line, parallel to the axis of the parabola and lying on its perpendicular bisecting plane

Bisection

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector In geometry, bisection is the division of something into two equal...

.

Formulas

Whitney's umbrella can be given by the parametric equation

Parametric equation

In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....

s in Cartesian coordinates
where the parameters u and v range over the real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s. It is also given by the implicit equation

Implicit function

The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval...

This formula also includes the negative z axis (which is called the handle of the umbrella).

Properties

Whitney's umbrella is a ruled surface

Ruled surface

In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...

 and a right conoid

Right conoid

In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly a fixed straight line, called the axis of the right conoid....

. It is important in the field of singularity theory

Singularity theory

-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...

, as a simple local

Local property

In mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points.-Properties of a single space:...

 model of a pinch point

Pinch point (mathematics)

right|frame|240px|Section of the [[Whitney umbrella]], an example of pinch point singularity.In geometry, a pinch point or cuspidal point is a type of singular point on an algebraic surface....

 singularity

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

. The pinch point and the fold singularity are the only stable local singularities of maps from R² to R³.

It is named after the American

United States

The United States of America is a federal constitutional republic comprising fifty states and a federal district...

 mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 Hassler Whitney

Hassler Whitney

Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...

.