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 ...

and

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

, a quasi-sphere is a generalization of the

hypersphere In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point ...

and the

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

to the context of a

pseudo-Euclidean space In mathematics and theoretical physics, a pseudo-Euclidean space of signature is a finite- dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vect ...

. It may be described as the set of points for which the

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

for the space applied to the displacement vector from a centre point is a constant value, with the inclusion of hyperplanes as a limiting case.

Notation and terminology

This article uses the following notation and terminology: * A pseudo-Euclidean vector space, denoted , is a real

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

with a

nondegenerate quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

with

signature A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...

. The quadratic form is permitted to be definite (where or ), making this a generalization of a

Euclidean vector 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'' ...

. * A pseudo-Euclidean space, denoted , is a real

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

in which

displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along ...

s are the elements of the space . It is distinguished from the vector space. * The

acting on a vector , denoted , is a generalization of the squared Euclidean distance in a Euclidean space.

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...

calls the ''scalar square'' of . * The

symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a biline ...

acting on two vectors is denoted or . This is associated with the quadratic form . * Two vectors are orthogonal if . * A normal vector at a point of a quasi-sphere is a nonzero vector that is orthogonal to each vector in the

tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...

at that point.

Definition

A quasi-sphere is a

submanifold In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...

of a pseudo-Euclidean space consisting of the points for which the displacement vector from a reference point satisfies the equation :, where and . Since in permitted, this definition includes hyperplanes; it is thus a generalization of generalized circles and their analogues in any number of dimensions. This inclusion provides a more regular structure under

conformal transformation In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\i ...

s than if they are omitted. This definition has been generalized to

s over

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s and

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

s by replacing the quadratic form with a

Hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear map, linear in each of its arguments, but a sesquilinear f ...

. A quasi-sphere in a quadratic space has a counter-sphere . Furthermore, if and is an isotropic line in through , then , puncturing the union of quasi-sphere and counter-sphere. One example is the

unit hyperbola In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative rad ...

that forms a quasi-sphere of the

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...

, and its conjugate hyperbola, which is its counter-sphere.

Geometric characterizations

Centre and radial scalar square

The ''centre'' of a quasi-sphere is a point that has equal scalar square from every point of the quasi-sphere, the point at which the

pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail of ...

of lines normal to the tangent hyperplanes meet. If the quasi-sphere is a hyperplane, the centre is the

point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...

defined by this pencil. When , the displacement vector of the centre from the reference point and the radial scalar square may be found as follows. We put , and comparing to the defining equation above for a quasi-sphere, we get :

p = -\frac b ,

r = p\cdot p - \frac c a.

The case of may be interpreted as the centre being a well-defined point at infinity with either infinite or zero radial scalar square (the latter for the case of a null hyperplane). Knowing (and ) in this case does not determine the hyperplane's position, though, only its orientation in space. The radial scalar square may take on a positive, zero or negative value. When the quadratic form is definite, even though and may be determined from the above expressions, the set of vectors satisfying the defining equation may be empty, as is the case in a Euclidean space for a negative radial scalar square.

Diameter and radius

Any pair of points, which need not be distinct, (including the option of up to one of these being a point at infinity) defines a diameter of a quasi-sphere. The quasi-sphere is the set of points for which the two displacement vectors from these two points are orthogonal. Any point may be selected as a centre (including a point at infinity), and any other point on the quasi-sphere (other than a point at infinity) define a radius of a quasi-sphere, and thus specifies the quasi-sphere.

Partitioning

Referring to the quadratic form applied to the displacement vector of a point on the quasi-sphere from the centre (i.e. ) as the ''radial scalar square'', in any pseudo-Euclidean space the quasi-spheres may be separated into three disjoint sets: those with positive radial scalar square, those with negative radial scalar square, those with zero radial scalar square. In a space with a positive-definite quadratic form (i.e. a Euclidean space), a quasi-sphere with negative radial scalar square is the empty set, one with zero radial scalar square consists of a single point, one with positive radial scalar square is a standard -sphere, and one with zero curvature is a hyperplane that is partitioned with the -spheres.

Notes

References

{{Dimension topics Multi-dimensional geometry Spheres