# Quasi-sphere

In mathematics and theoretical physics, a **quasi-sphere** is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the set of points for which the quadratic form 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**R**^{s,t}, is a real vector space with a nondegenerate quadratic form with signature (*s*,*t*). The quadratic form is permitted to be definite (where*s*= 0 or*t*= 0), making this a generalization of a Euclidean vector space.^{[lower-alpha 1]} - A
**pseudo-Euclidean space**, denoted E^{s,t}, is a real affine space in which displacement vectors are the elements of the space**R**^{s,t}. It is distinguished from the vector space. - The quadratic form
*Q*acting on a vector*x*∈**R**^{s,t}is denoted*Q*(*x*). - The symmetric bilinear form
*B*acting on two vectors*x*,*y*∈**R**^{s,t}is denoted*B*(*x*,*y*) or*x*⋅*y*. This is associated with the quadratic form*Q*.^{[lower-alpha 2]} - Two vectors
*x*,*y*∈**R**^{s,t}are**orthogonal**if*x*⋅*y*= 0. - A
**normal vector**at a point of a quasi-sphere is a nonzero vector that is orthogonal to each vector in the tangent space at that point.

## Definition

A **quasi-sphere** is a submanifold of a pseudo-Euclidean space E^{s,t} consisting of the points *u* for which the displacement vector *x* = *u* − *o* from a reference point *o* satisfies the equation

*ax*⋅*x*+*b*⋅*x*+*c*= 0,

where *a*, *c* ∈ **R** and *b*, *x* ∈ **R**^{s,t}.^{[1]}^{[lower-alpha 3]}

Since *a* = 0 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 transformations than if they are omitted.

This definition has been generalized to affine spaces over complex numbers and quaternions by replacing the quadratic form with a Hermitian form.^{[2]}

## Geometric characterizations

### Centre and radius squared

The *centre* of a quasi-sphere is a point that is equidistant from every point of the quasi-sphere – i.e., the quadratic form applied to the displacement vector (or radius vector) from the centre to a point of the quasi-sphere yields a constant, called the *radius squared*, or equivalently, the point at which the pencil of lines normal to the tangent hyperplanes meet. If the quasi-sphere is a hyperplane, the centre is the point at infinity defined by this pencil.

When *a* ≠ 0, the displacement vector *p* of the centre from the reference point and the radius squared *r*^{2} may be found as follows. We put *Q*(*x* − *p*) = *r*^{2}, and comparing to the equation above used to define a quasi-sphere, we get

The case of *a* = 0 may be interpreted as the centre *p* being a well-defined point at infinity with either infinite or zero radius squared (the latter for the case of a null hyperplane). Knowing *p* (and *r*^{2}) in this case does not determine the hyperplane's position, though, only its the orientation in space.

The radius squared may take on a positive, zero or negative value. When the quadratic form is definite, even though *p* and *r*^{2} may be determined from the above expressions, the set of vectors *x* satisfying the defining equation may be empty, as is the case in a Euclidean space for a negative radius squared.

### 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. *Q*(*x* − *p*)) as the *radius squared*, in any pseudo-Euclidean space the quasi-spheres may be separated into three disjoint sets: those with positive radius squared, those with negative radius squared, those with zero radius squared.^{[lower-alpha 4]}

In spaces with a positive-definite quadratic form (i.e. Euclidean space), the quasi-spheres with negative radius squared are the empty set, those with zero radius each consist of a single point, those with positive radius squared are standard *n*-spheres, and those with zero curvature are hyperplanes.

## Notes

- ↑ Some authors exclude the definite cases, but in the context of this article, the qualifier
*indefinite*will be used where this exclusion is intended. - ↑ The associated symmetric bilinear form of a (real) quadratic form
*Q*is defined such that*Q*(*x*) =*B*(*x*,*x*), and may may be determined as*B*(*x*,*y*) = 1/4(*Q*(*x*+*y*) −*Q*(*x*−*y*)). See Polarization identity for variations of this identity. - ↑ Though not mentioned in the source, we must exclude the combination
*b*= 0 and*a*= 0. - ↑ A hyperplane (a quasi-sphere with infinite radius or zero curvature) is partitioned with quasi-spheres to which it is tangent. The three sets may be defined according to whether the quadratic form applied to a vector that is a normal of the tangent hypersurface is positive, zero or negative. The three sets of objects are preserved under conformal transformations of the space.

## References

- ↑ Jayme Vaz, Jr.; Roldão da Rocha, Jr. (2016).
*An Introduction to Clifford Algebras and Spinors*. Oxford University Press. p. 140. ISBN 9780191085789. - ↑ Ian R. Porteous (1995),
*Clifford Algebras and the Classical Groups*, Cambridge University Press