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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.

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.^{[a]} - 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}, denoted*Q*(*x*), is a generalization of the squared Euclidean distance in a Euclidean space. Élie Cartan calls*Q*(*x*) the*scalar square*of*x*.^{[1]} - The symmetric bilinear form
*B*acting on two vectors*x*,*y*∈**R**^{s,t}is denoted*B*(*x*,*y*) or*x*⋅*y*.^{[b]}This is associated with the quadratic form*Q*.^{[c]} - 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.

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

*a**x*⋅*x*+*b*⋅*x*+*c*= 0,

where *a*, *c* ∈ **R** and *b*, *x* ∈ **R**^{s,t}.^{[2]}^{[d]}

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.^{[3]}

A quasi-sphere *P* = {*x* ∈ *X* : *Q*(*x*) = *k*} in a quadratic space (*X*, *Q*) has a **counter-sphere** *N* = {*x* ∈ *X* : *Q*(*x*) = −*k*}.^{[e]} Furthermore, if *k* ≠ 0 and *L* is an isotropic line in *X* through *x* = 0, then *L* ∩ (*P* ∪ *N*) = ∅, puncturing the union of quasi-sphere and counter-sphere. One example is the unit hyperbola that forms a quasi-sphere of the hyperbolic plane, and its conjugate hyperbola, which is its counter-sphere.

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 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 radial scalar square *r* may be found as follows. We put *Q*(*x* − *p*) = *r*, and comparing to the defining equation above for 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 radial scalar square (the latter for the case of a null hyperplane). Knowing *p* (and *r*) 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 *p* and *r* 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 radial scalar square.

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.

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 *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.^{[f]}

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 *n*-sphere, and one with zero curvature is a hyperplane that is partitioned with the *n*-spheres.

**^**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 symmetric bilinear form applied to the two vectors is also called their*scalar product*.**^**The associated symmetric bilinear form of a (real) quadratic form*Q*is defined such that*Q*(*x*) =*B*(*x*,*x*), and 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.**^**There are caveats when*Q*is definite. Also, when*k*= 0, it follows that*N*=*P*.**^**A hyperplane (a quasi-sphere with infinite radial scalar square 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.

**^**Élie Cartan (1981) [First published in 1937 in French, and in 1966 in English],*The Theory of Spinors*, Dover Publications, p. 3, ISBN 0486640701**^**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