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Quasi-sphere

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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.

Notation and terminology

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This article uses the following notation and terminology:

Definition

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A quasi-sphere is a submanifold of a pseudo-Euclidean space Es,t consisting of the points u for which the displacement vector x = uo from a reference point o satisfies the equation

a xx + bx + c = 0,

where a, cR and b, xRs,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 = {xX : Q(x) = k} in a quadratic space (X, Q) has a counter-sphere N = {xX : Q(x) = −k}.[e] Furthermore, if k ≠ 0 and L is an isotropic line in X through x = 0, then L ∩ (PN) = ∅, 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.

Geometric characterizations

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Centre and radial scalar square

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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(xp) = 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.

Diameter and radius

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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

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Referring to the quadratic form applied to the displacement vector of a point on the quasi-sphere from the centre (i.e. Q(xp)) 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.

See also

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Notes

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  1. ^ Some authors exclude the definite cases, but in the context of this article, the qualifier indefinite will be used where this exclusion is intended.
  2. ^ The symmetric bilinear form applied to the two vectors is also called their scalar product.
  3. ^ 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(xy)). See Polarization identity for variations of this identity.
  4. ^ Though not mentioned in the source, we must exclude the combination b = 0 and a = 0.
  5. ^ There are caveats when Q is definite. Also, when k = 0, it follows that N = P.
  6. ^ 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.

References

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  1. ^ Élie Cartan (1981) [First published in 1937 in French, and in 1966 in English], The Theory of Spinors, Dover Publications, p. 3, ISBN 0486640701
  2. ^ Jayme Vaz, Jr.; Roldão da Rocha, Jr. (2016). An Introduction to Clifford Algebras and Spinors. Oxford University Press. p. 140. ISBN 9780191085789.
  3. ^ Ian R. Porteous (1995), Clifford Algebras and the Classical Groups, Cambridge University Press