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.
 * 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 $x$.
 * The symmetric bilinear form $B$ acting on two vectors $x, y ∈ R^{s,t}$ is denoted $B(x, y)$ or $x &sdot; y$. This is associated with the quadratic form $Q$.
 * Two vectors $Q$ are orthogonal if $Q(x) = B(x, x)$.
 * 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 $B(x, y) = 1⁄4(Q(x + y) − Q(x − y))$ consisting of the points $x, y ∈ R^{s,t}$ for which the displacement vector $x &sdot; y = 0$ from a reference point $E^{s,t}$ satisfies the equation

where $u$ and $x = u − o$.

Since $o$ 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.

A quasi-sphere $a x &sdot; x + b &sdot; x + c = 0$ in a quadratic space $a, c ∈ R$ has a counter-sphere $b, x ∈ R^{s,t}$. Furthermore, if $b = 0$ and $a = 0$ is an isotropic line in $a = 0$ through $P = {x ∈ X : Q(x) = k}$, then $(X, Q)$, 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.

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 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 $N = {x ∈ X : Q(x) = −k}$, the displacement vector $Q$ of the centre from the reference point and the radial scalar square $k = 0$ may be found as follows. We put $N = P$, and comparing to the defining equation above for a quasi-sphere, we get


 * $$ p = -\frac b {2a}, $$
 * $$ r = p\cdot p - \frac c a. $$

The case of $k ≠ 0$ may be interpreted as the centre $L$ 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 $X$ (and $x = 0$) 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 $L ∩ (P ∪ N) = ∅$ and $a ≠ 0$ may be determined from the above expressions, the set of vectors $p$ 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. $r$) 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 $Q(x − p) = r$-sphere, and one with zero curvature is a hyperplane that is partitioned with the $a = 0$-spheres.