A pseudo-Euclidean space, denoted, is a real affine space in which displacement vectors are the elements of the space. It is distinguished from the vector space.
The quadratic form acting on a vector is denoted, called the quadrance of .
The symmetric bilinear form 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 at that point.
Definition
A quasi-sphere is a submanifold 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 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 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 that forms a quasi-sphere of the hyperbolic plane, and its conjugate hyperbola, which is its counter-sphere.
Geometric characterizations
Centre and radial quadrance
The centre of a quasi-sphere is a point that has equal quadrance from every point of the quasi-sphere – i.e., the quadratic form applied to the displacement vector from the centre to a point of the quasi-sphere yields a constant, called the radial quadrance, 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, the displacement vector of the centre from the reference point and the radial quadrance may be found as follows. We put, and comparing to the defining equation above for a quasi-sphere, we get The case of may be interpreted as the centre being a well-defined point at infinity with either infinite or zero radial quadrance. Knowing in this case does not determine the hyperplane's position, though, only its orientation in space. The radial quadrance 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 quadrance.
Diameter and radius
Any pair of points, which need not be distinct, 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, and any other point on the quasi-sphere 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 as the radial quadrance, in any pseudo-Euclidean space the quasi-spheres may be separated into three disjoint sets: those with positive radial quadrance, those with negative radial quadrance, those with zero radial quadrance. In a space with a positive-definite quadratic form, a quasi-sphere with negative radial quadrance is the empty set, one with zero radial quadrance consists of a single point, one with positive radial quadrance is a standard -sphere, and one with zero curvature is a hyperplane that is partitioned with the -spheres.
