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.
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).
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.
[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.