In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model, is a model of hyperbolic geometry in which the points are points on one sheet of a hyperboloid of two sheets. A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation:  (hyperboloid of one sheet), or  (hyperboloid of two sheets) If, and only if, , it is a hyperboloid of revolution. ...

If [t, x₁, ..., x_n] is a vector in real (n+1)-space, we may define the Minkowski quadratic form to be In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

The points of the n-dimensional hyperboloid model are then the vectors v such that Q(v) = 1, where t>0; that is, the upper or future sheet. Calling this U, the lines of the model are the intersections of planes through the origin with U, and in general the m-flats of the model are the intersection of the (m+1)-dimensional subspace through the origin with U. The flats of U are therefore a subset of the flats of n-dimensional projective space, making the usual identification of subspaces of real (n+1)-dimensional vector space (the Grassmanian) with flats of n-dimensional projective space; this leads to the related Klein model of hyperbolic geometry. In mathematics, a projective space is a fundamental construction from any vector space. ... In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ...

If C(r) is any parametrized curve on U, , then the length of C is

In terms of special relativity, if we use units of years for time and lightyears for space, then the points of four-diminsional U, which are the points in the model of three-dimensional hyperbolic geometry, are all of the points reached after one year of travel, ship time, in a straight line at a constant velocity. Hence they can also be indentified with constant velocites. Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ...

References

James W. Anderson, Hyperbolic Geometry, second edition, Springer 2005