Fig. 4-1b shows a unit hyperbola with sinh(a) and cosh(a), where a is likewise interpreted as twice the tinted area. Fig. 4-2 presents plots of the sinh, cosh, and tanh functions. For the unit circle, the slope of the ray is given by In the Cartesian plane, rotation of point into point by angle θ is given by In a spacetime diagram, the velocity parameter is the analog of slope. The rapidity, φ, is defined by where The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula; or in other words, The Lorentz transformations take a simple form when expressed in terms of rapidity. The γ factor can be written as Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called boosts. Substituting γ and γβ into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the may be written as and the inverse Lorentz boost in the may be written as In other words, Lorentz boosts represent hyperbolic rotations in Minkowski spacetime. The advantages of using hyperbolic functions are such that some textbooks such as the classic ones by Taylor and Wheeler introduce their use at a very early stage. 4‑vectors Four‑vectors have been mentioned above in context of the energy–momentum , but without any great emphasis. Indeed, none of the elementary derivations of special relativity require them. But once understood, , and more generally tensors, greatly simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that are manifestly relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance of Maxwell's equations in their usual form is not trivial, while it is merely a routine calculation (really no more than an observation) using the field strength tensor formulation. On the other hand, general relativity, from the outset, relies heavily on , and more generally tensors, representing physically relevant entities. Relating these via equations that do not rely on specific coordinates requires tensors, capable of connecting such even within a curved spacetime, and not just within a flat one as in special relativity. The study of tensors is outside the scope of this article, which provides only a basic