In special relativity it was first used by Gruner (1921) while developing Loedel diagrams. As already indicated in equations () in exponential form or () in terms of Cayley–Klein parameter, Lorentz boosts in terms of hyperbolic rotations can be expressed as squeeze mappings. Using asymptotic coordinates of a hyperbola ("u,v"), they have the general form (some authors alternatively add a factor of 2 or formula_89):  That this equation system indeed represents a Lorentz boost can be seen by plugging (1) into (2) and solving for the individual variables:  Lorentz transformation () of asymptotic coordinates have been used Laisant (1874) and Günther (1880/81) in relation to elliptic trigonometry, or by Lie (1879-81), Bianchi (1886, 1894), Darboux (1891/94), Eisenhart (1905) as Lie transform) of pseudospherical surfaces in terms of the Sine-Gordon equation, or by Lipschitz (1885/86) in transformation theory. From that, different forms of Lorentz transformation were derived: () by Lipschitz (1885/86), Bianchi (1886, 1894), Eisenhart (1905), trigonometric Lorentz boost () by Bianchi (1886, 1894) and Darboux (1891/94), and trigonometric Lorentz boost () by Eisenhart (1905). Lorentz boost () was rediscovered in the framework of special relativity by Hermann Bondi (1964) in terms of Bondi k-calculus, by which "k" can be physically interpreted as Doppler factor. Since () is equivalent to () in terms of Cayley–Klein parameter by setting formula_90, it can be interpreted as the 1+1 dimensional special case of Lorentz Transformation () stated by Gauss around 1800 (posthumously published 1863), Selling (1873), Bianchi (1888), Fricke (1891) and Woods (1895). Variables "u, v" in () can be rearranged to produce another form of squeeze mapping, resulting in Lorentz transformation () in terms of Cayley-Hermite parameter:  x_{1}^{\prime} & =-x_{0}\frac{2a}{1-a^{2}}+x_{1}\frac{1+a^{2}}{1-a^{2}} & & =\frac{-x_{0}2a+x_{1}\left(1+a^{2}\right)}{1-a^{2}}\\ x_{0} & =x_{0}^{\prime}\frac{1+a^{2}}{1-a^{2}}+x_{1}^{\prime}\frac{2a}{1-a^{2}} & & =\frac{x_{0}^{\prime}\left(1+a^{2}\right)+x_{1}^{\prime}2a}{1-a^{2}}\\  These Lorentz transformations were given (up to a sign change) by Laguerre (1882), Darboux (1887), Smith (1900) in relation to Laguerre geometry.