Hyperboloid Model - History

History

In 1880 Wilhelm Killing published "Die Rechnung in Nicht-Euclidischen Raumformen" in Crelle's Journal (89:265–87). This work discusses the hyperboloid model in a way that shows the analogy to the hemisphere model. Killing attributes the idea to Karl Weierstrass in a Berlin seminar some years before. Following on Killing’s attribution, the phrase Weierstrass coordinates has been associated with elements of the hyperboloid model as follows: Given an inner product on Rn, the Weierstrass coordinates of x ∈ Rn are:

compared to

for the hemispherical model. (See Elena Deza and Michel Deza (2006) Dictionary of Distances.)

According to Jeremy Gray (1986), Poincaré used the hyperboloid model in his personal notes in 1880. Gray shows where the hyperboloid model is implicit in later writing by Poincaré.

For his part, W. Killing continued to publish on the hyperboloid model, particularly in 1885 in his Analytic treatment of non-Euclidean spaceforms. Further exposure of the model was given by Alfred Clebsch and Ferdinand Lindemann in 1891 in Vorlesungen uber Geometrie, page 524.

The hyperboloid was explored as a metric space by Alexander Macfarlane in his Papers in Space Analysis (1894). He noted that points on the hyperboloid could be written

where α is a basis vector orthogonal to the hyperboloid axis. For example, he obtained the hyperbolic law of cosines through use of his Algebra of Physics.

H. Jansen made the hyperboloid model the explicit focus of his 1909 paper "Representation of hyperbolic geometry on a two sheeted hyperboloid". In 1993 W.F. Reynolds recounted some of the early history of the model in his article in the American Mathematical Monthly.

Being a commonplace model by the twentieth century, it was identified with the Geschwindigkeitsvectoren (velocity vectors) by Hermann Minkowski in his Minkowski space of 1908. Scott Walter, in his 1999 paper "The Non-Euclidean Style of Special Relativity" recalls Minkowski’s awareness, but traces the lineage of the model to Hermann Helmholtz rather than Weierstrass and Killing. In the early years of relativity the hyperboloid model was used by Vladimir Varićak to explain the physics of velocity. In his speech to the German mathematical union in 1912 he referred to Weierstrass coordinates.