Formulas for right-angled hyperbolic polygons (Q2702163)

The authors give explicit formulas for the hyperbolic cosine, sine and cotangent of the length of the side \(l\) of an hyperbolic \(N\)-sided polygon whose angles are all right. These formulas are given in terms of the lengths of the \(N-3\) sides different from \(l\) and its adjacent sides.NEWLINENEWLINENEWLINEThis work follows previous studies of the general problem of constructing hyperbolic convex polygons whose angles have prefixed measure [\textit{J. J. Etayo} and \textit{E. Martínez}, Math. Proc. Camb. Philos. Soc. 104, 261-272 (1988; Zbl 0662.30042) and \textit{A. F. Costa} and \textit{E. Martínez}, Geom. Dedicata 58, 313-326 (1995; Zbl 0839.51016)]. In the first article the authors have determined the number of parameters from which the construction depends. For some value of this parameters one can obtain the construction given by \textit{A. F. Beardon} [J. Lond. Math. Soc., II. Ser. 20, 247-254 (1979; Zbl 0407.51007)]. The second article also studied the construction of right angle polygons in terms of the length of \(N-3\) sides, and a procedure to express the hyperbolic cosine. NEWLINENEWLINENEWLINEIn this paper the authors obtain the formulas from the article by Costa and Martínez but now they use the same procedure as in [\textit{P. Buser}, ``Geometry and spectra of compact Riemannian surfaces'', (Prog. Math. 106. Birkhäuser, Boston) (1992; Zbl 0770.53001)] i.e. product of \(3\times 3\) matrices, which can be easily determined using a computer. The formula for the hyperbolic cosine is obtained independently from the formula for the hyperbolic sine or the cotangent.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00066].