Hyperbolic Trigonometry

Aim of the work is to propose the main terms of the EXCEL-orientated procedures for calculating the values of elementary and special functions with interval argument that is assigned on the hyperbolic form. The results of the work. The methods of presenting the interval values in the hyperbolic form and the rules of addition, subtraction, multiplication, and division of this values were considered. The procedures of calculating the function values, whose arguments can be degenerate or interval values were described. Namely, the direct and the reverse functions of the linear trigonometry, the direct and the reverse functions of the hyperbolic trigonometry, exponential function, arbitrary exponential function and power function, Gamma-function, incomplete Gamma-function, digamma-function, trigamma-function, tetragamma-function, pentagamma-function, Beta-function and its partial derivatives, integral exponential function, integral logarithm, dilogarithm, Frenel integrals, sine integral, cosine integral, hyperbolic sine integral, hyperbolic cosine integral. The basic terms of the EXCEL-orientated procedures for calculating the values of elementary and special functions with interval argument that is assigned on the hyperbolic form were proposed. The numerical examples were provided, that illustrate the application of the proposed methods.

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The Trigonometry of Hyperbolic Tessellations

Canadian Mathematical Bulletin ◽

10.4153/cmb-1997-019-0 ◽

1997 ◽

Vol 40 (2) ◽

pp. 158-168 ◽

Cited By ~ 7

Author(s):

H. S. M. Coxeter

Keyword(s):

Unit Circle ◽

Hyperbolic Plane ◽

Hyperbolic Trigonometry ◽

The Right ◽

Positive Integers

AbstractFor positive integerspandqwith (p− 2)(q− 2) > 4 there is, in the hyperbolic plane, a group [p, q] generated by reflections in the three sides of a triangleABCwith angles π/p, π/q, π/2. Hyperbolic trigonometry shows that the sideAChas length ψ, where cosh ψ =c/s, c = cos π/q,s= sin π/p. For a conformal drawing inside the unit circle with centreA, we may take the sidesABandACto run straight along radiiwhileBCappears as an arc of a circle orthogonal to the unit circle. The circle containing this arc is found to have radius 1/ sinh ψ =s/z, where z =, while its centre is at distance 1/ tanh ψ = c/z fromA. In the hyperbolic triangleABC, the altitude fromABto the right-angled vertex C is ζ, where sinh ζ = z.

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