Many of the results in hyperbolic geometry take on simpler, and sometimes familiar form
when expressed in terms of the hyperbolic sine of half of
the hyperbolic lengths involved.
some notation
For a hyperbolic triangle ABC, we let
a = d(B,C), b = d(A,C), c = d(A,B),
ca = cosh(a), cb = cosh(b), cc = cosh(c),
α = sinh(½a), β = sinh(½b), γ = sinh(½c),
A = angle at A, B = angle at B, C = angle at C.
We have met the polynomials
F(x,y,z) = 3 + 2(xy+yz+zx) -2(x+y+z) - (x2+y2z2),
Δ2(x,y,z) = 1 + 2xyz - (x2+y2+z2).
From the theorems of Heron and Brahmagupta, we define
H(x,y,z) = (x+y+z)(x+y-z)(y+z-x)(z+x-y).
It is easy to see that there is a euclidean triangle
with sides
of length x,y,and z if and only if H(x,y,z) > 0.
Heron's Theorem says that the triangle has area ¼H½(x,y,z).
B(x,y,z,t) = (x+y+z-t)(x+y+t-z)(x+z+t-y)(y+z+t-y).
We met F(ca,cb,cc) in our search for the circumcircle.
If we work instead in the sinh, we find that
F(ca,cb,cc) = 4H(α,β,γ).
Unfortunately Δ2 does not simplify, but
Δ2(ca,cb,cc)-F(ca,cb,cc) = (4αβγ)2.
Thus, in hyperbolic geometry,
The circumcircle
exists if and only if H(α,β,γ) > 0, and
the circumradius r satisfies sinh2(r) = 4(αβγ)2/H(α,β,γ).
Note. It is reasonable to find sinh(r) rather than sinh(½r) -
the euclidean analogue is usually expressed in terms of 2r.
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