Solutions: Non -Euclidean Geometry Problems

1) Given that a = 1, b= 2, k = 1 and we need to find c.

 

We have:  cosh(c/k) = cosh (a/k) cosh(b/k)

 

Or: Letting k = 1:

 

cosh(c) = cosh (a) cosh(b)

cosh(c) = cosh(1) cosh(2) = (1.543)(3.762) = 5.804

 

Take the inverse hyperbolic cosine:

 

cosh^-1(c) = cosh^-1(5.804)  = c

 

Or:

 

c =  2.444

 

The ratio of the sines of the non-Euclidean angles A, B is given by:

 

sin (A)/ sin(B) = {(sinh(a/k)/ sinh(b/k)}

 

so that, since k = 1:

 

 sin (A)/ sin(B) = {(sinh(a) / sinh(b)}

 

sin (A)/ sin(B) = {sinh (1)/ sinh (2)}

 

where: sinh(1) = 1.175 and sinh(2) = 3.626

 

Therefore:

sin (A)/ sin(B) =  (1.175)/ (3.626) =  0.323

 

2) We compute the ratio x/k = 2 (i.e. 1.0 / 0.5 = 2.0;  0.5/ 0.25 = 2.0 etc.)

 

Then: exp (- x/ k) =   exp (-2) =  0.1353=     1 / (exp(2))

 

But the curvature: k = -1/ k²

 

To conform with: 1 / (exp(x/k)) = 1 (exp (2)) we must have k = 1 (positive real no.)

 

Therefore: k = -1/ k²    =  -1 / (1) ²  =  -1

 

Hence, the curvature is negative, and the space is hyperbolic.