How to correct for non-standard curvature?

[rg0now]

Dear all,

I am given a hyperbolic space of constant negative curvature 'k' and a circle of radius 'R'. How do I get the circumference and the area of the circle? My problem is that I know the answer only for the case when 'k=-1', but my space can have any possible (negative) curvature, possibly different from '-1'. I have the feeling that these formulas must somehow be corrected for non-standard curvature.

And if we are at it: Is there a generic formula for circle circumference and area, which is somehow valid for spherical as well as Euclidean and hyperbolic spaces of different curvature?

Thanks,
Gabor

[rg0now]

For the archive: if the curvature is '-k', then the circumference is:

c(r) = 2 * PI / sqrt(-k) * sinh( sqrt(-k) * r)

and the area is:

a(r) = 2 * PI / (-k) * (cosh(sqrt(-k) * r) - 1)

Besides, I am still curious what the answer to the second question could be.

Regards,
Gabor

[gudi]

Please read Roberto Bonola's book, 'non-euclidean geometry'

[gudi]

On Aug 29, 9:51 pm, rg0now <retv...@tmit.bme.hu> wrote:

> And if we are at it: Is there a generic formula for circle circumference and area, which is somehow valid for spherical as well as Euclidean and hyperbolic spaces of different curvature?

The cases are generalizations of the euclidean geometry.That is same
as saying, euclidean geometry is a special case of the hyperbolic and
elliptic geometries. For example:

The Law of Sines applicable for a triangle:

Elliptic (spherical) geometry
sin(a) / sin(A) = sin(b) / sin(B) =sin(c) / sin(C) = E

Euclidean Geometry
a / sin(A) = b / sin(B) = c / sin(C) = 2 Circum-circle radius R

Hyperbolic geometry
sinh(a) / sin(A) = sinh(b) / sin(B) =sinh(c) / sin(C) = H

It can be seen that small values of 'a' produce euclidean geometry
from the two 'deeper' geometries as special cases.

It seems as yet no one knows the geometrical meaning of the triangle
invariant property for either E or H.

Regards,

Narasimham