How Can You Determine the Radius of Curvature Without Knowing R?

[skippy1729]

Given a triangle on a hyperbolic surface with all angles and edge length known the area is given by R^2 x(PI - a - b - c), where a, b and c are the angles and R is the radius of curvature of the surface. What if you don't know R?

Same question for a triangle on a spherical surface where R is unknown.

Equivalent question: How do you measure R using LOCAL length and angle measurements? Assume that you know only that the surface has a constant curvature.

Thanks, Skippy

 

[Reb]

How do you measure R

If I understand your question, you can find [tex]R=1/\sqrt{|K|}[/tex] by means of the Gaussian curvature [tex]K={\rm det}(II)/{\rm det}(I)[/tex], [tex]I[/tex] and [tex]II[/tex] being the first and second fundamental forms of the surface.

However, you can always invoke the Gauss-Bonnet formula:

http://mathworld.wolfram.com/Gauss-BonnetFormula.html


Especially if you restrict your study to geodesic triangles, you can derive the formulas for their areas.