# Calculations with Curvature of geodesic circles on surface with constant curvature

A closed oval with constant curvature $$\kappa_g$$ of geodesic circles on a surface with constant curvature $$K$$ has perimeter $$p$$ and surface area $$A$$

How can we find $$(p,A)$$ in terms of $$(\kappa_g, K,u)?$$ connecting them to geodesic radial co-ordinate $$u$$ in a relation?

Flat case

$$p = 2 \pi u, \quad A = \pi u^2$$

Positive Curvature ($$R$$ radius of equator of sphere)

$$A= \frac{2 \pi}{K} (1- \cos (u \sqrt K) ); \, p= 2 \pi R \sin (\sqrt K u)$$

Negative Curvature ($$R$$ radius of cuspidal equator of pseudo-sphere)

$$A= \frac{2 \pi}{|K|} (1- \cosh (u \sqrt {|K| }); \, p= 2 \pi R \sinh (\sqrt {|K|} u)$$

Are these correct? Can we eliminate $$R$$ to find $$(K, \kappa_g)$$ relation? If so how? How are they derived from their corresponding metrics? Thanking you in advance.