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.


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