If $\kappa_n, \tau_g$ represent normal curvature and geodesic torsion respectively of a curve on a surface in $\mathbb R^3 $ where $(h,k,R )$ are constants, what is represented when a special Mohr's Circle curvature relationship:
$$ (\kappa_n- h)^2 + (\tau_g -k)^2 = R^2 \tag 1 $$
holds?
It is not a general property or identity of 3d curves/surfaces.
If $(\kappa_1,\kappa_2)$ of normal curvature of Euler relation are given,
$$\kappa_n=\kappa_1 \cos^2 \psi +\kappa_2\sin ^2 \psi \tag 2 $$
differentiation with respect to Mohr circle arc gives
$$ \kappa_n^{'} = -\kappa_1 2 \cos \psi \sin\psi + \kappa_2 2 \cos\psi \sin\psi =-2 \psi^{'} \tau_g $$ from definition of $\tau_g = \dfrac{\kappa_1-\kappa_2}{2}$ we have an important combination relation:
$$ \boxed{ \frac{d \kappa_n}{d \psi}+ 2 \tau_g=0 }\tag 3 $$
Trivial solutions
Great circles on a sphere with constant $ (\kappa_n = c_1,\tau_g=0) $ trivially satisfy the above Mohr general equation.
How we can integrate the above ordinary differential equation for some general or interesting particular solutions?
The curvatures are encountered through bipolar coordinate representation, in stereographic projection of concentric cylinders intersecting a sphere etc., but the general solution is sought.
An arbitrarily chosen surface of revolution loxodrome of constant rhumb angle $ \psi= \pi/4$ with constant curvatures is chosen for easier calculation per details given. It is a pseudosphere $ K=\frac{-7}{4} $ but does not obey Mohr curvature relation.
$$\kappa_n= 1/\sqrt 2, \tau_g =1/2 $$ $$\kappa_n= (\kappa_1+\kappa_2)/2;\; \tau_g=(\kappa_1-\kappa_2)/2 $$
$$\kappa_1= \kappa_n+ \tau_g;\;\kappa_2= \kappa_n- \tau_g;\;K= \kappa_n^2- \tau_g^2;$$
Numerical example B.C. $ \;\psi_{start}= 0.7, slope =45^0,$
Thanks for all thoughts/comments.