On arbitrary axisymmetric patches geodesics with polar symmetry and common Clairaut's constant $ r_o$ minimum radius of fiber tangency have the differential equation:
$$ r_o= r \, \sin \psi= const $$
Orthogonal trajectories of these geodesic family are given by $ \psi\rightarrow \pi-\psi $
$$ r_b= r \, \cos \psi= const $$
which can be called 3D involutes as orthogonal trajectories. On the following surface, geodesics can be seen on the outside and equidistant tubes drawn for 3D clarity on the inside.
Just as in the plane case, the width of cyclic involutes is constant, a concept inherent with radial geodesic polar coordinate parameter.. right from Leibnitz's time.
(fig to be soon loaded for plane involutes case)