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

$$ r_b= r \, \cos \psi= const $$

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. 

[![Constant Width Equidistant Curves][1]][1]

Just as in the plane case, the width of cyclic involutes is constant, a concept inherent with radial geodesic polar coordinate parameter..  from Leibnitz's time.

fig to be loaded

  [1]: https://i.sstatic.net/owFkK.png