Generally speaking, when do constant K (Gauss curvature) and zero K surfaces intersect
to produce lines of constant geodesic curvature $ k_g $ ? Small circles on a sphere
Or more specifically the question is:
How should a plane be bent and relatively positioned in order to intersect a constant K surface along a constant $k_g$ line ?
EDIT1: A more complicated example with a cone type pseudosphere and a geodesic circle:
EDIT2: A hyperbolic sphere (with cuspidal edge not shown) cut by an almost flat bent cylinder:
In both cases constant $k_g$ geodesic circles are seen rotated around axis of symmetry.
MY BACKGROUND CONJECTURE:
"A suitably bent right circular cylinder intersects constant Gauss curvature surfaces in $R^3$ along lines of constant geodesic curvature.
It remains to disprove my conjecture and, if validated define what that bending is.
EDIT4: In the particular images above generators of cylinder are perpendicular to symmetry axis of revolution.
I suppose this edit would make it easier to define case of moving red ring: The ring is made to slide on surface of revolution of K as a special case.
Let a thin parallel circle of ring be mounted on the surface. Let us imagine it is made of an inextensible but flexible material.It can be cut and dislocated to any arbitrary latitude,cut ends repaired to be again glued back elsewhere on the surface.
Perimeter length and $ k_g $ do not change after such a transportation. Find cordinates of this oval on an axial plane when one axis of the oval is projected on the axial plane as a function of the new arbitrary latitude.