A curved line segment of constant curved length $ L=PQ$ rotates around fixed point O contacting a smooth rigid surface. "Diameters" $P-Q,P1-Q1,P2-Q2...$ are sliding on, or skinning along the surface. They bend or twist according to line curvature of the surface.
Can we define some examples of line/surface combination of given boundary length $\lambda L $ for constant $\lambda ?$
Only a trivial case of a straight diameter line L sweeping out around center point $O$ a circle C in a plane with $ \lambda =\pi $ comes to mind. So also is the case with geodesic circles using geodesic polar coordinates where a pole at center is fixed and radii are of fixed length in rotation without sliding.
EDIT 1:
To start with a simple spherical cap when $O$ is not at the center of the cap we have differential arcs $s_1,s_2$ on opposite sides of O
$$ d \psi= \frac{ds_1}{\rho}=\frac{ds_2}{L-\rho}=\frac{d(s_1+s_2)}{L} $$
where $\psi$ is rotation from a fixed reference line, $\rho$ is radial length OP in the manner of geodesic polar coordinates. But could not proceed further.
Thanks in advance for all comments about handling such situations.
