[Blood of Tyrants]

He made a point I had never considered, that a line through a globe could be parallel with a curved line on the surface.

[equaviator]

"a line through a globe could be parallel with a curved line on the surface"

The applicable term for that condition or circumstance would be "coplanar".

A condition of parallelism would require that at least two defining points of the line and two perpendicular points on the surface (section) curve be "equidistant".

Therefore, your example could only be true if your surface "curve" falls onto a planar surface that runs parallel to your line, and it also means that your surface can't have curvature and/or contour to it. So rather than a globe, your 3D shape would have to be at least a cube or a cylinder.