With certain types of 3D curves (notably helixes), the Follow Me tool rotates the profile as it “climbs” the curve. This usually produces an undesired effect, such as twisting a helical (spiral) stair rail. Generally, this type of geometric problem requires something like @eneroth3’s Upright Extruder in lieu of the Follow Me tool.
However, this quirky behavior can be used to produce any number of interesting shapes. Any curve on the surface of a sphere (made up of many line segments, of course), can be used with the Follow Me tool to produce a contour that remains radial from the center of the sphere.
For example, the intersection of these two surfaces produces a 3D curve:
Add a very thin rectangle at one end:
Using the Follow Me tool produces this (a solid, BTW):
Making the single shape a component and replicating it about the center of the sphere:
Adding a mirrored copy:
Model with 360 degree animation: sphere_5a.skp (643.5 KB)