This item got one answer after some hours on stackexchange, and I have a feeling I should solicit whatever variety of opinions may be out there:

Draw a line from a point on a sphere, which let us call the north pole, through another point on the sphere, to a plane parallel to the plane tangent to the sphere at the north pole.  That last point is the stereographic projection of the typical point on the sphere onto that plane.  Then the same thing gets done in higher dimensions and the same term --- "stereographic projection" --- is used.

No problem so far.

But I hesitate to use that term when it's from a circle to a line, because "ster-" or "stere-" usually means "solid" or "three-dimensional".

Are there opinions on the propriety of that usage?

Also, is there a name for the inverse mapping from the line or plane or hyperplane to the sphere?