That is a neat idea for a very specific use case - thank you.
To be honest, I was fitting the # of lathe segments in my example to the # of vertical lines of that specific sample texture, to show the effect in maximum clarity. Actually, the kink in mapping a straight vertical line to a quadrilateral will be seen for any number of lathe segments which is different from an exact integer multiple of the number of visible texture map subdivisions, i.e. vertical white lines.
Ultimately I’m aiming at getting more realistic reflective mapping of natural textures, which typically lack any regular vertical traits. So the nature and cause of the problem would have been much more difficult to illustrate.