Jacob:

Irrelevant...the motion of the shadow cast by the Sun will never be linear...!
If you want to play with angles and surfaces to produce a linear response then plan on different formulas for location/construction for every day of the year...it's a pointless exercise.
However...!
On the day of the two solstices the "tip" of the shadow will follow a straight line from west to east in the northern hemisphere...opposite down under at any location on this planet.


Best.
Clive