Back towards the stone age we go!
That's actually a pretty good idea but then the sun needs a non-linear path. Otherwise "sundials" become "sunlines"
Well no.
If the sun didn't travel...
It's hard to explain, damn astrophysics.
It depends on the angle of the sun's path in the sky offset to its path as it would be perpendicular to the ground. On the Equator on the first day of Spring and Fall, the Sun will travel perpendicular to the ground at exactly noon.