In a nutshell:
Given a standard (angular) GPS measurement and heading, relative geographic north. If i take my GPS coordinates and turn them into UTM coordinates, do I also have to convert my heading? I can't find this topic brought up anywhere, so I'm guessing that Im missing something important here.
As the UTM coordinates deviates the further away you get from the center planar projection point (centered around the prime meridian), the heading must also deviate. Since the prime meridian is the only line that is parallell to the geographic north (except on the equator), the error in heading would increase the further i move away from the prime meridian.
Look at the above image, showing a very large plane projected on a sphere. The gray and red latitudinal lines represents geographic north in point p1 and p2 respectively. On the prime meridian, in p1, the geographic north reading is the same in angular and UTM coorinates. When moving away from the prime meridian, as in p2, the UTM y-axis does not line up with my GPS reading of geographical north. The error is thus the angle between my UTM y-axis and my geographic north.
Look at the second image. Subtending angle c would be 90-latitude, subtending angle a would be longitude - UTM prime meridian. Spherical angle B is 90 degrees (orthogonal spherical coordinate axis)
From the cosine rule it can be derived that
cos(a)*cos(B) = cot(c)*sin(a) - cot(C)*sin(B)
Thus, spherical angle C becomes
C = atan(sin(B) / (cot(c)*sin(a) - cos(a)*cos(B)))
And my final error in heading becomes 90-C degrees.
Am I correct in that there is an error present when looking at geographical north from a planar projection, and would this be it?