# [Proj] Convergence

Kees Krikke kees.krikke at orange.nl
Mon Sep 15 11:28:05 EDT 2008

```Hello,

I've made a piece of s/w that uses proj's convergence calculation to
find the true north for a given grid north.

When testing with a utm and lcc projection i found some confusing
results regarding the sign (direction) of the convergence.

To demonstrate this i modified proj.exe so it prints the convergence (in
deg) at the end of the line when using the -S flag:

C:\proj-4.6.1\src>proj -vS +proj=utm +datum=WGS84 +lon_0=3
#Universal Transverse Mercator (UTM)
#       Cyl, Sph
#       zone= south
# +proj=utm +datum=WGS84 +lon_0=3 +ellps=WGS84 +towgs84=0,0,0
2 52
431350.30       5761510.32      <0.999658 0.999658 0.999316 1.20783e-006
0.999658 0.999658 -0.788041>
4 52
568649.70       5761510.32      <0.999658 0.999658 0.999316 8.54066e-007
0.999658 0.999658 0.788041>
4 52.1
568496.62       5772632.28      <0.999658 0.999658 0.999315 0 0.999658
0.999658 0.789115>

From the above results it seems that for utm the convergence is
negative when west and positive when east of lon_0.
If you calculate two utm positions along the same longitude east of
lon_0, you see that the easting reduces when going northwards.

From this is seems that TrueNorth = GridNorth + convergence

I used the same points for a lcc projection:

C:\proj-4.6.1\src>proj -vS +proj=lcc +datum=WGS84 +lat_0=52 +lon_0=3
+lat_1=50 +lat_2=54
#Lambert Conformal Conic
#       Conic, Sph&Ell
#       lat_1= and lat_2= or lat_0
# +proj=lcc +datum=WGS84 +lat_0=52 +lon_0=3 +lat_1=50 +lat_2=54 +ellps=WGS84
# +towgs84=0,0,0
2 52
-68634.11       472.08  <0.999392 0.999392 0.998785 0.000355057 0.999395
0.999389 0.788173>
4 52
68634.11        472.08  <0.999392 0.999392 0.998785 0.000355708 0.999395
0.999389 -0.788173>
4 52.1
68481.15        11591.10        <0.999393 0.999393 0.998787 0.000434814
0.999397 0.99939 -0.788173>

The convergence sign west and east of the origin are opposite of the
signs found with the utm. But what confuses me is that when i calculate
two positions along a longitude east of the origin, the easting reduces
when going northwards just like with the utm projection.

From this is seems that TrueNorth = GridNorth - convergence

I think that either the convergence for lcc or utm has the wrong sign.
But i can't tell which one because i don't know how convergence is defined.