geodesy - How to calculate the distance between UTM-projected coordinates?

My coordinates are projected using the following projection:

proj= {"UTMZone32", {"GridOrigin" -> {500000, 0}, "CentralScaleFactor" -> 0.9996}};

Now I wish to calculate the distance between two points (ignoring elevation), e.g.

p1= GeoGridPosition[{359577, 5.51291*10^6,0}, proj]
p2= GeoGridPosition[{509108, 5.972*10^6,0}, proj]

When I try GeoDistance

GeoDistance[p1,p2]

it fails with the error message

GeoDistance::invparam: "Invalid parameters \!\(\"GeoGridPosition[{359577, 5.51291*^6, 0},
{\\\"UTMZone32\\\", {\\\"GridOrigin\\\" -> {500000, 0}, 

 \\\"CentralScaleFactor\\\" -> 0.9996}}]\"\). "

Also, the GeoPositionXYZ function, as in

GeoPositionXYZ[p1]

fails with the error messages

ToString::nonopt: Options expected (instead of InputForm) beyond position 2 in 
ToString[None,{GridOrigin->{500000,0},CentralScaleFactor->0.9996},InputForm]. 
An option must be a rule or a list of rules. >>


GeoGridPosition::invparam: "Invalid parameters ToString[\!\(None, {
 \"GridOrigin\" -> {500000, 0}, \"CentralScaleFactor\" -> 0.9996`}, InputForm\)]."

GeoPositionXYZ::invcoord: "\!\(\"GeoPosition[GeoGridPosition[{359577, 5.51291*^6, 0}, 
{\\\"UTMZone32\\\", {\\\"GridOrigin\\\" -> {500000, 0}, \\\"CentralScaleFactor\\\" -> 
0.9996}}]]\"\) is not a valid coordinate specification."

Both functions work, however, when I switch proj to the string UTMZone32.

Do I need to get the full projection specification to work?

EDIT: After some further googling, I realized that in UTM coordinates the distance between two points is simply

Norm[{p1[[1,1;;2]]-p2[[1,1;;2]]}]

so I would answer my own question with no.

Answer

As @Sjoerd states in the comments, your projection system (UTMZone32) has a defined set of parameters. You can check these using GeoProjectionData:

GeoProjectionData["UTMZone32"]

{"TransverseMercator", {"Centering" -> {0, 9}, "CentralScaleFactor" -> 0.9996, "GridOrigin" -> {500000, 0}, "ReferenceModel" -> "WGS84"}}

These coincide with the ones you are trying to set.

To define your own projection system similar to UTM (based on Transverse Mercator), you can simply specify those in GeoGridPosition:

GeoGridPosition[{1000000, 1000000}, 
 {"TransverseMercator", {"Centering" -> {0, 0}, "CentralScaleFactor" -> 0.95,
 "GridOrigin" -> {500000, 0}, "ReferenceModel" -> "WGS84"}}]

This now can be easily converted to LatitudeLongitude.

So, since this is a projected coordinate system, and as you state at the end of the question, can be easily calculated using Norm or EuclideanDistance or whatever:

Norm[{359577, 5.51291*10^6, 0} - {509108, 5.972*10^6, 0}]

EuclideanDistance[{359577, 5.51291*10^6, 0}, {509108, 5.972*10^6, 0}]

482828.

482828.

But we can also use the built-in GeoDistance which in v10 returns a Quantity:

pos1 = GeoGridPosition[{359577, 5.51291*10^6, 0}, "UTMZone32"];
pos2 = GeoGridPosition[{509108, 5.972*10^6, 0}, "UTMZone32"];
GeoDistance[pos1, pos2]~UnitConvert~"Meters"

482985. m

Sadly, they're 157 meters apart.

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