Why is a projection conversion off by factor of SQRT(2)?

Question

I am attempting to draw a 3 km ring around downtown Minneapolis. Let downtown be defined as Government Plaza (-93.2661, 44.9765). I am using this ring to calculate census tracts within a distance of a point, so I am using the Census Bureau's projection; the census shapes are given in lat/lon coordinates, so I have to convert back and forth to calculate shape intersects on one hand, and measure distance on the other.

In Python, using shapely and pyproj:

import shapely.geometry as shpgeo, pyproj

lon, lat = (-93.2661, 44.9765)

proj = pyproj.Proj(proj="utm", ellps="GRS80", datum="NAD83")

# project into UTM to calculate distance in meters
point = shpgeo.Point(*proj(lon, lat))

# get a radius 3000 meter circle
bufferShape = point.buffer(3000)

# convert from utm back to lon/lat
coords = shp.__geo_interface__['coordinates']
# this works if shp.__geo_interface__['type'] == Polygon
newCoords = [[proj(*point, inverse=inv) for point in linring] for linring in coords] 
bufferShape = shpgeo.shape({'type': shpType, 'coordinates': tuple(newCoord)})

Now, let us test this shape, to see if we got what we wanted:

print(bufferShape.centroid)
print(bufferShape.bounds)

POINT (-93.26610002878017 44.97650391330887)
(-93.29303779998378, 44.95738308641549, -93.23916202385492, 44.99562317086678)

The centroid is just about dead on through the conversion to and from UTM; at least accurate enough for my purposes. The bufferShape should be a circle around the starting point; the max_x coord and the centroid's y coord should be the north-most point of the circle. That point is (44.9956, -93.2661); which according to distance checks with third party software is 2.12 km away; not 3 km.

How did this happen? The number 2.12 looked suspicous to me, so I changd this line:

 bufferShape = point.buffer(3000 * 1.41421)

 POINT (-93.26610005756029 44.97650782661781)
 (-93.30419574627818, 44.94946643426645, -93.22800390139928, 45.003546080298214)

The new north-most point is (-93.2661, 45.0035). Multiplying the distance by the square root of 2 in UTM, I get the correct coordinates in lon/lat.

Why is this happening?

Answer 1

The most likely answer here is that the UTM initialization is incorrect. Universal Transverse Mercator is a family of projections and by not specifying the correct zone the pyproj assumes central meridian at -183°:

Proj('+proj=tmerc +lat_0=0 +lon_0=-183 +k=0.9996 +x_0=500000 +y_0=0 +datum=NAD83
 +units=m +no_defs', preserve_units=True)

Which is likely why the grid coordinates and buffer are incorrect - this central meridian is a long way from where you are working. Instead, try specifying the correct UTM zone as part of the proj call:

proj = pyproj.Proj(proj="utm", zone=15, ellps="GRS80", datum="NAD83")

or

proj = pyproj.Proj("+init=EPSG:26915")