Why is hexbin grid distorted in polar stereo projection

Question

I am trying to understand why a hexbin plot in a north or south polar stereo projection shows squashed hexagons, even though the area of the grid is square and the projection is approximately equal area.

I've tried both north and south polar stereo projections using basemap.

import numpy as np
from numpy.random import uniform
import matplotlib.pyplot as plt 
from mpl_toolkits.basemap import Basemap

fig = plt.figure(figsize=(12,10)) # width, height in inches
ax =fig.add_axes([-0.02,0.1,0.74,0.74]) 

m = Basemap(epsg='3413',lon_0=0.,resolution='l',width=6000000,height=6000000)

m.drawcoastlines()

m.drawmapscale(0.,90.,0.,90.,1000)

npts=2000
lats = uniform(60.,80.,size=npts)
lons = uniform(0.,360.,size=npts)
data = uniform(0.,4800.,size=npts)

x,y=m(lons, lats)

thiscmap=plt.cm.get_cmap('viridis')

p=m.hexbin(x,y,C=data,gridsize=[10,10],cmap=thiscmap)

plt.show()

Answer 1

I don't know why you get squashed hexagons. But you can adjust the hexagon shape by setting appropriate values of gridsize. Here I modify your code and get better plot.

import numpy as np
from numpy.random import uniform
import matplotlib.pyplot as plt 
from mpl_toolkits.basemap import Basemap

fig = plt.figure(figsize=(12,10)) # width, height in inches
ax =fig.add_axes([-0.02, 0.1, 0.74, 0.74]) 

# North polar stereographic projection epsg='3413'; ***large areal distortion***
#m = Basemap(epsg='3413', lon_0=0., resolution='c', width=6000000, height=6000000)

# 'laea':  Lambert Azimuthal Equal Area
# Thematic mapping with ground surface data should be plotted on 'equal-area' projection
m = Basemap(projection='laea', lon_0=0., lat_0=90, resolution='l', width=6000000, height=6000000)

m.drawcoastlines(linewidth=0.5)

m.drawmapscale(0.,90.,0.,90.,1000)  # 1000 km?

npts = 2000
lats = uniform(60.,80.,size=npts)  # not cover N pole
lons = uniform(0.,360.,size=npts)  # around W to E
data = uniform(0.,4800.,size=npts)

x,y = m(lons, lats)

thiscmap = plt.cm.get_cmap('viridis')

# To get 'rounded' hexagons, gridsize should be specified appropriately
# need some trial and error to get them right
#p=m.hexbin(x, y, C=data, gridsize=[10,10], cmap=thiscmap)  # original code
m.hexbin(x, y, C=data, gridsize=[16,11], cmap=thiscmap)     # better

plt.colorbar()  # useful on thematic map

plt.show()

The projection you use (epsg:3413) is stereographic projection which has large areal distortion. More appropriate projection for thematic mapping in this case is Lambert Azimuthal Equal Area.

The resulting plot: