Cartopy: Convert Point From Axes Coordinates To Lat/lon Coordinates

I am working on a map plot with Cartopy (TransverseMercator projection). I have a point in axes coordinates (p_a = (0.1, 0.9)) and I need its lat/lon coordinates (i.e., those in PlateCarree projection). How can I achieve this?

For this you need to transform the point from axes coordinates to display coordinates, then to data coordinates, and finally to lat/ lon coordinates. Thus you need transformations from matplotlib and cartopy.

The point p_a = (0.1, 0.9) seems to be outside of valid lat/ lon coordinates (for the default ccrs.TransverseMercator()). Therefore I use p_a = (0.6, 0.6)

import cartopy.crs as ccrs
import matplotlib.pyplot as plt

proj = ccrs.TransverseMercator()
proj_cart = ccrs.PlateCarree() 

f, ax = plt.subplots(subplot_kw=dict(projection=proj))
ax.coastlines()

# define point
p_a = (0.6, 0.6)

# plot point in Axes coordinates
ax.plot(*p_a, transform=ax.transAxes, marker='o', ms=10)

# convert from Axes coordinates to display coordinates
p_a_disp = ax.transAxes.transform(p_a)

# convert from display coordinates to data coordinates
p_a_data = ax.transData.inverted().transform(p_a_disp)

# convert from data to cartesian coordinates
p_a_cart = proj_cart.transform_point(*p_a_data, src_crs=proj)

# make sure we are correct
ax.plot(*p_a_cart, transform=proj_cart, marker='x', ms=10)

This yields the following figure: