How to transform from lat/lon to cartesian coordinates for setting plot limits

I’ve been unable to find online information that clarifies for me how to do the following seemingly simple task:

Given user-specified (central_lat, central_lon) and geographic radius in units of distance (or more conveniently, degrees latitude DR), set the plotting extent for a stereographic projection centered on (central_lat, central_lon) and with square plot limits just enclosing the specified radius.

I’ve done this sort of thing easily with simpler map plotting systems, in which the orthographic projection maps lat/lon to a cartesian coordinate system with the origin at (central_lat, central_lon).

In pseudocode:

  1. Set orthographic project centered on (central_lat, central_lon)
  2. Find cartesian coordinates (X0, Y0) at (central_lat, central_lon)
  3. Find cartesian coordinates (X0, Y0+DY) at (central_lat + DR, central_lon)
  4. Set plot extent to [X0-DY, X0+DY, Y0-DY, Y0+DY)

I’ve tried various combinations of projections and transform methods, but I seem not to have found the one that gets me from lat/lon to cartesian (not display) coordinates and that can be used to set the plot extent.

Here are some code fragments to help orient anyone reading:

  clat = 40
  clon = -90
  DR = 20    # plot has a radius equivalent to DR degrees 

  proj = ccrs.Orthographic(central_latitude=clat, central_longitude=clon)
  data_proj = ccrs.PlateCarree()

  fig = plt.figure(figsize=(14, 14))
  m = plt.axes(projection=proj)

  ....  # get transformed limits here somehow

  extent = [X0, X1, Y0, Y1]   # Cartesian coordinates   
  m.set_extent(extent, proj) # ???  or some other projection here

I’m a bit confused, as sometimes you’re talking about a Stereographic plot and sometimes an Orthographic one. And then sometimes a radius and then sometimes a square plot.