Calculating the Max N/S/E/W of polygon
While Darren's code only checks one coordinate axis to determine if the current vertex being examined is more extreme in a certain direction, he actually replaces the point associated with that compass direction with the full vertex point coordinate pair from the polygon boundary. So all of the points listed in his table are real coordinates that fall on the polygon boundary.
I believe the extent pairs of min/max coordinates are correct for a single axis of each point in Dennis's list, but the other axis of the coordinate does not have to fall on the extent rectangle boundary corners. So while the westerly most point has to match the min X value of the extent, the Y value can be anywhere along the western edge of the extent polygon (min Y <= West point Y <= max Y) as long as it is on a vertex of the original polygon (or a point along a true curve). So as long as there are no true curves it seems like you could search through the vertices of the polygon and find any and all points that matched on the appropriate axis from the extent (which in the end should be what Dennis's code effectively does). It can be accomplished by creating the extent rectangle for each polygon and doing an intersect to get the polygon points that actually touch the extent edge as a multipoint, but Python is easier since each polygon has to only intersect its own extent which would require iteration in ModelBuilder (inefficient). See the picture below. She wants the red points for the polygon below.
If the polygon included any true curves that could touch an edge of the polygon extent between the ends of the curve, you would have to use the intersect tool or the geometry intersect approach to find the real point for that compass direction, since the point along the extent boundary would not be a vertex of the polygon in that case.
Here's how you can do it using Python:
1.) Add fields to polygon feature class (could also do in Python, if you want):
2.) Run script
>>> fc = 'all_aoi' # change to your layer name or feature class path
... fields = ['SHAPE@','N_X','N_Y','E_X','E_Y','S_X','S_Y','W_X','W_Y'] # pay attention to order
... with arcpy.da.UpdateCursor(fc,fields) as cursor: # loop through polygons
... for row in cursor: # for each polygon
... N = arcpy.Point(0,float('-inf')) # dummy coordinates
... E = arcpy.Point(float('-inf'),0) # dummy coordinates
... S = arcpy.Point(0,float('inf')) # dummy coordinates
... W = arcpy.Point(float('inf'),0) # dummy coordinates
... for part in row[0]: # loop through polygon parts
... for pnt in part: # loop through vertices
... if pnt.Y > N.Y: # test coordinates
... N = pnt # remember coordinates if vertex is more extreme
... if pnt.X > E.X: # test coordinates
... E = pnt # remember coordinates if vertex is more extreme
... if pnt.Y < S.Y: # test coordinates
... S = pnt # remember coordinates if vertex is more extreme
... if pnt.X < W.X: # test coordinates
... W = pnt # remember coordinates if vertex is more extreme
... new_row = [row[0],N.X,N.Y,E.X,E.Y,S.X,S.Y,W.X,W.Y] # build new row data, same order as fields list
... cursor.updateRow(new_row) # write row
