Shapefiles can be projected to a polar stereographic projection with true scale at 71°S using ArcGIS. The steps are outlined below.
1 Start ArcToolbox, go to Data Management Tools then Projections.
2 If the shapefiles do not have a defined projection (i.e. a .prj file) then it is necessary to use the Define Projection Wizard (shapefiles, geodatabase) to create such a file. If the shapefiles have a defined projection, go to step 3.
(a) Open the Define Projection Wizard (shapefiles, geodatabase).
(b) Follow the steps in the wizard which involve:
(i) navigating to and selecting a shapefile or shapefiles (the projection of more than one shapefile can be defined at once if they have the same projection);
(ii) defining the coordinate system (a predefined coordinate system can be selected or modified, a coordinate system can be imported from another geodataset such as a shapefile or a new coordinate system can be defined);
(iii) accepting a summary of the coordinate system defined.
3 (a) Open the Project Wizard (shapefiles, geodatabase).
(b) Follow the steps in the wizard which involve:
(i) navigating to and selecting a shapefile or shapefiles (more than one shapefile can be projected at once), the path and coordinate system(s) of the shapefiles will then be displayed;
(ii) nominating the name of the new shapefile to be created and where it is to be saved;
(iii) defining the coordinate system:
Select Coordinate System...
Select...
Projected Coordinate Systems
Polar
South Pole Stereographic.prj and then Add
Modify...
Change the Scale_Factor parameter from 1 to 0.972769012892 (this is the appropriate scale factor for the WGS84 ellipsoid; it ensures that true scale will be at 71°S rather than 90°S)
The Central_Meridian parameter can be changed if necessary (the data will be oriented so that this meridian is 'up the page')
OK
(iii) accepting a summary of the coordinate system defined and the extents of the new shapefile.
In ArcMap, right click in your map window, either in the data view or layout view, and select Properties.
Select the Coordinate System tab.
Under Select a coordinate system, expand the selection under Predefined, Projected Coordinate Systems, Continental, Africa, and select Africa Lambert Conformal Conic.
On the right hand side of this window, select Modify...
A new window will appear.
Next to Name, change the name to something that describes your area, e.g. Heard_Lambert_Conformal_Conic.
Change 3 values under the column Value.
Central_Meridian - Change to a longitude value that runs down the centre of your map.
Standard_Parallel_1 - Change to a latitude value that runs across your map about 1/3 down your map.
Standard_Parallel_2 - Change to a latitude value that runs across your map about 2/3 down your map
Select OK.
Back in the Data Frame Properties window, select the button Add to Favourites. In future then the Lambert Conformal Conic projection can be selected from Favorites.