Extremities of the Australian Coastline

From a list of latitude and longitude coordinates of points on the mainland Australian coastline, it is easy to determine the extremities in North, South, East, and West directions. Coastline coordinates at 1:10m scale can be downloaded from Natural Earth in GeoJSON format and extracted using the Python json Library. The 9,463 points obtained, however, have insufficient resolution for exact location of the extremities. Higher resolution coordinates can be downloaded as land polygons from Data Derived from OpenStreetMap in Shapefile format and extracted using the Python PyShp Library. The polygon for Australia has 606,907 points and gives excellent accuracy for the extremities.

Extremities can be found in directions other than the cardinal directions. Each direction is specified in θ compass degrees measured clockwise from North. The extremity in that direction is the point with longitude x and latitude y which maximises xsinθ + ycosθ. For the cardinal and intercardinal directions, the results are:-

Directionθlatitude and longitude of extremitymap
North  0-10.6871474 142.531555410°41'13.73"S 142°31'53.60"ECape York
NorthEast 45-10.7496663 142.610367610°44'58.80"S 142°36'37.32"ECape York
East 90-28.6360097 153.638673128°38'09.63"S 153°38'19.22"ECape Byron
SouthEast135-37.5179563 149.964395037°31'04.64"S 149°57'51.82"ECape Howe
South180-39.1367387 146.373834939°08'12.26"S 146°22'25.81"ESouth Point
SouthWest225-34.3755982 115.133544234°22'32.15"S 115°08'00.76"ECape Leeuwin
West270-26.1522827 113.156063526°09'08.22"S 113°09'21.83"ESteep Point
NorthWest315-21.8741489 113.996313221°52'26.94"S 113°59'46.73"ENorth West Cape

The cardinal continental extremities are also listed by Geoscience Australia but, surprisingly, their locations are hundreds of meters away from the coastline according to Google Maps and OpenStreetMap.

The above directional formula applies to a Cartesian plane, although the latitude and longitude coordinates are of course expressed in a spherical coordinate system. In such a system it’s not obvious how to define what is meant by extremities in non-cardinal directions like North Easterly. There are inconsistencies: equally Northerly and Southerly points lie on small circles (except for the equator) whereas equally Easterly and Westerly points lie on great circles. A possible approach might involve the rhumb lines of the Mercator projection. It’s better, however, to define the extremities of a landmass in a way which is independent of its latitude and longitude and the method of projection. Otherwise, if Australia was moved towards the South Pole, it’s Southernmost point could lie inside the Great Australian Bight.

This is a gnomonic projection centred on an imaginary South pole at latitude -33.36, longitude 129.9, showing a circle of constant latitude close to this imaginary pole. Using sight lines from coastal points achieves the desired independence. This would only be possible in practice if no coastal points were further apart than the distance to the horizon, about 5 kilometres. Otherwise, we can suppose that the line of sight would follow a geodesic, or string stretched over the globe between coastal points.

A string stretched around the Australian coastline would contact the mainland at these and other extremities. The effective path would be a geodesic connecting the contact points. The string would follow the sides of a convex hull enclosing the non-convex coastline. The convex hull can be calculated using the Python scipy.spatial Library. This convex hull algorithm, however, assumes that points are joined by straight lines on a flat Cartesian plane. To get a correct result, the coastline coordinates can be converted using these formulae to a gnomonic projection, in which geodesics on the sphere are projected as straight lines. These are the coordinates of the 61 vertices of the corrected convex hull, with the above extremities shown in red:-

-10.6871474 142.5315554Cape York
-10.6871870 142.5316444Cape York
-10.6873049 142.5318117Cape York
-10.6874926 142.5320769Cape York
-10.6875673 142.5321783Cape York
-10.7256884 142.5826739Cape York
-10.7260315 142.5831252Cape York
-10.7496663 142.6103676Cape York
-22.4460593 150.7536412Reef Point
-25.2762852 152.9081745Urangan Pier
-25.9322646 153.1922704Double Island Point
-25.9328455 153.1924889Double Island Point
-28.2573144 153.5856849Cudgen Creek Headland
-28.6333542 153.6385685Cape Byron
-28.6335778 153.6385953Cape Byron
-28.6360097 153.6386731Cape Byron
-28.6369490 153.6385953Cape Byron
-28.8434741 153.6090633Flat Rock
-31.0564500 153.0645600Hat Head
-31.0569700 153.0644200Hat Head
-31.4762011 152.9379795Tacking Point
-32.4434358 152.5398567Sugarloaf Point
-37.5045900 149.9772920Cape Howe
-37.5179563 149.9643950Cape Howe
-37.6850029 149.6773850Little Rame Head
-37.7257162 149.5978160Sandpatch Wilderness
-39.1306095 146.4256874South East Point
-39.1307323 146.4251509South East Point
-39.1342838 146.3948956Wilsons Promontory
-39.1367387 146.3738349South Point
-35.0607074 116.6297280Near Aldridge Cove
-35.0605998 116.6293447Near Aldridge Cove
-35.0166614 116.4820718Bottleneck Bay
-34.8424630 116.0022580Point D'Entrecasteaux
-34.8423766 116.0020264Point D'Entrecasteaux
-34.3760865 115.1341653Cape Leeuwin
-34.3757056 115.1336617Cape Leeuwin
-34.3755982 115.1335442Cape Leeuwin
-34.2712770 115.0348967Near Elephant Rock Beach
-34.2702619 115.0341739Near Elephant Rock Beach
-34.2541764 115.0243043Knobby Head
-34.2528980 115.0235799Knobby Head
-26.1522827 113.1560635Steep Point
-26.1505485 113.1561101Steep Point
-24.2250942 113.3912229Cape Cuvier
-22.5776932 113.6537093Ningaloo
-22.5595778 113.6598194Ningaloo
-22.5589437 113.6600339Ningaloo
-22.5514606 113.6630888Ningaloo
-21.9737528 113.9335147North West Cape
-21.9731161 113.9338285North West Cape
-21.9540217 113.9458019North West Cape
-21.8745920 113.9959377North West Cape
-21.8741489 113.9963132North West Cape
-21.8513320 114.0241063North West Cape
-21.8506150 114.0250625North West Cape
-21.8156679 114.0797836North West Cape
-21.8148263 114.0811783North West Cape
-13.9265332 126.0095948Hat Point
-11.1293700 131.9751000Vashon Head
-11.1252832 131.9937577Vashon Head
This convex hull is drawn on a map produced with Leaflet. You can pan and zoom the map using the mouse. Leaflet draws polygon edges as straight lines in the WGS 84 Web Mercator projection, so these would not exactly represent the geodesic paths of the string. They were drawn correctly using the Leaflet.Geodesic plugin.

Extremities for some other land masses are shown in gnomonic projection here:-

Christopher B. Jones 2024-04-25