Spherical geodesy
The previous chapter introducedellipsoidal geodesy functions.
The package has also a set of spherical geodesy tools, also ported from thelibrary developed in JavaScript by Chris Veness (seeMovable Type Scriptsweb site and sourcecodeon GitHub).
These geodesy functions are based on calculations on a spherical earth model.Distance, bearing, destination and other functions are provided both for greatcircle paths and rhumb lines. All calculations use simple sphericaltrigonometric algorithms.
Actually the earth is slightly ellipsoidal, not spherical. However errors aretypically up to 0.3% (see notes byMovable Type Scripts)when using a spherical model instead of an ellipsoidal.
🔢 Ellipsoidal vs spherical geodesy
You can try both and choose!
Ellipsoidal geodesy functions are moreaccurate, but spherical geodesy functions are a bit faster as theimplementation is much simpler.
// Ellipsoidal and spherical geodesy functions to calculate distances.final greenwich=Geographic.parseDms(lat:'51°28′40″ N', lon:'0°00′05″ W');final sydney=Geographic.parseDms(lat:'33.8688° S', lon:'151.2093° E');// How to calculate distances using ellipsoidal Vincenty, spherical// great-circle and spherical rhumb line methods is shown below.// The distance along a geodesic on the ellipsoid surface (16983.3 km).greenwich.vincenty().distanceTo(sydney);// By default the WGS84 reference ellipsoid is used but this can be changed.greenwich.vincenty(ellipsoid:Ellipsoid.GRS80).distanceTo(sydney);// The distance along a spherical great-circle path (16987.9 km).greenwich.spherical.distanceTo(sydney);// The distance along a spherical rhumb line path (17669.8 km).greenwich.rhumb.distanceTo(sydney);🧐 Great circle vs rhumb line
As already mentioned forspherical geodesy the package provides two methods,calculations alonggreat circle paths andrhumb lines.
According to Wikipedia, agreat circle ororthodrome is the circular intersection of a sphere and a plane passingthrough the sphere’s center point. Arhumb line orloxodrome is an arc crossing all meridians of longitude at the same angle,that is, a path with constant bearing as measured relative to true north.
Differences between a rhumb line (blue) compared to a great-circle arc (red) asdescribed by Wikipedia arevisualized in the illustration (top: orthographic projection, bottom: Mercatorprojection) showing paths from Lisbon, Portugal to Havana, Cuba.
The rhumb line path is slightly longer than the path along the great circle.Rhumb lines are sometimes used in marine navigation as it’s easier to follow aconstant compass bearing than adjusting bearings when following a great circlepath.
🌐 Great circle paths
Examples usinggreat circle paths (orthodromic) on a spherical earth model:
// sample geographic positionsfinal greenwich=Geographic.parseDms(lat:'51°28′40″ N', lon:'0°00′05″ W');final sydney=Geographic.parseDms(lat:'33.8688° S', lon:'151.2093° E');// decimal degrees (DD) and degrees-minutes (DM) formatsconst dd=Dms(decimals:2);const dm=Dms.narrowSpace(type:DmsType.degMin, decimals:2);// prints: 16987.9 kmfinal distanceKm= greenwich.spherical.distanceTo(sydney)/1000.0;print('${distanceKm.toStringAsFixed(1)} km');// prints (bearing varies along the great circle path): 60.94° -> 139.03°final initialBearing= greenwich.spherical.initialBearingTo(sydney);final finalBearing= greenwich.spherical.finalBearingTo(sydney);print('${dd.bearing(initialBearing)} ->${dd.bearing(finalBearing)}');// prints: 51° 31.28′ N, 0° 07.50′ Efinal destPoint=greenwich.spherical.destinationPoint(distance:10000, bearing:61.0);print(destPoint.latLonDms(format: dm));// prints: 28° 33.97′ N, 104° 41.62′ Efinal midPoint= greenwich.spherical.midPointTo(sydney);print(midPoint.latLonDms(format: dm));// intermediate points along the great circle between Greenwich and Sydney// prints 10 intermediate geographic positions:// 0.0: 51° 28.67′ N, 0° 00.08′ W// 0.1: 56° 33.44′ N, 24° 42.13′ E// 0.2: 55° 50.76′ N, 52° 19.42′ E// 0.3: 49° 39.17′ N, 75° 34.08′ E// 0.4: 40° 00.39′ N, 92° 22.91′ E// 0.5: 28° 33.97′ N, 104° 41.62′ E// 0.6: 16° 14.46′ N, 114° 29.30′ E// 0.7: 3° 31.26′ N, 123° 05.85′ E// 0.8: 9° 16.56′ S, 131° 28.24′ E// 0.9: 21° 51.83′ S, 140° 28.86′ E// 1.0: 33° 52.13′ S, 151° 12.56′ Efor (var fr=0.0; fr<1.0; fr+=0.1) {final ip= greenwich.spherical.intermediatePointTo(sydney, fraction: fr);print('${fr.toStringAsFixed(1)}:${ip.latLonDms(format:dm)}');}// prints: 0° 00.00′ N, 125° 18.98′ Efinal intersection= greenwich.spherical.intersectionWith(bearing:61.0,other:constGeographic(lat:0.0, lon:179.0),otherBearing:270.0,);if (intersection!=null) {print(intersection.latLonDms(format: dm));}🗺️ Rhumb line paths
Examples usingrhumb line paths (loxodromic) on a spherical earth model:
// prints: 17669.8 kmfinal distanceKm= greenwich.rhumb.distanceTo(sydney)/1000.0;print('${distanceKm.toStringAsFixed(1)} km');// prints (bearing remains the same along the rhumb line): 122.49° -> 122.49°final initialBearing= greenwich.rhumb.initialBearingTo(sydney);final finalBearing= greenwich.rhumb.finalBearingTo(sydney);print('${dd.bearing(initialBearing)} ->${dd.bearing(finalBearing)}');// prints: 51° 25.80′ N, 0° 07.26′ Efinal destPoint=greenwich.spherical.destinationPoint(distance:10000, bearing:122.0);print(destPoint.latLonDms(format: dm));// prints: 8° 48.27′ N, 80° 43.98′ Efinal midPoint= greenwich.rhumb.midPointTo(sydney);print(midPoint.latLonDms(format: dm));More examples are provided in the API documentation andtest cases.