Navigation

WGS-84 has become the de facto standard for satellite navigation since its adoption by the Navstar Global Positioning System (GPS) and the USA’s permission for civilian use of GPS in 1983.

U.S. President Ronald Reagan permitted civilian use of GPS after the airliner KAL 007 was shot down by Soviet interceptor aircraft; the aircraft had strayed into prohibited airspace due to a navigation error.

As a result of these developments, modern aircraft navigation standards now require aircraft to follow geodesic segments on the WGS-84 ellipsoid, see RTCA DO-283C / Eurocae ED-323.

What is a geodesic segment?

A geodesic segment is a curve representing the shortest path between two points over the surface of an ellipsoid.

Current satellite navigation systems are capable of extraordinary accuracy. For example, GPS receiver positions are accurate to within 1.82 metres 95% of the time, see GPS Accuracy.

Unfortunately, the high accuracy of satellite navigation positions is often lost in navigation calculations due to the demanding nature of accurately calculating geodesic segments on the surface of the WGS-84 ellipsoid.

Map projection

Traditionally, navigators plotted courses as straight lines on flat paper charts, i.e., on map projections of the Earth’s surface.

For example, the Mercator projection was originally developed to aid navigation with a magnetic compass, as a straight line on a Mercator chart represents a path of constant bearing, known as a rhumb line.

Projection distortion

Map projections transform 3D latitude-longitude coordinates into 2D x-y coordinates. Unfortunately, all map projections distort the Earth’s surface, since it is impossible to represent a curved surface on a flat surface without distortion.

Different map projections suffer from different distortions, see Tissot’s indicatrix. This distortion causes inaccuracies in navigation calculations, such as calculating the distance between two points.

Situation display projection

Many Air Traffic Control (ATC) situation displays use a stereographic projection, see: which map projection is typically used on ATC radar screens?.

Stereographic projection was used in the original US National Airspace System (NAS), the Eurocontrol ARTAS system, and the Thales TopSky system.

Calculation accuracy

Some form of map projection is necessary to display global position data on a planar display screen.
However, some systems also use a map projection to perform navigation calculations,
causing projection distortion to reduce the accuracy of their calculations.

Spherical geometry

The Earth is almost a perfect sphere (the flattening of the WGS-84 ellipsoid is just over 1/3%), so modelling the Earth as a sphere is good enough for many purposes.

Navigators have modelled the Earth as a sphere for centuries. For example, the shortest (great circle) distance between two points can be calculated using the spherical trigonometry Haversine formula.

A sphere can also be modelled by 3D vectors:

• a point on the surface of the sphere can be represented by a vector from the centre,
• and a pole of a great circle can be represented by a vector from the centre, perpendicular to the plane of the great circle.

Ironically, some navigation solutions are simpler using spherical 3D vectors than planar 2D vectors. For instance, the intersection of two great circles can simply be calculated from the cross product of their pole vectors.

Modern computers can process vectors in parallel (see SIMD). Spherical vector geometry can harness linear-algebra libraries to solve navigation calculations faster than spherical trigonometry.

Geodesic geometry

Unlike straight lines and great circles, geodesics are non-linear, which makes tasks such as calculating distances much more complicated.

The original geodesic formulas were developed by Andoyer in the 1930s. They are relatively quick to calculate and were used in the LORAN radio navigation system.

Thaddeus Vincenty published his formulas in 1975. His goal was to minimize program length on a 1970s desk calculator. His formulas are more accurate than Andoyer’s, but the iterative techniques often require many iterations and do not always converge to a solution. They are the recommended formulas in the current air navigation standards.

Charles F F Karney reformulated Vincenty’s formulas in 2013 to produce formulas that are more accurate, require fewer iterations, and always converge to a solution. Karney’s formulas are widely recognised as the most accurate geodesic formulas available today.

Geodesic intersections

Karney’s 2023 paper Geodesic Intersections is a complete solution to the intersections of two geodesic paths on the surface of an ellipsoid.

For example, it considers the issue of coincident geodesic paths, which had been overlooked by previous algorithms, such as Baselga & Martinez-Llario’s 2017 paper: Intersection and point-to-line solutions for geodesics on the ellipsoid.

Geodesic algorithms

Our geodesic algorithms combine Karney’s Geodesic Intersections algorithms with spherical vector geometry to calculate:

• distance along a geodesic segment: Along Track Distance (ATD);
• distance across a geodesic segment: Cross Track Distance (XTD);
• and distances along a pair of geodesic segments to their closest intersection point.

Note: the algorithms only provide solutions for geodesic segments on the WGS-84 ellipsoid, which are the shortest paths between points on the surface of the ellipsoid.

They do NOT calculate solutions along complete geodesic paths or on ellipsoids with a larger flattening than the WGS-84 ellipsoid.