Which is the most accurate method of measuring distance?

I aim to analyse the accuracy of the various methods of geodesic calculations (distance between two points on Earth)based on the three assumed models of Earth - flat, spherical & ellipsoidal. I need a reliable index to compare my results to measure for accuracy. Will using the WGS84 geoid with Vincenty's formula yield the most accurate results?

Let me focus just on the question of the most accurate way to determine distances on the WGS84 ellipsoid.

The accuracy of Vincenty's method is about 0.1 mm providing it converges. It fails to converge for nearly antipodal points.

My library GeographicLib is accurate to about 15 nanometers and converges everywhere; the algorithm is published in my paper Algorithms for geodesics. I've also published a Test set for geodesics which gives the distances between selected points on the WGS84 ellipsoid accurate to about 0.1 picometers. For more background information see the wikipedia page Geodesics on an ellipsoid.

For the sphere you can use the SLC (Spherical Law of Cosines). The HF (Haversine Formula) will work for this as well. You can use both of these on the ellipsoid and errors are typically very low (< 0.3%).

If you desire an oblate spheroid then yes use VF (Vincenty's Formula).

For flat I am not sure it matters, do you mean a plane? If so then just use planar distance.

The Geoid varies greatly so in some locations SLC or HF will be best but all other things being equal VF will be best. Basically though, no best exists.

I need a reliable index to compare my results to measure for accuracy.

I doubt one exists. You could I suppose use a calculator that accounts for the WGS 84 geoid for error measurement (http://seismo.cqu.edu.au/CQSRG/VDistance/) but this itself is bases on VF but may work if you are just using spheroids, ellipsoids, and a plane.

This answer may somewhat help deciding between SLC and HF. Why is law of cosines more preferable than haversine when calculating distance between two latitude-longitude points?

Regarding the case of the "flat Earth". The answer depends on three key things:

• the map projection used
• whether or not you apply a scale factor correction to the calculated Euclidean distance
• the separation and orientation of the two points.

Generally stated,

• all map projections -- transforms from geographic to planar coordinates -- introduce linear distortions (except in certain directions)
• the distortion increases with distance
• each projection is different.

Some projections have readily-obtainable corrections (scale factors) that, when applied to short distance calculations, yield results that approximate the spheroidal geodesic calculations.

See these related questions: