Transverse Mercator with an accuracy of a few nanometers

TL;DR

This paper presents two highly accurate algorithms for the transverse Mercator projection, achieving near machine precision, with one method based on exact equations and the other on an extended series, suitable for high-precision geospatial applications.

Contribution

It introduces a new exact algorithm and an improved series-based method for the transverse Mercator projection with nanometer-level accuracy.

Findings

Exact method achieves 9 nm accuracy over the entire ellipsoid.

Series method has errors less than 5 nm within 3900 km of the central meridian.

Series method is computationally competitive, while the exact method is slower.

Abstract

Implementations of two algorithms for the transverse Mercator projection are described; these achieve accuracies close to machine precision. One is based on the exact equations of Thompson and Lee and the other uses an extension of Krueger's series for the projection to higher order. The exact method provides an accuracy of 9 nm over the entire ellipsoid, while the errors in the series method are less than 5 nm within 3900 km of the central meridian. In each case, the meridian convergence and scale are also computed with similar accuracy. The speed of the series method is competitive with other less accurate algorithms and the exact method is about 5 times slower.