The application of mathematics and computers for map projections development

Mathematics and computers are the indispensable allies of map projection. Indeed it would be beyond the abilities of today's geospatial professionals to produce a map (projection) without a computer. And embedded in the software will be mathematical algorithms. Without maths and computers, teaching map projections is reduced to imagination and hand-waving; but with them, students (and staff) can extend theory into deep practical knowledge giving them tools to develop or improve computer software. To demonstrate the connections we describe a set of equations developed by L. Krueger in 1912 for the transverse Mercator (TM)projection of the ellipsoid. These equations are undergoing a renaissance due entirely to Computer Algebra Systems (CAS)that have extended them so that computed errors in position are less than 5 nanometres withing 3900 km of the central meridian of the projection. These extended series should completely replace current TM formula that are unnecessarily complicated and if used beyond their limited range are wildly inaccurate. We show how these series may be derived using Maxima (an early CAS) and hopefully demonstrate that the tedium of mathematical development can be side-stepped.