Monday, April 21, 2014
Comparison of Spherical Triangulation with Plane Triangulation
We can make a comparison of the results for triangulation on a sphere with triangulation in a plane. To do so we need to convert from angular distances to ordinary distances and take into account the change in the length of a degree of longitude with changes in latitude. The angles and distances are easier to convert and compare if we assume a unit sphere.
The unknown position X' for the plane triangulation is shifted about 1.1 minutes of arc from the previous spherical triangulation position X. The arc lengths, α1 and α2, and the distances from the known points, ΔsM and ΔsR, are similar in value but also differ slightly too.
So we see that due to the map distortions plane triangulation does not work as well at a distance of about 30 kilometers.
Edit (Apr 21): corrected the error on deg of longitude which used sine instead of cosine and got a better agreement of the results. Also made minor changes to the text.