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, Δs_M and Δs_R, 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.