2-D versus 3-D: Sine of Squared Angular Error and Weighted Perpendicular Fixing

摘要

If (1) two dimensional algorithms work on variables from a gnomonic plane, and (2) gnomonic planes are based on the same point of tangency both within a single algorithm and between the two and three dimensional algorithms then it is reasonable to ask the question: How much difference does it make if the gnomonic projections are taken before of after fixing. This question will be put into mathematical form and analyzed in this memo. The procedure used will be asymptotic expansion, on the assumption that error would go to zero as the radius of the Earth became infinite if the same gnomonic projected coordinates were involved. In this way the dependence of the difference of projections on the scale restriction is a significant known parameter, best accounted for in this way. For analysis purposes it is also relevant to identify how error models cross to the gnomonic plane. They also may or may not correspond well with two dimensional error models. The two fix techniques being analyzed are the sine of squared angular error and its approximation, the weighted perpendicular. The weighted perpendicular technique is used by the Improved Guardrail V (IGRV) program, and differs from the sine of squared angular error technique in interesting ways that will be discussed later.