Introduction
The direct geodetic problem is:
Given the geodetic latitude and longitude of a station, the geodetic azimuth of a line to the second station, the ellipsoid distance to a second point, and two defining parameters of the ellipse,
Find the geodetic latitude and longitude of the second point and the geodetic azimuth going from the second station to the first station.
This problem is shown in Figure 1. As the book indicates on page 220, there have been several methods developed to solve this problem. The first was to express the equations
f2 = f(S)
l2 = l(S)
a2 = a(S)
For a differential right triangle where S is on the geodesic of an ellipsoid, the following equations hold
Defining the terms c and V as
Then M and N can be written as
The differentiate equation (1), we need the derivative of V with respect to latitude and S.
where
With the above expressions utilized, the following working equations can be developed.
And that's the rest of the story! These equations are accurate to only 100 km.
A second method involves using numerical integration techniques with the following formulas.
Numerical integration techniques are discussed in SUR 345, Numerical Methods in Adjustment Computations.
A third approach has been to approximate the ellipsoid with a sphere
where computations are rigorously performed and then transformed
back onto the ellipsoid. The first set of equations, we were originally
derived by Puissant, and used by the U.S. Coast and Geodetic Survey.
The second set of equations were developed by Bowring. These
equations may be safely used for baselines up to 100 km in length.
To derive the necessary equations, consider a sphere of radius N1 tangent along the parallel through the first point. For a "short" distance the sphere will be approximately coincident with the second point. This derivation assumes that the azimuth and distance are the same on the sphere and the ellipsoid, and is depicted in Figure 2. The co-latitudes of points 1 and 2 are arcs on a sphere of radius N1 tangent at point 1. With these approximations the direct problem solved with the following set of equations.
Note that the function for the azimuth (in seconds) produces the value for the convergence of the meridians. That is, this is the amount that the meridians of point 1 and 2 are converging. Also note that in the equation for the change in latitude, the unknown Df is also in the third term on the right-side of the equation. This value can initially be solved by using only the first, second and fourth terms. Using this initial value Df, the equation can be computed.
In 1981 Bowring derived a set of equations for lines up to 150 km in length. This derivation was discussed in ACSM's Surveying and Mapping Journal. The method uses a conformal projection of the ellipsoid on a sphere called the Gaussian projection of the second kind. The procedure for the direct and inverse solution is non-iterative.
Direct Problem Equations
A final method we will look at is to convert the geodesic distance, S, to a chord distance, and then compute the second station's coordinates in a local coordinate system. The conversion of a geodesic distance to a chord distance is given by
The conversion going from the chord distance to the geodesic distance is given by
The accuracy of the formulas depends primarily on the length of the line. For a line of length of 200 km, the equations will have an error under 9 mm.
The coordinate differences would be
It can now be shown that
where V is the dip angle of the chord. Once the changes in the coordinates are known, the second stations coordinates can be computed as
If the computations are correct, then the resultant coordinates (x2, y2, z2) should satisfy the equation for the ellipse, or:
If V is incorrect the right hand side will equal something other than 0, say h. h is a correction to V that can be computed as:
After iterating V to reach a final solution, the coordinates can be inversed to determine the geodetic position of the second station with the following equations.
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