8.2 Geodetic Position Computations: Direct and Inverse Problems on the Reference Ellipsoid

Geodetic Position Computations: Direct and Inverse Problems

In plane surveying, positions are computed using simple 2D Cartesian trigonometry ($X = X_0 + S \sin \theta$). On an oblate reference ellipsoid, however, lines of latitude and longitude converge toward the poles, and the surface curvature varies dynamically with latitude. Position computations across the curved ellipsoidal surface are classified into two fundamental problems: the Direct Geodetic Problem and the Inverse Geodetic Problem.

---

1. Classification of Ellipsoidal Geodetic Problems

The Direct Geodetic Problem (Forward Problem)

• Given: Geodetic coordinates of Station 1 ($\phi_1, \lambda_1$), geodetic distance along the geodesic ($s$), and forward geodetic azimuth ($\alpha_{12}$) from Station 1 to Station 2.
• Compute: Geodetic coordinates of Station 2 ($\phi_2, \lambda_2$) and the reverse (back) geodetic azimuth ($\alpha_{21}$).
• Applications: Extending control survey traverses, projecting forward station coordinates, establishing new triangulation hubs.

The Inverse Geodetic Problem (Reverse Problem)

• Given: Geodetic coordinates of two known stations ($\phi_1, \lambda_1$) and ($\phi_2, \lambda_2$).
• Compute: Ellipsoidal geodetic distance ($s$), forward geodetic azimuth ($\alpha_{12}$), and reverse geodetic azimuth ($\alpha_{21}$).
• Applications: Checking baseline lengths between network stations, boundary computations, determining true spatial vectors between GNSS monuments.

---

2. Geodesic Lines on the Reference Ellipsoid

A geodesic is defined as the curve of shortest distance connecting two points on a surface. On a sphere, a geodesic is a segment of a great circle. On an oblate ellipsoid of revolution, a geodesic is a complex double-curved line.

Clairaut's Law for Geodesics

At any point along a geodesic curve on an ellipsoid of revolution, the product of the parallel radius of curvature $r$ and the sine of the geodetic azimuth $\alpha$ is constant:

$r \sin \alpha = C = \text{constant}$

Since the radius of the parallel circle is $r = N \cos \phi$, Clairaut's relation is written as:

$N \cos \phi \sin \alpha = C$

Where:

• $N$ = Radius of curvature in the prime vertical at latitude $\phi$.
• $\phi$ = Geodetic latitude.
• $\alpha$ = Forward geodetic azimuth of the geodesic.
• $C$ = Characteristic constant of the specific geodesic line.

[!IMPORTANT] A key consequence of Clairaut's Law is that as a geodesic approaches higher latitudes (increasing $\phi$), $\cos \phi$ decreases, causing $\sin \alpha$ to increase. At maximum latitude $\phi_{\max}$, the geodesic turns back toward the equator where $\alpha = 90^\circ$ or $270^\circ$ (so $\sin \alpha = 1$).

---

3. Computational Methods: Puissant's Formulae

For line lengths up to approximately $100\text{ km}$, Puissant's method provides adequate accuracy for primary triangulation networks. It relies on Taylor series expansions expressed in terms of Puissant coefficients derived from the ellipsoid radii of curvature $M$ and $N$.

Puissant's Direct Problem Formulas

$\Delta \phi'' = \frac{s \cos \alpha_{12}}{M_m \arcsin 1''} - \frac{s^2 \sin^2 \alpha_{12} \tan \phi_1}{2 M_m N_1 \arcsin 1''} + \dots$ $\Delta \lambda'' = \frac{s \sin \alpha_{12}}{N_2 \cos \phi_2 \arcsin 1''} + \dots$ $\Delta \alpha'' = \Delta \lambda'' \sin \left( \frac{\phi_1 + \phi_2}{2} \right) \sec \left( \frac{\Delta \phi}{2} \right) + \dots$

Where:

• $M_m$ is the meridian radius of curvature evaluated at mean latitude $\phi_m = \frac{\phi_1 + \phi_2}{2}$.
• $N_1, N_2$ are prime vertical radii of curvature at stations 1 and 2.
• $\Delta \alpha = \alpha_{21} - (\alpha_{12} \pm 180^\circ)$ is the meridian convergence.

Convergence of Meridians

Meridians converge toward the poles. Because of this convergence, the forward azimuth $\alpha_{12}$ from Station 1 to Station 2 and the reverse azimuth $\alpha_{21}$ from Station 2 to Station 1 do not differ by exactly $180^\circ$. The exact relationship is:

$\alpha_{21} = \alpha_{12} \pm 180^\circ + \Delta \alpha$ $\Delta \alpha \approx \Delta \lambda \sin \phi_m$

---

4. Modern High-Precision Methods: Vincenty's Formulae

While Puissant's series truncated terms degrade in precision over long lines ($> 100\text{ km}$), Vincenty's formulae (1975) use an iterative technique on an auxiliary sphere to achieve millimeter-level accuracy over global distances. Vincenty's method evaluates ellipsoidal geodesics regardless of line length.

---

5. Worked Examples

Worked Example 8.2.1: Convergence of Meridians Calculation

Problem: Two stations in Northern Luzon have geodetic longitudes $\lambda_1 = 120^\circ 15' 00.00''\text{ E}$ and $\lambda_2 = 121^\circ 05' 00.00''\text{ E}$. The mean latitude of the line is $\phi_m = 17^\circ 30' 00.00''\text{ N}$. Calculate the approximate meridian convergence $\Delta \alpha$ in arc-seconds.

Solution:

1. Compute the longitude difference $\Delta \lambda$: $\Delta \lambda = 121^\circ 05' 00'' - 120^\circ 15' 00'' = 0^\circ 50' 00'' = 3000''$
2. Apply the meridian convergence approximation formula: $\Delta \alpha \approx \Delta \lambda \sin \phi_m$
3. Evaluate $\sin(17^\circ 30')$: $\sin(17.5^\circ) = 0.3007058$
4. Calculate $\Delta \alpha$: $\Delta \alpha = 3000'' \times 0.3007058 = 902.1174''$
5. Convert to degrees, minutes, seconds: $902.1174'' / 60 = 15.03529' = 15' 02.12''$ $\Delta \alpha \approx +15' 02.12''$

Worked Example 8.2.2: Reverse Geodetic Azimuth

Problem: Using the line from Example 8.2.1, if the forward geodetic azimuth $\alpha_{12} = 65^\circ 20' 10.00''$, compute the reverse geodetic azimuth $\alpha_{21}$.

Solution:

1. Use the reverse azimuth equation: $\alpha_{21} = \alpha_{12} + 180^\circ + \Delta \alpha$
2. Substitute the values: $\alpha_{21} = 65^\circ 20' 10.00'' + 180^\circ + 15' 02.12''$ $\alpha_{21} = 245^\circ 20' 10.00'' + 15' 02.12'' = 245^\circ 35' 12.12''$
3. The reverse geodetic azimuth $\alpha_{21}$ is $245^\circ 35' 12.12''$.