1. Geodetic Control Network Lecture 4. Computations on the ellipsoid

2. Outline The differential equation of the geodesic Reduction of observations to the ellipsoid Projections according to: Pizetti’s method Helmert’s method Reduction to the ellipsoid (j 1, j 2, j 3 )

3. The differential equations of the geodesic In some cases the position, orientation, size or the parameters of the reference ellipsoid must be changed. What happens with the coordinates? Let’s suppose that: P 1 (  1, 1 ); P 2 (  2, 2 ),  12 ;  21, s a, f ellipsoidal parameters

4. The differential equations of the geodesic When the aforementioned parameters change with an infinitesimal amount, then:

5. The differential equations of the geodesic   s a f

6. The differential equations of the geodesic

7. Projections to the ellipsoid – Pizetti vs Helmert Helmert Pizetti

8. Reduction to the ellipsoid The ellipsoidal normals have a skewness, therefore when the target is projected along the ellipsoidal normal, then the projected point is not on the normal section anymore.

9. Reduction to the ellipsoid – j 1 Instead of the normal section the geodesic should be used -> correction to the geodesic: Combined:

10. Reduction to the ellipsoid – j 2 The standing axis is aligned with the local vertical instead of the ellipsoidal normal:

11. Reduction to the ellipsoid – j 3

12. Projections to the ellipsoid – Pizetti vs Helmert Helmert = j 1 +j 2 Pizetti = j 1 +j 2 +j 3

13. Thank You for Your Attention!