1. Adjustment of Triangulation

2. Introduction Triangulation was the preferred method for horizontal control surveys until the EDM was developed Angles could be measured to a high level of accuracy Measured baseline distances were included every so often to strengthen the network

4. Azimuth Observation Equation

5. Arctangent Function for Azimuth xiyixjyjxj-xiyj-yiatan2(dy,dx)atan(dx/dy) 000.5000.8660.5000.86630 000.8660.5000.8660.50060 001.0000.0001.0000.00090#DIV/0! 000.866-0.5000.866-0.500120-60 000.500-0.8660.500-0.866150-30 000.0000.0001800 00-0.500-0.866-0.500-0.866-15030 00-0.866-0.500-0.866-0.500-12060 000.0000.000-90#DIV/0! 00-0.8660.500-0.8660.500-60 00-0.5000.866-0.5000.866-30 000.0001.0000.0001.00000

6. Azimuth Examples

7. Correction Term Even if we use a full-circle arc tangent function we may still need a correction term This can happen where the azimuth is near ±180° Check the K-matrix term (measured minus computed) If it is closer to ±360° than it is to 0°, correction is needed

8. Linearizing the Azimuth Equation

9. Other Partials

10. Linearized Azimuth Observation Equation

11. Angle Observation Equation

13. Linearized Form

14. Example 14.1

15. First – Initial Approximations

16. Approximations - Continued

18. Determine Computed Values for Angles and Distances

19. Computed Values - Continued

20. Set Up Matrices First, we need to define the Backsight, Instrument, and Foresight stations for the observed angles. angleBIF θ 1 URS θ 2 RSU θ 3 UST θ 4 STU

21. J Matrix Note: Rho (ρ) is the conversion factor from radians to seconds. This complication can be avoided by keeping all angles in radian units (for example, in the K matrix).

22. K Matrix If this was in radians, we wouldn’t need Rho. Also, the second value should be zero. (why?)

23. Compute Solution and Update Coords Note: Further iterations produce negligible corrections.

24. Compute Statistics Residuals: V = J X - K S0S0

25. Coordinate Standard Errors

26. Other Angle Networks Resection – more than 3 points is redundant Triangulated quadrilaterals Other geometric shapes

27. Resection

28. Triangulated Quadrilateral