1. Lecture 13 – Geodetic Reference Systems GISC-3325 3 March 2008

2. Update Exam scheduled for 13 March 2008 –Covers lectures, labs, homework, required readings and chapters 5-7. Required reading: –Burkholder 3D Cogo article

3. 3D Coordinate Systems Geodetic (Curvilinear) Coordinates –Latitude, longitude and ellipsoid height –Right-handed, earth-centered earth-fixed, positive east Geocentric (Cartesian) Coordinates –X, Y, Z –Likewise, ECEF, right-handed, –Orthogonal

4. GPS vectors Difference in geocentric coordinates. Both difference in geocentric coordinates and changes in local geodetic horizon coordinates.

5. Local Geodetic Horizon (LGH) ECEF, right-handed, orthogonal, 3-D Origin at any point specified –N in meridian plane oriented twd N pole –U normal to ellipsoid at origin –E perpendicular to meridian plane

6. LGH

7. Geodetic azimuth. Mark-to-mark slant range. Vertical or zenith angle.

8. Geodetic to Geocentric Coordinate Conversions

9. Geocentric to Geodetic

10. We use ellipsoid parameters (a, f -1 ) Calculate preliminary values (set: h = 0) –Lat 1 = atan( (Z / sqrt(x 2 +y 2 ))*(1/(1-e 2 )) –N 1 = a / sqrt(1-e 2 *sin(Lat 1 ) 2 ) –h 1 = (sqrt(x 2 +y 2 )/cos(Lat 1 ))-N 1 We iterate using these starting values We stop iterating when the shift in ellipsoid height is within our accuracy goal.

11. 2D-Coordinate Transformations Given –x = r * cos(γ) –y = r * sin(γ) Rotate coordinate system byΘ –x’ = r * cos(γ – Θ) –y’ = r * sin(γ – Θ) Use trig identities to solve recollect –cos(γ – Θ) = cos γcos Θ+sin Θsin γ –sin(γ – Θ) = sin γ cos Θ – cos γ sin Θ

12. Translation If we shift the origin we can update coordinates by merely adding/subtracting shift from matching coordinate. –x’ = x – tx –y’ = y - ty

13. Scale change We can scale coordinates to account for issues like m to ft. –x’ = s * x –y’ = s * y

14. Four-parameter transformation Combines rotations, translations and scale in one operation. –x’ = s*(x*cos Θ+y*sin Θ) + tx –y’ = s*(-x*sin Θ+y*cos Θ)+ty Matrix form is simpler

15. Three-Dimensional Transformation 7-parameters –scale –rotations along X,Y,Z axes –translations in X,Y,Z

16. Euler matrices and 7-parameter Matrix D for rotation on Z axis Matrix C for rotation of Y axis Matrix B for rotation of X axis

17. Euler matrices

18. Application of 7-para transf