Lecture 14 – More damned mathematics

1. Lecture 14 – More damned mathematics GISC-3325 5 March 2008

2. Update • Scheduled lab changed from web page due to NGS server updates effecting CORS data access. • Exam scheduled next Wednesday 12 March 2008 • http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-215Fall-2006/CourseHome/index.htm

3. Linear algebra Info • Review of Linear Algebra Covering Vectors and Matrices, Solving Linear Equations, Vector Spaces, Eigenvalues and Vectors, Rotation Matrices • http://ocw.mit.edu/NR/rdonlyres/Earth--Atmospheric--and-Planetary-Sciences/12-215Fall-2006/B9777836-A797-4FC8-B68C-84636829A29C/0/12_215_lec08.pdf

4. Review Rotation Problem 2D • Since x = r * cos(γ) and y = r * sin(γ) • Rotation requires application of the trig difference (for CCW rotation) formula • x’ = r( cos γcos Θ + sin Θsin γ ) • x’ = r( cos γcos Θ) + r(sin γ sin Θ) • y’ = r (sin γ cos Θ – cos γ sin Θ) • y’ = r (sin γ cos Θ) – r(cos γ sin Θ) • x’ = x cos Θ y sin Θ • y’ = -x sin Θ y cos Θ

5. Matrix form • [x’; y’] = [ cosΘ sin Θ; -sinΘ cos Θ] ×[ x; y] • where “ ; ” indicates new row, column elements separated by spaces • Matrices can only be multiplied when inner dimensions agree (m=rows n= columns) • if amn = 3×1 and bmn = 3×3 • we cannot multiply a×b but can multiply b×a can be because in a n=1 and in b m =3 in first case

6. Local to geocentric • Done using geocentric coordinates. • [ e; n; u ] <-> [ x; y; z ] • Rotations • align local system with geocentric • e-axis local with x-axis geocentric • u-axis local with z-axis geocentric • Translation • Origin of local system are the geocentric coordinates of the origin.

8. LGH to XYZ

9. XYZ to LGH

10. Problem • Given starting geodetic coordinate: • Lat: 39d 34m 54s • Lon: 078d 53m 51s • h: 100.000 m • We want to compute new point given: • forward geod. azimuth: 310d 44m 51s • zenith angle: 89d 47m 57s • slant range: 283.505 m • NAD 83 coordinates

11. What is the transformation matrix? • What are we trying to do? New XYZ local geodetic coordinates Scale Rotation Matrix Geocentric coordinates of starting point We multiply the rotation matrix by local geodetic horizon coordinate vector first then add geocentric coordinates of starting point

12. Solve for ENU • Given: • forward geod. azimuth: 310d 44m 51s • zenith angle: 89d 47m 57s • slant range: 283.505 m

13. Compute XYZ • As I specified NAD 83, we use the GRS80 reference ellipsoid parameters • a = 6378137 m, 1/f = 298.257222101. Solve for e2 • We were provided the following station coordinates: • Lat: 39d 34m 54s • Lon: 078d 53m 51s • h: 100.000 m

16. Matrix to non-matrix multiplication • We can take the matrix form and convert it to one equation for each parameter. • for example: • x = -sin(lat)∙e + (-sin(lat) ∙cos(lon) ∙n) + (cos(lat) ∙cos(lon) ∙u) • Matrix multiplication is done as follows: • [ g; h ] = [ c d; e f ] ∙ [ a; b ] • g = c∙a + d∙b • h = e∙a + f∙b

17. Another way …