Lecture 14 – More damned
mathematics
GISC-3325
5 March 2008
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/12215Fall-2006/CourseHome/index.htm
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/12215Fall-2006/B9777836-A797-4FC8B68C-84636829A29C/0/12_215_lec08.pdf
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
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
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.
LGH to XYZ
XYZ to LGH
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
What is the transformation matrix?
• What are we trying to do?
New
XYZ
Scale
local geodetic
coordinates
Rotation Matrix
We multiply the rotation matrix by local
geodetic horizon coordinate vector first then
add geocentric coordinates of starting point
Geocentric
coordinates of
starting point
Solve for ENU
Given:
forward geod. azimuth: 310d 44m
51s
zenith angle: 89d 47m 57s
slant range: 283.505 m
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
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
Another way …