Lab Description
CIVE 4381 – Advanced Geomatics and Geosensing Lab #1 – Coordinate Computations
The purpose of this lab is to introduce the methods for 3D coordinate conversions in an ellipsoidal reference frame.
Problem: Cartesian to Geodetic and Geodetic to Cartesian Coordinate Conversions
The transformation between Cartesian and curvilinear coordinates plays an important role in geodesy because terrestrial measurements are usually expressed in curvilinear geodetic coordinates (f, l, h) while satellite observations are usually referred to a Cartesian (x,y,z) coordinate system.
Cartesian to Curvilinear Conversion
The conversion from Curvilinear to Cartesian coordinates is given by the closed form solution:
(1) where:
æ X öCT æ (N + h)cosf cos l ö çY÷ =ç(N+h)cosfsinl÷
ç Z ÷ ç[(1- e2 )N + h]sinf ÷ èøPè ø
a 1-e2 sin2 f
a2 -b2
(2) e2 = (3)
N=
a2 Curvilinear to Cartesian Conversion
There is no linear relationship to transform Cartersian to curvilinear coordinates. Therefore, in order to express curvilinear coordinates in terms of Cartesian coordinates, an iterative solution has to be applied.
From Equation (1), l can be obtained directly by:
(6) p= x2 +y2 =(N+h)cosfÞh= p -N pp cosf
The last equation in (1), can be divided by p, and reordered to solved for tanf, and f.
(5)
However, f and h have to be iterated. From the first two equations in (1) we obtain:
tanl = xyp p
(7)
tanφ = zp + Ne2 sinφ p
−1)zp # Ne2 sinφ&, ⇒ φ = tan + p %1+ z (.
*$ p ‘- The procedure to determine f, l, h from x, y, z is therefore:
a) Compute l from (5).
b) Determine Initial Estimate, f0. Refer to the class notes.
c) Iteration
Using f0, compute N(f0) from (3)
Use N(f0) and f0 in (6) to get f1
Repeat this step until converge is achieved.*
d) Compute h from (6).
*In general, the convergence criterion should be set for N, and should correspond to the resolution of the initial x, y, z coordinates.
Assignment
(1) Write a program (in Matlab, C++, Visual Basic) or develop a spreadsheet (Excel) application that will perform both the curvilinear to Cartesian, and Cartesian to curvilinear coordinate conversions. The program or spreadsheet will also allow the user to select a reference ellipsoid for the conversions. Two reference ellipsoids will be available, GRS80, and Clarke 1866. Their parameter values are given in Table 1.
Table 1: Reference Ellipsoid Definitions
(2) Using the application you developed in (1), convert the following coordinates to their curvilinear or
Cartesian equivalent, on the specified reference ellipsoid.
(a) f = 38-53-42.38757(N) l = 077-02-11.57375(W) h = -23.610, GRS80
(b) f = 39 13 26.71218(N) l = 098 32 31.74604(W) h = -573.979, Clarke 1866 (c) X = -742529.80 m Y = -5462904.79 m Z = 3196804.62 m, GRS80
Ellipse
Semi-Major Axis (a) in meters
1/flattening
Clarke 1866
6378206.4
294.9786982
GRS 80
6378137.0
298.257222101
(d) X = -2304648.273 m Y = -3638659.7069 m Z = 4688686.3639, Clarke 1866
Submit: Hard copy of the program or spreadsheet application developed in (1), along with converted coordinate solutions for all four points in (2).
Due Date: February 13th, 2020 at 11:30 am (hand in hard copy at start of class)