Math Problem Statement

%Program to compute SLD99 values from accepted parameters of SLD99 report. %Coordinate Transformation From WGS84 To SLD99 system; %Entered Latitudes and Longitutes value in degrees, Minutes, Seconds formatin %a text file clc clear %Reading the text file format long [PI,PM,PS,LI,LM,LS,HI]=textread('wgsLP.txt','%f %f %f %f %f %f %f'); NPoints = min(length(PI)); for i=1:NPoints; phi=(PI(i)+(PM(i)/60)+(PS(i)/3600)); lambda=(LI(i)+(LM(i)/60)+(LS(i)/3600)); h=HI(i); % Constants for WGS84 ellipsoid a = 6378137.0; % semi-major axis in meters f = 1/298.257223563; % flattening e2 = (2*f-f^2); % square of first eccentricity b = a * (1 - f); % semi-minor axis

% Transformation parameters from WGS84 to Kandawala
deltaX =   0.2933; % Change in X (m)
deltaY = -766.9499; % Change in Y (m)
deltaZ =-87.713; % Change in Z (m)
delta_a = -860.655; % Change in semi-major axis (m)
delta_f = -0.2836136808*10^(-4); % Change in flattening


% Calculate radii of curvature
RM = a * (1 - e2) / (1 - e2 * sind(phi)^2)^(3/2);
RN = a / sqrt(1 - e2 * sind(phi)^2);

% Apply the transformation formulas
delta_phi = (1 / (RM + h)) * (-deltaX * sind(phi) * cosd(lambda) - deltaY * sind(phi) * sind(lambda) + deltaZ * cosd(phi) + ...
            (RN / a) * e2 * sind(phi) * cosd(phi) * delta_a  + ...
            sind(phi) * cosd(phi) * ((RN * b) / a + (RM * a) / b) * delta_f);
delta_lambda = (1 / ((RN + h) * cosd(phi))) * (-deltaX * sind(lambda) + deltaY * cosd(lambda));
delta_h = deltaX * cosd(phi) * cosd(lambda) + deltaY * cosd(phi) * sind(lambda) + deltaZ * sind(phi) - ...
          (a / RN) * delta_a + (RN * b) * (sind(phi)^2 / a) * delta_f;

% Calculate new coordinates in Kandawala
phi_kandawala = phi + rad2deg(delta_phi); % Convert back to degrees
lambda_kandawala = lambda + rad2deg(delta_lambda); % Convert back to degrees
h_kandawala = h + delta_h;




fprintf('\n %f %f %.3f  ',phi_kandawala,lambda_kandawala,h_kandawala);

end

Solution

The provided MATLAB script performs a coordinate transformation from the WGS84 (World Geodetic System 1984) to the SLD99 (Sri Lanka Datum 1999) system by reading latitude, longitude, and height values from a text file (wgsLP.txt) and applying a series of mathematical operations to convert the coordinates. Let's break down the program and the key components:

Key Sections of the Code:

  1. Reading the Text File: The textread function is used to read latitudes, longitudes, and height data (in degrees, minutes, and seconds format) from a text file (wgsLP.txt). The variables PI, PM, PS, LI, LM, LS, HI represent parts of latitude and longitude.

  2. Conversion from DMS to Decimal Degrees: The latitude (phi) and longitude (lambda) values in Degrees, Minutes, and Seconds (DMS) format are converted into decimal degrees using: ϕ=PI+PM60+PS3600\phi = PI + \frac{PM}{60} + \frac{PS}{3600} λ=LI+LM60+LS3600\lambda = LI + \frac{LM}{60} + \frac{LS}{3600}

  3. Constants for WGS84 Ellipsoid: The WGS84 ellipsoid is defined by:

    • a: Semi-major axis (radius at the equator)
    • f: Flattening (difference between the semi-major and semi-minor axes)
    • e2: First eccentricity squared, derived from the flattening. The semi-minor axis b is calculated as b=a×(1f)b = a \times (1 - f).
  4. Transformation Parameters: These are the parameters for transforming coordinates from WGS84 to the SLD99 datum:

    • deltaX, deltaY, deltaZ: Translation components in meters.
    • delta_a: Change in the semi-major axis.
    • delta_f: Change in the flattening.
  5. Radii of Curvature:

    • RM: Meridian radius of curvature.
    • RN: Prime vertical radius of curvature. These are used in geodetic transformations.
  6. Coordinate Transformation Equations:

    • delta_phi: Change in latitude.
    • delta_lambda: Change in longitude.
    • delta_h: Change in height. These values are calculated using transformation formulas, which involve trigonometric operations and depend on the translation parameters.
  7. Calculate Kandawala Coordinates: After applying the transformation formulas, the new coordinates in the Kandawala system are:

    • phi_kandawala: Transformed latitude.
    • lambda_kandawala: Transformed longitude.
    • h_kandawala: Transformed height.
  8. Output: The transformed coordinates are printed out using fprintf, displaying the latitude and longitude in degrees and the height in meters.

Improvements and Considerations:

  • Error Handling: If the text file contains incomplete or incorrect data, the script might crash. It would be useful to add error checks.
  • Precision: The output may require specific precision handling depending on the coordinate system’s use case.
  • Input File Format: Ensure the wgsLP.txt file has data correctly formatted for this script.

Do you have any questions or would you like further details? Here are 5 relative questions for expansion:

  1. How is the flattening parameter related to the eccentricity of the ellipsoid?
  2. What are the practical applications of converting from WGS84 to SLD99?
  3. How could the script handle larger datasets efficiently?
  4. Can this code be extended to other coordinate transformation systems, and how?
  5. How do translation parameters (deltaX, deltaY, deltaZ) affect the final coordinates?

Tip: Always check the ellipsoid and datum definitions carefully when transforming coordinates between systems, as small errors can lead to significant positional inaccuracies.

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Math Problem Analysis

Mathematical Concepts

Coordinate transformation
Geodesy
Ellipsoid geometry
Trigonometry

Formulas

φ = PI + (PM/60) + (PS/3600)
λ = LI + (LM/60) + (LS/3600)
RM = a * (1 - e²) / (1 - e² * sin²(φ))^(3/2)
RN = a / sqrt(1 - e² * sin²(φ))
Δφ, Δλ, Δh transformation formulas

Theorems

WGS84 ellipsoid definition
Datum transformation formulas

Suitable Grade Level

University/Advanced Geodesy