To create a MATLAB code for finding the transformation parameters and their corresponding standard deviations based on the Veis transformation model as depicted in the provided image, we will follow these steps:

Steps:

1. Define the Veis Transformation Model based on the information given in the image.

   • The rotation matrix $R_V$ and translation vector $T$ will be utilized to find the transformation parameters.
   • From the equations, we will use least squares adjustment to compute the unknown parameters $t_x, t_y, t_z, \Delta S$, etc.

2. Matrix Form of the Model:

   • Eq. (41) provides a system of linear equations that can be represented in the form: $V = A X - L$
   • Use the least-squares method to solve this system for the unknowns.

3. Covariance Matrix and Standard Deviations:

   • Once the least squares solution is obtained, compute the covariance matrix to find the standard deviations of the transformation parameters.

Here’s the MATLAB code outline based on the Veis transformation model from the image:

matlabCopy code
% MATLAB code to find transformation parameters in the Veis model

% Data points in source datum (Everest 1830)
% [X1, Y1, Z1]: Coordinates of source points in the source datum
% [X2, Y2, Z2]: Coordinates of corresponding points in the target datum
% These can be input as example coordinates or from a dataset

X1 = [ ... ]; % Fill in source X coordinates
Y1 = [ ... ]; % Fill in source Y coordinates
Z1 = [ ... ]; % Fill in source Z coordinates

X2 = [ ... ]; % Fill in target X coordinates
Y2 = [ ... ]; % Fill in target Y coordinates
Z2 = [ ... ]; % Fill in target Z coordinates

% Number of points
n = length(X1);

% Construct matrix A based on Eq. (41)
A = zeros(3*n, 7); % For transformation parameters (tx, ty, tz, eta, epsilon, deltaS, lambda)

for i = 1:n
    A(3*i-2:3*i, :) = [
        1, 0, 0, Y1(i)*sin(phi)*cos(lambda) - Z1(i)*cos(phi)*sin(lambda), Z1(i)*cos(lambda), -Y1(i)*sin(lambda), X1(i);
        0, 1, 0, X1(i)*sin(lambda)*cos(phi) - Z1(i)*cos(lambda), -Z1(i)*sin(phi), X1(i)*cos(lambda), Y1(i);
        0, 0, 1, -X1(i)*sin(lambda) + Y1(i)*cos(lambda), Z1(i)*cos(phi)*cos(lambda), -Y1(i)*sin(phi), Z1(i);
    ];
end

% Construct vector L based on Eq. (41)
L = zeros(3*n, 1);
for i = 1:n
    L(3*i-2:3*i) = [X2(i) - X1(i); Y2(i) - Y1(i); Z2(i) - Z1(i)];
end

% Solve using least squares
X = A \ L; % X contains the transformation parameters [tx, ty, tz, eta, epsilon, deltaS, lambda]

% Compute residuals and covariance matrix
V = A * X - L;
sigma_0 = sqrt((V' * V) / (3*n - 7)); % Standard error of unit weight

% Covariance matrix of the unknown parameters
Q = inv(A' * A);
Cov_X = sigma_0^2 * Q;

% Standard deviations of the parameters
std_dev = sqrt(diag(Cov_X));

% Display the results
fprintf('Transformation parameters:\n');
fprintf('tx = %.6f ± %.6f\n', X(1), std_dev(1));
fprintf('ty = %.6f ± %.6f\n', X(2), std_dev(2));
fprintf('tz = %.6f ± %.6f\n', X(3), std_dev(3));
fprintf('eta = %.6f ± %.6f\n', X(4), std_dev(4));
fprintf('epsilon = %.6f ± %.6f\n', X(5), std_dev(5));
fprintf('deltaS = %.6f ± %.6f\n', X(6), std_dev(6));
fprintf('lambda = %.6f ± %.6f\n', X(7), std_dev(7));

Breakdown of Code:

• Input Data: The coordinates $X1, Y1, Z1$ and $X2, Y2, Z2$