Bursa-Wolf, Coordinate Frame, and Position Vector Transformations

The fundamental relationship between Bursa-Wolf, Coordinate Frame, and Position Vector transformations relates to how the parameters used for coordinate transformation between geodetic datums are applied and interpreted. These three transformation methods are used during the transformation of geodetic coordinates from one reference system to another and are based on similar mathematical foundations, but have different interpretations and ways of application.

Bursa-Wolf Transformation

The Bursa-Wolf transformation is a seven-parameter model used for coordinate transformation between geodetic datums. These parameters consist of three translations (\(t_X, t_Y, t_Z\)), three rotation angles (\(\epsilon_X, \epsilon_Y, \epsilon_Z\)), and a scale factor (\(k\)). This transformation expresses the relationships between different geodetic systems on the earth.

Coordinate Frame Transformation

The Coordinate Frame transformation provides the transformation of coordinates from one reference system to another. This transformation directly uses the rotation angles (\(\epsilon_X, \epsilon_Y, \epsilon_Z\)) and generally expresses the transformation of the coordinate frame.

Transformation Equation:

\[ \begin{pmatrix} X_{\text{new}} \\ Y_{\text{new}} \\ Z_{\text{new}} \end{pmatrix} = (1 + k) \begin{pmatrix} 1 & \epsilon_Z & -\epsilon_Y \\ -\epsilon_Z & 1 & \epsilon_X \\ \epsilon_Y & -\epsilon_X & 1 \end{pmatrix} \begin{pmatrix} X_{\text{old}} \\ Y_{\text{old}} \\ Z_{\text{old}} \end{pmatrix} + \begin{pmatrix} t_X \\ t_Y \\ t_Z \end{pmatrix} \]

Position Vector Transformation

The Position Vector transformation provides the transformation of coordinates from one reference system to another. This transformation uses the rotation angles (\(\epsilon_X, \epsilon_Y, \epsilon_Z\)) with the opposite sign and generally expresses the transformation of the position vector.

Transformation Equation:

\[ \begin{pmatrix} X_{\text{new}} \\ Y_{\text{new}} \\ Z_{\text{new}} \end{pmatrix} = (1 + k) \begin{pmatrix} 1 & -\epsilon_Z & \epsilon_Y \\ \epsilon_Z & 1 & -\epsilon_X \\ -\epsilon_Y & \epsilon_X & 1 \end{pmatrix} \begin{pmatrix} X_{\text{old}} \\ Y_{\text{old}} \\ Z_{\text{old}} \end{pmatrix} + \begin{pmatrix} t_X \\ t_Y \\ t_Z \end{pmatrix} \]

Basic Relationship and Differences

• Parameters and Usage: All three transformations use the translation (\(t_X, t_Y, t_Z\)), rotation (\(\epsilon_X, \epsilon_Y, \epsilon_Z\)), and scale (\(k\)) parameters.
• Signs of Rotation Angles: In the Coordinate Frame transformation, the rotation angles (\(\epsilon_X, \epsilon_Y, \epsilon_Z\)) are used directly, while in the Position Vector transformation, these angles are used with the opposite sign (\(-\epsilon_X, -\epsilon_Y, -\epsilon_Z\)).
• Interpretation and Application: The Coordinate Frame transformation expresses the transformation of the coordinate frame itself, while the Position Vector transformation expresses the transformation of the position vector. The Bursa-Wolf model forms the basis of both transformation methods as a general transformation model.

Summary

1. Bursa-Wolf Transformation: It is the general model used for coordinate transformation between geodetic datums and is based on seven parameters.

2. Coordinate Frame Transformation: It expresses the transformation of the coordinate frame, rotation angles are used directly.

3. Position Vector Transformation: It expresses the transformation of the position vector, rotation angles are used with the opposite sign.

These transformation methods ensure the accurate transformation of location data between different reference systems on the earth.