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Transformation between Terrestrial Frames: Difference between revisions
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{{Article Infobox2 | {{Article Infobox2 | ||
|Category=Fundamentals | |Category=Fundamentals | ||
|Authors=J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain. | |||
|Authors=J. Sanz Subirana, | |||
|Level=Intermediate | |Level=Intermediate | ||
|YearOfPublication=2011 | |YearOfPublication=2011 | ||
|Title={{PAGENAME}} | |||
}} | }} | ||
From elemental linear algebra, all transformations between two Cartesian coordinate systems can be decomposed in a shift vector <math> \left ( \Delta \mathbb{\mathbf X}=[\Delta x, \Delta y, \Delta z] \right )</math>, three consecutive rotations around the coordinate axes (<math> \theta_1</math>, <math>\theta_2</math>, <math>\theta_3</math>), and a scale factor (<math>\alpha</math>). That is, they can be described by the following equation, which involves 7 parameters: | From elemental linear algebra, all transformations between two Cartesian coordinate systems can be decomposed in a shift vector <math> \left ( \Delta \mathbb{\mathbf X}=[\Delta x, \Delta y, \Delta z] \right )</math>, three consecutive rotations around the coordinate axes (<math> \theta_1</math>, <math>\theta_2</math>, <math>\theta_3</math>), and a scale factor (<math>\alpha</math>). That is, they can be described by the following equation, which involves 7 parameters: | ||
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<math> | <math> | ||
\begin{array}{l} | \begin{array}{l} | ||
\mathbb{\mathbf | \mathbb{\mathbf R}_1[\theta]=\left [ | ||
\begin{array}{ccc} | \begin{array}{ccc} | ||
1 & 0 & 0\\ | 1 & 0 & 0\\ | ||
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where <math>T_1</math>, <math>T_2</math>, <math>T_3</math> are three translation parameters, <math>D</math> is a scale factor and <math>R_1</math>, <math>R_2</math> and <math>R_3</math> are three rotation angles. | where <math>T_1</math>, <math>T_2</math>, <math>T_3</math> are three translation parameters, <math>D</math> is a scale factor and <math>R_1</math>, <math>R_2</math> and <math>R_3</math> are three rotation angles. | ||
Transformation parameters from ITRF2000 to | Transformation parameters from ITRF2000 to past ITRFs are listed in table 4.1 of IERS Conventions (2003) [Denis et al., 2004].<ref> [Denis et al., 2004] Denis, D., McCarthy and Petit, G., 2004. IERS Conventions (2003). IERS Technical Note 32. IERS Convention Center., Frankfurt am Main.</ref> | ||
Latest revision as of 11:41, 23 February 2012
| Fundamentals | |
|---|---|
| Title | Transformation between Terrestrial Frames |
| Author(s) | J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain. |
| Level | Intermediate |
| Year of Publication | 2011 |
From elemental linear algebra, all transformations between two Cartesian coordinate systems can be decomposed in a shift vector [math]\displaystyle{ \left ( \Delta \mathbb{\mathbf X}=[\Delta x, \Delta y, \Delta z] \right ) }[/math], three consecutive rotations around the coordinate axes ([math]\displaystyle{ \theta_1 }[/math], [math]\displaystyle{ \theta_2 }[/math], [math]\displaystyle{ \theta_3 }[/math]), and a scale factor ([math]\displaystyle{ \alpha }[/math]). That is, they can be described by the following equation, which involves 7 parameters:
- [math]\displaystyle{ \mathbb{\mathbf X}_{TRF2}=\Delta\mathbb{\mathbf X}+\alpha \; \mathbb{\mathbf R}_1[\theta_1] \; \mathbb{\mathbf R}_2[\theta_2] \; \mathbb{\mathbf R}_3[\theta_3] \; \mathbb{\mathbf X}_{TRF1} \qquad\mbox{(1)} }[/math]
where:
[math]\displaystyle{ \begin{array}{l} \mathbb{\mathbf R}_1[\theta]=\left [ \begin{array}{ccc} 1 & 0 & 0\\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \\ \end{array} \right ] \;;\;\; \mathbb{\mathbf R}_2[\theta]=\left [ \begin{array}{ccc} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0\\ \sin \theta &0 & \cos \theta \\ \end{array} \right ]\\ \\ \mathbb{\mathbf R}_3[\theta]=\left [ \begin{array}{ccc} \cos \theta & \sin \theta &0\\ -\sin \theta & \cos \theta & 0\\ 0 & 0 & 1\\ \end{array} \right ] \end{array} \qquad\mbox{(2)} }[/math]
Adopting the convention used by IERS, the previous equation (1) can be written as follows:
- [math]\displaystyle{ \left ( \begin{array}{c} x\\ y\\ z\\ \end{array} \right )_{_{TRF2}} = \left ( \begin{array}{c} x\\ y\\ z\\ \end{array} \right )_{_{TRF1}} + \left ( \begin{array}{c} T_1\\ T_2\\ T_3\\ \end{array} \right ) + \left ( \begin{array}{ccc} D & -R_3 & R_2\\ R_3 & D & -R_1\\ -R_2 & R_1 & D\\ \end{array} \right ) \left ( \begin{array}{c} x\\ y\\ z\\ \end{array} \right )_{_{TRF1}} \qquad\mbox{(3)} }[/math]
where [math]\displaystyle{ T_1 }[/math], [math]\displaystyle{ T_2 }[/math], [math]\displaystyle{ T_3 }[/math] are three translation parameters, [math]\displaystyle{ D }[/math] is a scale factor and [math]\displaystyle{ R_1 }[/math], [math]\displaystyle{ R_2 }[/math] and [math]\displaystyle{ R_3 }[/math] are three rotation angles.
Transformation parameters from ITRF2000 to past ITRFs are listed in table 4.1 of IERS Conventions (2003) [Denis et al., 2004].[1]
References
- ^ [Denis et al., 2004] Denis, D., McCarthy and Petit, G., 2004. IERS Conventions (2003). IERS Technical Note 32. IERS Convention Center., Frankfurt am Main.