Transformation parameters

The parameters of the coordinate equations required for a coordinate transformation of a surveying network or earth model are referred to as transformation parameters ; with other math. Transformations, however, the term is not used. The more complex the transformation, the higher the number of parameters. In three-dimensional space, for example, requires

• a parallel shift (translation) 3 parameters (1 each in the coordinates x, y and z)
• a rotation 2–3 parameters ( only 1 in the 2D plane)
• a uniform scale factor 1 parameter,
• a combination of all 3 processes (in which the shape of the transformed structure is retained) 7 parameters, see 7-parameter transformation or Helmert transformation .

The latter transformation is often used in geodesy, for example for the integration of a land survey into a global reference system . The shape must not be changed, as it would otherwise contradict the precise measurement data.

The Galileo transformation , which is often used in physics , differs from the Helmert's one by further 3 parameters of a uniform movement (a movement vector ), so it requires the determination of 10 transformation parameters .

If deformations of the structure are also permitted, additional parameters are added - depending on the complexity of the permissible change in shape. Examples are the shear (three parameters) and the affine transformation (at least eight parameters), which, however, anticipates some of the seven parameters mentioned above.

Example: 7-parameter transformation

Helmert's 7-parameter transformation is often used in geodesy and navigation , for example for converting GPS to national coordinates. The meaning of the parameters is shown in coordinate notation. Converted are Cartesian coordinates X, Y, Z of a system A to the system B, wherein the angle of rotation , and are to be used with its value in radians. In addition, each coordinate receives a shift ( , and ), and system B is given a scale factor µ , which is close to 1: ${\displaystyle r_{x}}$ ${\displaystyle r_{y}}$ ${\displaystyle r_{z}}$ ${\displaystyle c_{x}}$ ${\displaystyle c_{y}}$ ${\displaystyle c_{z}}$

${\displaystyle {\begin{matrix}X_{B}=c_{x}+\mu \cdot (X_{A}+r_{z}\cdot Y_{A}-r_{y}\cdot Z_{A})\\Y_{B}=c_{y}+\mu \cdot (-r_{z}\cdot X_{A}+Y_{A}+r_{x}\cdot Z_{A})\\Z_{B}=c_{z}+\mu \cdot (r_{y}\cdot X_{A}-r_{x}\cdot Y_{A}+Z_{A})\\\end{matrix}}}$

The 7 parameters are determined for the respective region (Operat, Federal State etc.) with 3 or more "identical points" for both systems. In general, the inevitable small contradictions (usually only a few cm) are balanced out using the least squares method , i.e. eliminated in the most statistically plausible way.

literature

• Bernhard Heck, calculation methods and evaluation models for national surveying . Wichmann-Verlag, Karlsruhe 1987, ISBN 3-87907-173-X .
• KP Schwarz (ed.), Geodesy beyond 2000 , Part 3 ( Advances in Theory and Numerical Techniques ). Proceedings, IAG General Ass. Birmingham, Springer-Verlag 2000
• MapRef.org , specialist literature and links on 2D and 3D coordinate transformations