Transformation of coordinates

The GROMA program has built in one universal transformation of coordinates used by all the calculation jobs which need the transformation.

Identical points

According to the transformation type at least two or three so called identical points are required. The term "identical points" isn't very suitable and does not describe the point's essence (generally while a transformation, these points are not identified in both systems). But since this term is used in standard practice, we will use here for better understanding.

Identical points are points whose coordinates are known in both systems, and can therefore be used for the calculation of the transformation key. The used transformation is linear; in a general case, both systems are identified only in one point - in the center of gravity of the identical points. For any other points for which the transformation key applies, it was determined that their coordinates trans-formed from one system into another are not generally equal to the coordinates in the other system (they only meet the condition of the minimum sum of the error square); therefore, they are not truly "identical".

Transformation key

The transformation key, along with the list of identical points, is stored in a file and later loaded back into the program. The majority of the calculation functions using the transformation allow displaying the resulting transformation parameters.

Among the transformation parameters are the following:

  • Transformation key:

    • Shift in the direction of the axis X,

    • Shift in the direction of the axis Y,

    • System rotation,

    • One or two scale coefficients

  • Identical point:

    Coordinates of the point where both systems are identified (i.e. the centre of gravity of the points, from which was generated the transformation key).

The transformation parameters are determined based on the smallest squares method with the condition to minimize the sum of the squares of the coordinate's adjustment at the identical points.

The majority of jobs using the transformation of coordinates contain the button (Key), which displays a box with transformation parameters. This box contains buttons (Store key) and (Load key), which serve to write the defined transformation key into a file and to its repetitive loading. If the files with the transformation key are mutually compatible - for example - in the transformation of co-ordinates, which was defined during the orthogonal method (and vice versa), the key can be loaded and used.

Identity transformation

Identity transformation is the linear transformation of coordinates which preserves the coordinates' dimensions (the scale coefficient equals 1). Therefore, the transformation has three parameters (two translations and rotation); to establish transformation keys at least two identical points are required.

Similarity transformation

Similarity transformation is the linear transformation of coordinates which uses one scale coefficient for the direction of both the X and Y axes. Therefore, the transformation has four parameters (two translations, rotation and scale coefficient); to establish transformation keys at least two identical points are required.

Affine transformation - 5 degrees of freedom

The affine transformation has, unlike the similarity transformation, two different scales for the directions of the X and Y axes. The number of transformation parameters is five (two translations, rotation and two scale coefficients); therefore, the minimum number of identical points increases to three.

Affine transformation - 6 degrees of freedom

In this type of transformation, geometric transformation parameters are not established, but the elements of the transformation matrix are directly established. The total number of the transformation parameters is six (two translations and four elements of the transformation matrix). In this case, the elements of the transformation matrix are not mutually interconnected by geometric relations; there-fore, geometric transformation parameters can't be established. The minimum number of the identical points for this type of transformation is three.

User's transformation

If you choose this type of transformation, you don't have to enter any identical points, but you can directly enter transformation equations for calculation of the coordinates of the points.