Download Coordinate lists

Chapter 10. Calculations
(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 coordinates, 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; therefore, geometric transformation parameters can't be established. The minimum number of the identical points for this type of transformation is three.
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