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Affine/Triangle-based Transformation

Za slovensko verzijo te datoteke glej PREBERIME.md.

Theory

Affine transformation only works with cartesian coordinates (GK x,y ⇔ TM x,y). You need minimum 3 control points (for example a triangle), each with 4 coordinates (Xs, Ys, Xt, Yt), where Xs,Ys are coordinates in the source system and Xt,Yt coordinates in the target system.

For each new point Xi,Yi transformation is calculated via the following equation:

Xo = a⁎Xi + b⁎Yi + c
Yo = d⁎Xi + e⁎Yi + f

where parameters a..f have the following meaning:

a: Scale X
e: Scale Y
d: Rotation X
b: Rotation Y
c: Translation X
f: Translation Y

To calculate parameters a..f for a given triangle you have to solve the system of linear equations with 6 unknowns, represented with the following matrices:

| Xs1 Ys1 1 0 0 0 | | a |   | Xt1 |
| Xs2 Ys2 1 0 0 0 | | b |   | Xt2 | 
| Xs3 Ys3 1 0 0 0 | | c | = | Xt3 | 
| 0 0 0 Xs1 Ys1 1 | | d |   | Yt1 | 
| 0 0 0 Xs2 Ys2 1 | | e |   | Yt2 | 
| 0 0 0 Xs3 Ys3 1 | | f |   | Yt3 |

This system can be solved using elimination method (row operations ⇒ identity matrix) or with other methods (Jacobi, Gauss-Seidel, ...).

Pre-calculation

For Slovenia we have 899 reference virtual tie points (v3.0) in both coordinate systems (GK and TM), which form 1776 triangles:

We can pre-calculate parameters a..f for every triangle and store their values in a table to be directly used in a program.

To calculate triangles from the initial list of reference points we use the existing program triangle from Jonathan Shewchuk (UCB), which generates exact Delaunay triangulations.

Exact description of process

  • get the reference virtual tie points (already supplied):
    ⇒ file virtualne_vezne_tocke_v3.0.txt
  • extract GK and TM coordinates from the reference file
  • use triangle program to create triangles
  • use the supplied program ctt to create tables to be included in a program:
    ⇒ files aft_gktm.h and aft_tmgk.h
    ctt solves the above mentioned system of linear equations for each triangle.

All this can be automated using the supplied scripts cvvt.sh (for Unix) or cvvt.bat (for Windows).

Usage in a program

For each cartesian coordinate you have to loop through the array of 1776 pre-calculated triangles and check if the coordinate is contained in the triangle. This check should be very accurate, otherwise you'll end up with a point, which lies somewhere "between" the triangles (see geo_api.md).

When/if such triangle is found, a simple transformation using parameters a..f is applied (see Theory above).