6. Two-Dimensions vs Three-Dimensions

While the discussion here is limited to horizontal (two dimensional) coordinate systems, it is relatively easy to extend this into the third dimension by incorporating elevations or Z coordinates. An additional translation parameter would be needed for the elevation component, TElev.

The rotation angle, ρ, for a two dimensional transformation is a horizontal angle rotated about the vertical, or Z, axis. In a three dimensional transformation there would three rotation angles, one about each axis. Figure J-25 shows X-Y-Z as the From system and E-N-Elev as the To system. There are three rotations and three translations. How about scale?

Figure J-25
Three-dimensional transformation

Scale could again be Unitary, Uniform, or Differential. In the latter case, there would be three scale factors.

The total number of parameters in a three dimensional transformation could be:

  • Six: Three translations, three rotations, 1:1 scale
  • Seven: Three translations, three rotations, uniform scale
  • Eight: Three translations, three rotations, one scale for horizontal, one scale for vertical
  • Nine: Three translations, three rotations, three scales

The eight-parameter transformation could be used where there is a quality difference between horizontal and vertical positions in one system or the other.