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, T_Elev.

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?

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.