Coordinate Transformation
Source
Adjustment Computations
Statistics and Least Squares in Surveying and GIS
Paul R. Wolf and Charles D. Ghilani
Copyright 1997 by John Wiley & Sons, Inc
Chapter 17
Introduction
The transformation of points from one coordinate system to another is a common problem encountered in surveying and mapping.
Two-Dimensional Conformal Coordinate Transformation
The two-dimensional conformal coordinate transformation is also known as the four-parameter similarity transformation:
- 1 rotation to make the reference axes of the 2 systems parallel
- 2 translations to create a common origin for the 2 systems
- 1 scale factor to create equal dimensions in the 2 systems
It is commonly used in surveying (convert separate surveys into a common reference coordinate system). It requires a minimum of 2 common points.
The mathematical model for this conformal transformation is:
or
Two-Dimensional Affine Coordinate Transformation
The two-dimensional affine coordinate transformation is also known as the six-parameter transformation:
- 1 rotation to make the reference axes of the 2 systems parallel
- 2 translations to create a common origin for the 2 systems
- 2 scale factors, one for each reference axis
- 1 coefficient for nonorthogonality of the transformed axis
It is commonly used in photogrammetry (transform arbitrary measurement photo coordinate system to camera fiducial system). It requires a minimum of 3 common points.
The mathematical model for this affine transformation is:
Two-Dimensional Projective Coordinate Transformation
The two-dimensional projective coordinate transformation is also known as the eight-parameter transformation. It is appropriate to use when a one two-dimensional coordinate system is projected onto another non-parallel system. It is used in photogrammetry (relation between image and world coordinate systems) as well as to transform NAD27 coordinates into NAD83 system. It requires a minimum of 4 common points.
The mathematical model for this projective transformation is:
Three-Dimensional Conformal Coordinate Transformation
The three-dimensional conformal coordinate transformation is also known as the seven-parameter similarity transformation:
- 3 rotations
- 3 translations
- 1 scale factor
The mathematical model for this conformal transformation is:
The coefficients are the elements of a single rotation matrix where:
The 3 rotation angles can be easily visualized with the use of an intermediate coordinate system x’y’z’. This system x’y’z’ is parallel to the XYZ system but has its origin at the origin of the xyz system. The 3 sequential two-dimensional rotations ω, Φ, κ convert coordinates from x’y’z’ to xyz.
The rotation ω about the x’ axis expressed in matrix form is:
The rotation Φ about the y1 axis expressed in matrix form is:
The rotation κ about the z2 axis expressed in matrix form is:
The final rotation matrix is where:
It requires a minimum of 2 control points with known Z-Y and x-y coordinates plus a minimum of 3 control points with known Z and z coordinates.
If there are more than the minimum number of control points, a least squares solution can be used. The following linearized equations can be written:
Coordinate Transformation
Catherine LeCocq, SLAC – June 2005 – Page 4