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 fourparameter 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:
  a xb y c
 b xa yd
or
  k (cos x  sin  y)  t
  k (sin  x  cos y)  t 
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
Coordinate Transformation
Catherine LeCocq, SLAC – June 2005 – Page 1
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:
  a xb y c
  d xe y f
Two-Dimensional Projective Coordinate Transformation
The two-dimensional projective coordinate transformation is also known as the eightparameter 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:
a1 x  b1 y  c1
a3 x  b3 y  1
a x  b2 y  c2
 2
a3 x  b3 y  1

Three-Dimensional Conformal Coordinate Transformation
The three-dimensional conformal coordinate transformation is also known as the sevenparameter similarity transformation:
- 3 rotations
- 3 translations
- 1 scale factor
The mathematical model for this conformal transformation is:
  k (m11x  m21 y  m31z )  t
  k (m12 x  m22 y  m32 z )  t
  k (m13 x  m23 y  m33 z )  t
Coordinate Transformation
Catherine LeCocq, SLAC – June 2005 – Page 2
 m11 m12
The coefficients mi j are the elements of a single rotation matrix   m21 m22

m31 m32
where:
m11  cos cos 
m12  sin  sin  cos   cos  sin 
m13   cos  sin  cos   sin  sin 
m13 
m23 
m33 
m21   cos sin 
m22   sin  sin  sin   cos  cos 
m23  cos  sin  sin   sin  cos 
m31  sin 
m32   sin  cos 
m33  cos  cos 
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: 1  1'
The rotation Φ about the y1 axis expressed in matrix form is: 2  2 1
The rotation κ about the z2 axis expressed in matrix form is:    3  2
The final rotation matrix is    3 21 where:
 x'
   y '
 z ' 
'
 x1 
1   y1 
 z1 
0
0 
1

1  0 cos  sin  
0  sin  cos  
 x2 
 2   y2 
 z2 

   
  
cos  0  sin  
 2   0
1
0 
 sin  0 cos  
 cos 
 3   sin 
 0
sin 
cos 
0
0
0
1
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 3

 m11x  m21 y  m31z
k

 m12 x  m22 y  m32 z
k

 m13 x  m23 y  m33 z
k

0


  k (m13 x  m23 y  m33 z )


 k (m12 x  m22 y  m32 z )


 k ( sin  cos  x  sin  sin  y  cos  z )


 k (sin  cos  cos  x  sin  cos  sin  y  sin  sin  z )


 k ( cos  cos  cos  x  cos  cos  sin  y  cos  sin  z )


 k (m21x  m11 y )


 k (m22 x  m12 y )


 k (m23 x  m13 y )

Coordinate Transformation
Catherine LeCocq, SLAC – June 2005 – Page 4