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:
a xb y c k (cos x sin y ) t
or
b xa yd 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
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 xb yc
d xe y f
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:
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 seven-
parameter similarity transformation:
- 3 rotations
- 3 translations
- 1 scale factor
The mathematical model for this conformal transformation is:
k ( m11 x m21 y m31 z ) t
k (m12 x m22 y m32 z ) t
k (m13 x m23 y m33 z ) t
m11 m12 m13
m23 where:
The coefficients mi j are the elements of a single rotation matrix m21 m22
m31 m32 m33
m11 cos cos
m12 sin sin cos cos sin
m13 cos sin cos sin sin
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 21 where:
x' x1 x2
y '
'
1 y1 2 y2
z ' z1 z2
1 0 0 cos 0 sin cos sin 0
1 0 cos sin 2 0 1 0 3 sin cos 0
0 sin cos sin 0 cos 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:
m11 x m21 y m31 z
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 ( m21 x m11 y )
k ( m22 x m12 y )
k (m23 x m13 y )
