Description of 2D and 3D Coordinate Systems and
Derivation of their Rotation Matrices
Conventions:
In a 3D coordinate system, Xs, Ys, Zs will be used for object coordinates in the scanner
coordinate system. This is the coordinate system from which the transformation is made.
Xc ,Yc ,Zc will be used for the object coordinates expressed in the camera coordinate system
after they have been scaled by the camera lens. The Zc value for the object will be the
location of the image plane in the camera, the CCD sensor and will be equal to the focal
length of the camera, f (at CCD sensor, Zc = f). This is the coordinate system to which the
scanner coordinates are transformed. Xi, Yi, Zi will be used to designate the object coordinates
after they are transformed to a coordinate system which has as its origin, the camera
coordinate system origin but before scaling by the lens occurs. Xi, Yi, Zi are unaffected by the
camera lens optics. x, y, z will be used for the translation needed to move the origin of the
scanner coordinate system to the origin of the camera coordinate system.
The derivation will first be explained used a 2D example. In a 2D planar coordinate system
Xs and Ys will be used for the coordinate system that corresponds to the scanner coordinates,
that is the system from which the transformation is made. Xc and Yc will be used for the
coordinate system that corresponds to the camera coordinates, that is the system to which the
X and Y coordinates are transformed. The terms camera and scanner are used here only to
maintain continuity between the 2D and the 3D derivations. A 2D transformation does not
apply to an actual transformation between scanner and camera coordinates.
In a 3D coordinate system Omega (ω) will describe rotation about the X-axis, Phi (Ф) will
describe rotation about the Y-axis, and Kappa (κ) will describe rotation about the Z-axis.
Theta (θ) will describe rotation in a 2D planar coordinate system.
Derivation of 2D transformation
In a 2D planar coordinate system a counter-clockwise rotation from the scanner coordinates to
the camera coordinates can be accomplished with the following transformation matrix
assuming that the origins of the two coordinate systems are located at the same spot. In a
counter-clockwise system positive rotation is in the counter-clockwise direction. Negative
rotation is in the clockwise direction.
Counter-clockwise rotation will be used in all transformations. This appears to be the most
commonly used convention, although it is possible to perform the transformations equally as
well with a convention that uses a clockwise rotation. This may be the preferable convention
for rotation because the cross-product of two vectors which have been defined relative to
each other in a counter-clockwise convention has the positive direction of the result of the
cross-product pointing toward the viewer. That is, with the two vectors in the horizontal
plane, the cross-product is positive if it points up.
The black coordinate axes in Figure 1 are rotated counter-clockwise to align them with the
blue coordinate axes.
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Figure 1. Counterclockwise rotation of coordinate systems,
transformation of X, Y to x, y. Point P is transformed from the
black coordinate system to the blue coordinate system.
Conversion from one coordinate system to the other is derived below.
Y = Xb + bP
x = oa + ab + bx
= X*cosθ + Xb*sinθ + bP*sinθ
= X*cosθ + (Xb + bP)*sinθ
= X*cosθ + Y*sinθ
y = -cY + od = -aX + od
= -X*sinθ + Y*cosθ
The above equations for x and y are expressed below in matrix notation.
 x   cos 
 y    sin 
  
sin    X 
cos    Y 
If the two coordinate systems have their origins separated by ΔX and ΔY distances, then a
translation of the origin of the XY coordinate system must be made to the xy origin by adding
the difference between their locations.
 x   cos 
 y    sin 
  
sin    X  X 
cos   Y  Y 
 cos
The transformation matrix M  
 sin 
sin  
is a 2X2 matrix.
cos 
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A clockwise rotation from the scanner coordinates to the camera coordinates will use the
following transformation matrix. The only difference is the signs for sinθ are reversed.
 x  cos 
 y    sin 
  
 sin    X  X 
 X  X 
 M




cos    Y  Y 
 Y  Y 
Derivation of the 3D transformation matrix
In a 3D coordinate system the terms right and left hand coordinate systems are used. This
method of defining a 3D coordinate system has the positive direction of the X-axis aligned
with the thumb, the positive direction of the Y-axis aligned with the index finger, and the Zaxis aligned with the middle finger. The thumb and index finger are spread at 90o and aligned
with the plane of the palm. The middle finger is bent at 90o to the palm.
If the coordinate system aligns with the appropriate fingers of the left hand it is called a lefthand coordinate system. And vice-versa for the right-hand coordinate system. There is no
rotation possible that will align a left-hand coordinate system with a right-hand coordinate
system. However, the problem is solved by simply negating (multiplication by -1) the values
of one of the axes of one of the coordinate systems. It makes no difference, which system or
which axis.
Note: The Riegel Laser Scanner uses a right-hand coordinate system. A camera with the
positive direction of the Z-axis exiting the lens toward the object, X being the horizontal
dimension with positive to the right, and Y being the vertical dimension with positive being
up, uses a left-hand coordinate system. The camera can be converted to a right-hand system
by negating the values on the Z-axis.
Positive rotation about an axis is determined by aligning the thumb of the hand (right or left)
in the positive direction of the axis and curling the fingers. The direction of the curled fingers
is the direction of positive rotation. A right hand rotation has the same matrix form as a
counter-clockwise rotation in a 2D coordinate system. A left hand rotation has the same
matrix form as a clockwise rotation in a 2D coordinate system.
Having three axes, a 3D coordinate system will require a 3X3 transformation matrix. The
matrices to transform the scanner coordinates to the camera coordinates are described below.
Right-hand rotation will be used. The equations could as well be derived using a left-hand
rotation.
0
0 
1

Mx  0 cos  sin  
0  sin  cos  
Rotation about the camera x-axis
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cos  0  sin  
My   0
1
0 
 sin  0 cos  
 cos  sin  0
Mz   sin  cos  0
 0
0
1
Rotation about the camera y-axis
Rotation about the camera z-axis
Note: Below the negation of the Z-axis value is disregarded, the camera and the scanner
coordinates are assumed to use the same coordinate system. This will be corrected at the end
of the derivation to reflect the reality of the Riegl scanner and the camera coordinate systems.
A single rotation about the X-axis would have the form shown below.
 X i  1
0
0   X s  x
X i 
 X s  x
 Y   Mx *  Y  y    Y   0 cos  sin    Y  y 
 i 
 s

 i
 s







 Z i 
 Z s  z 
Z
0

sin

cos

Z

z
s

  
Rotation about the three camera axes can be done in any order, but must then be done
consistently throughout the computations in the same order. Generally, matrix multiplication
is not transitive, that is [M] * [N] does not equal [N] * [M]. Rotation about the X-axis
followed by rotations about the Y and Z-axes is the common sequence that is used. Thus, a
transformation from the scanner coordinate system to the camera coordinate system is
accomplished with the following matrix operations.
The full transformation would have the form shown below.
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X i 
 X s  x
 Y   Mz * My * Mz  Y  y 
 i
 s

 Z i 
 Z s  z 
 X i   cos 
 Y    sin 
 i 
 Z i   0
sin 
cos 
0
or, expressed in complete form below
0 cos  0  sin   1
0
0   X s  x




0  0
1
0  0 cos  sin    Ys  y 
1  sin  0 cos   0  sin  cos    Z s  z 
The combined transformation matrix M can be expressed as M  Mz * My * Mx . And
the transformation from scanner to camera coordinates can be expressed as
X i 
 X s  x
 Y   M Y  y 
 i
 s

 Z i 
 Z s  z 
Multiplying out the rotation matrices Mz, My, and Mx is shown below
 cos  cos  cos  sin   sin  sin  cos  sin  sin   cos  sin  cos  
M   cos  sin  cos  cos   sin  sin  sin  sin  cos   cos  sin  sin  
 sin 

 sin  cos 
cos  cos 
Each element of the matrix is named according to its row/column position.
m11  cos  cos 
m12  cos  sin   sin  sin  cos 
m13  sin  sin   cos  sin  cos 
m 21   cos  sin 
m 22  cos  cos   sin  sin  sin 
m 23  sin  cos   cos  sin  sin 
m31  sin 
m32   sin  cos 
m33  cos  cos 
 X i   m11 m12 m13   X s  x 
 Y   m21 m22 m23  Y  y 
 i 
 s

 Z i   m31 m32 m33  Z s  z 
In order to bring the camera and the scanner coordinate systems into the right-hand coordinate
system of the scanner, it is necessary to multiply Zi by -1.
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 X i  1 0 0   m11 m12 m13   X s  x 
 Y   0 1 0  m21 m22 m23  Y  y 
 i  

 s

 Z i  0 0  1  m31 m32 m33  Z s  z 
In order to scale the coordinates of the camera to that of the image plane (Xc, Yc) in the
camera, Xi and Yi must be scaled by the ratio of the focal length (f) to that of the Zi value for
each point. The first and second rows of the matrix equation are multiplied by the scaling
factor f/Zi. The third row of the matrix equation is left unchanged.
 f
  f
Z Xi  Z
 Xc   i
  i
f
 Y  Y 0
 c  Z i 
 Z i   i  
 Zi
0

 

 
0
f
Zi
0

0
  m11 m12 m13   X s  x 
0  m21 m22 m23  Ys  y 




 1  m31 m32 m33  Z s  z 


Refer to Derivation of Tranformation Parameter Computation for a 2D/3D Coordinates
System From Laser Scanner Coordinates to Camera Coordinates for a more complete
explanation of the transformation to camera focal plane coordinates.
Note: The magnification of a camera lens system is the ratio of the focal length to the distance
to the object. If a camera lens has a focal length of 25 mm and this is considered to be 1X,
then replacing the 25 mm lens with a 50 mm lens will increase the magnification by a factor
of two and reduce the area that was imaged by a factor of ½. Assume that the CCD sensor
has a width of 25 mm (this is very close to the actual width of high quality Digital Single Lens
Reflex (DSLR) cameras. A lens with a focal length of 25 mm is used. The distance to the
outcrop is 100 m. The picture that is taken will cover 100 m of width of the outcrop,
assuming that the camera is aligned perpendicular to the outcrop and that the outcrop runs
straight left to right. If a lens with a focal length of 50 mm is put on the camera, the
magnification will be about 2X and will only cover 50 m of the outcrop. Likewise, if a lens
with a focal length of 100 mm is put on the camera, the image will appear to be magnified by
4X the first image and will cover only 25 m of the outcrop.
Derivation of the Collinearity Equations (basis for scanner to camera transformation)
Below is a 3D transformation matrix without scaling by the camera lens.
 X i   m11 m12 m13   X s  x 
 Y   m21 m22 m23  Y  y 
 i 
 s

 Z i   m31 m32 m33  Z s  z 
The camera coordinate system has the origin located at the rear nodal point of the lens system.
The Z-axis of the camera is parallel to the axis of the lens system and is perpendicular to the
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CCD sensor in the camera. The positive direction of the Z-axis is toward the object being
photographed. Since the CCD sensor is located behind the focal point the CCD sensor is
located at a Z-coordinate of (–f ), [f: focal length of the lens]. The CCD sensor represents
the X-Y plane of the camera coordinate system and is orthogonal to the Z-axis. Note that with
the CCD sensor (X-Y plane) located behind the focal point, inversion in the X and Y
dimensions occurs. This is corrected by virtually moving the CCD sensor a distance (f) in
front of the focal point. This places the image on the CCD sensor in the same orientation that
the object has in space. It would be the same as taking a picture of the object and then
holding the picture up in front of the camera. What is on the left side of the object appears on
the left side of the picture. What is on the upper side of the object appears on the upper side
of the picture.
The above transformation matrix transforms the scanner coordinates to camera coordinates
but does not scale the dimensions to the scale of the CCD sensor. That is, the dimensional
reduction that occurs due to the camera lens is not reflected in the transformation matrix at
this point.
An image dimension is related to the object dimension by the ratio of the focal length of the
lens and the distance to the object. Using the relationship of similar triangles as shown below,
Xc = (f/Zi)*Xs, where f is the focal length of the camera, Zc is the distance from the rear nodal
point of the camera lens to the object and Xs is the dimension of the object.
The Xi and Yi values will be scaled by the ratio f/Zi yielding Xc and Yc which are the
coordinates on the CCD sensor. Thus, the matrix equation above when translated to the plane
of the CCD sensor can be expressed as
f   f


0 0
X i Z  Z
i
X c  
  c
  m11 m12 m13   X s  x 
f
f
Y   Y
 0
0 m21 m22 m23  Ys  y 
 c  iZ  

Z
i
c
 Z i  





Zi
0
0 1  m31 m32 m33  Z s  z 

 


 

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The above complete transformation matrix is expressed in equation form as
Xc 
f
m11 X s  x   m12Ys  y   m13Z s  z 
Zi
Yc 
f
m21 X s  x   m22Ys  y   m23Z s  z 
Zi
Z i  m31 X s  x   m32Ys  y   m33Z s  z 
What are called the collinearity equations are formed by substituting for Zi in the equations
for Xc and Yc. The collinearity equations are
f m11 X s  x   m12Ys  y   m13Z s  z 
Xc 
m31 X s  x   m32Ys  y   m33Z s  z 
f m21 X s  x   m22Ys  y   m23Z s  z 
m31 X s  x   m32Ys  y   m33Z s  z 
Yc 
The scanner and camera systems are not compatible, one is left handed the other is right
handed. They are made compatible by negating the values of one of the axes. By negating
the value of the Z axis, the equations become
Xc 
Yc 
 f m11 X s  x   m12Ys  y   m13Z s  z 
m31 X s  x   m32Ys  y   m33Z s  z 
 f m21 X s  x   m22Ys  y   m23Z s  z 
m31 X s  x   m32Ys  y   m33Z s  z 
The following assignments are made in order to simplify the notation later, refer to the
numerators and denominators of the above equations.
U   f m11 X s  x   m12Ys  y   m13Z s  z 
V   f m21 X s  x   m22Ys  y   m23Z s  z 
W  m31 X s  x   m32Ys  y   m33Z s  z 
Xc and Yc are shown below expressed in terms of U, V, W.
U
W
V
Yc 
W
Xc 
Xc and Yc are the coordinates of a point on the camera CCD sensor with the origin of the
coordinate system at the center of the CCD sensor. These are not UV coordinates. UV
coordinates have the origin at the upper left corner of a photograph and have values between 0
and 1.0. Xc and Yc have dimensions which relate to the size of the CCD sensor and range
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from -1/2 the sensor width to +1/2 the sensor width and from -1/2 the sensor height to +1/2
the sensor height. In solving for the transformation parameters (ω, Φ, κ, x, y, and z) four
points are identified common between the photograph and the 3D model. Thus we have four
sets of values for Xc, Yc, Xs, Ys, and Zs. We do not have values for Zi in the camera
coordinate system. By substituting for Zi in the equations for Xc and Yc, the need to solve for
Zi is eliminated.
A different derivation of the collinearity equations is provided by Mikhail and is
described below.
Starting with the 3D transformation below, the system is turned into a 2D/3D transformation
by adding a uniform scaling factor (a) and setting the image Z value, Zi, to –f, the focal length
of the camera.
 X i   m11 m12 m13   X s  x 
 Xc 
 m11 m12 m13   X s  x 
 Y   m21 m22 m23  Y  y  becomes  Y   a m21 m22 m23  Y  y 
 i 
 s

 c 

 s

 Z i   m31 m32 m33  Z s  z 
 f 
 m31 m32 m33  Z s  z 
Writing this in equation form
X c  am11 X s  x   m12Ys  y   m13Z s  z 
Yc  am21 X s  x   m22Ys  y   m23Z s  z 
 f  am31 X s  x   m32Ys  y   m33Z s  z 
Division of the first two equations by the third equation eliminates the scaling factor a.
Xc 
Yc 
 f m11 X s  x   m12Ys  y   m13Z s  z 
m31 X s  x   m32Ys  y   m33Z s  z 
 f m21 X s  x   m22Ys  y   m23Z s  z 
m31 X s  x   m32Ys  y   m33Z s  z 
The least squares solutions for these equations are provided in the other documents.
References:
Mikhail, E.M., Bethel, J.S., McGlone, J.C.; Introduction to Modern Photogrametry, 479
pages, John Wiley and Sons, 2001.
Larson, R.E., Edwards, B.H.; Elementry Linear Algebra, 568 pages, D.C. Heath and
Company, 3rd Edition, 1996.
Wolf, P.R., Dewitt, B.A.; Elements of Photogrammetry with Applications in GIS, Edition,
McGraw Hill, Inc., 3rd Edition, 2000,
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