Symbolic – Numeric Algebra
in Geodesy
R. Lewis, B. Paláncz, P. Zaletnyik, J. Awange
1
Geodesy and Surveying
Geodesy and Surveying are concerned with measuring the
Earth on different scales.
Variables such as distances, angles and height differences are
measured and other geometrical variables (i. e. coordinates of
location) are derived from these measurements.
Many geodetic problems can be represented as systems of
multivariable polynomials.
GPS Positioning (GPS Ranging)
In
GPS
Positioning
pseudo
distances are measured between a
GPS receiver and minimum 4 GPS
satellites to calculate the position of
the receiver (X,Y,Z) and the
receiver’ clock bias (T).
For each satellites:
( X  ai )2  (Y  bi )2  (Z  ci )2  (T  Pi )2  0
where ai, bi, ci are the coordinates of the
satellites and Pi the measured distances.
3D Helmert similarity transformation
In geodesy is a frequent task to calculate the coordinates of points from
one to another different coordinate system. In the 7-parameter
similarity transformation 3 translations, 3 rotations and 1 scale
parameter are used.
z
Z
P
Z
Y
X
x
0
 xi 
 Xi   X0 
 
   
y

m

R

 i
 Yi    Y0 
[ 33]
z 
Z  Z 
 i
 i  0
i
0
X
0
Y
y
where R is the rotation matrix, m is the
scale parameter and X0, Y0, Z0 are the 3
translation parameters.
Solving Algebraic Computational
Problems in Geodesy
The book of J.L. Awange and E.W. Grafarend
can be considered as a milestone in this
research area.
The main algebraic methods they applied:
Groebner Basis
Resultants (Sylvester, Macaulay and
Sturmfels’ Formulations)
Gauss-Jacobi combinatorial solution
2005
Improvement of models and methods
Geodesy needs more precise measurements,
more precise computations with more general and
enhanced models.
Computational algebra provides more effective
algorithms and methods to carry out these
computations.
3D affine transformation
The 3D affine transformation is the generalization of the 3D Helmert
transformation, using 3 different scale parameters according to the
coordinate axes, instead of one. The 9 transformation parameters (3
translation, 3 rotations and 3 scale parameters) should be determined
using control points with known coordinates in both coordinate system.
z
Z
P
Z
Y
X
x
0
 xi 
 Xi   X0 
 
   
 R   Yi    Y0 
 yi   [ M
33 ] [ 33 ]
z 
Z  Z 
 i
 i  0
i
0
X
0
Y
y
where R is the rotation matrix, M the
diagonal matrix of the scale parameters
and X0, Y0, Z0 are the 3 translation
parameters.
Rotation matrix
z
Z
The rotation matrix in general is
given by using 3 axial rotation
angles (, ,  - Cardan angles),
leading to the next rotation matrix:
P
Z
Y
0
i
0
X
0
Y
X
x
cos  cos


R( ,  ,  )   sin  sin  cos  cos sin 
 cos sin  cos  sin  sin 

cos  sin 
sin  sin  sin   cos cos
cos sin  sin   sin  cos
Instead of the traditional form, the rotation
matrix can be expressed with a skewsymmetric matrix (S) also, and this facilitates
the symbolic solution of the problem.
 sin 


sin  cos  
cos cos  
 0 c b 


S  c
0  a
b a
0 

R  ( I 3  S )1 I 3  S 
y
Scale parameters
In 3D affine transformation 3 scale parameters (m1, m2, m3) are
used according to the 3 coordinate axis:
 m1

M  0
0

0
m2
0
0 

0 
m3 
In further, instead of the scale parameters (m1, m2, m3), we will use
their reciprocals (1, 2, 2) to get more simple equations.
introducing:
  M 1
1
1
1
1  ,  2 
, 3 
m1
m2
m3
0 
 1 0


   0 2 0 
0 0  
3

The algebraic equations
Using
 1 instead of M and
R  ( I 3  S )1 I 3  S  we get:
 xi 
 Xi   X0 
 
   
1
1
 yi     I 3  S   I 3  S    Yi    Y0 
z 
Z  Z 
 i
 i  0
0 
 xi 
 Xi 
 X0 
 
 
 
 






0

I

S



y

I

S

Y

I

S



 
 i
 i 
 Y0 
3
3
3
0 
z 
Z 
Z 
 
 i
 i
 0
c  b   1 0
0  xi   1  c b   X i   1
c  b   1 0 0  X 0 
0   1
  

  
   

 
a  0  2 0  yi    c
1  a    Yi     c 1
a  0  2 0  Y0 
0     c 1
 0   b  a 1  0 0   z    b a
1   Z i   b  a 1  0 0  3  Z 0 
3  i  
  

The 3-point problem
As for one point with known coordinates in both coordinate systems 3
equations can be written, to determine the 9 parameters of the
transformation 3 points are required.
In this case the following 9 equations can be written:
f 1   X 1  cY1  bZ1  x1 1  X 0 1  cy1 2  cY0 2  bz1 3  bZ0 3
f 2  cX 1  Y1  aZ 1cx1 1  cX 0 1  y1 2  Y0 2  az1 3  aZ0 3
f 3  bX 1  aY1  Z 1  bx1 1  bX 0 1  ay1 2  aY0 2  z 1 3  Z 0 3
f 4   X 2  cY2  bZ2  x 2 1  X 0 1  cy2 2  cY0 2  bz2 3  bZ0 3
f 5  cX 2  Y2  aZ 2 cx2 1  cX 0 1  y 2 2  Y0 2  az2 3  aZ0 3
f 6  bX 2  aY2  Z 2  bx2 1  bX 0 1  ay2 2  aY0 2  z 2 3  Z 0 3
f 7   X 3  cY3  bZ3  x3 1  X 0 1  cy3 2  cY0 2  bz3 3  bZ0 3
f 8  cX 3  Y3  aZ 3 cx3 1  cX 0 1  y 3 2  Y0 2  az3 3  aZ0 3
f 9  bX 3  aY3  Z 3  bx3 1  bX 0 1  ay3 2  aY0 2  z 3 3  Z 0 3
Simplification of the equations
g 1  f 1  f7
g 2  f 4  f7
g3  f2  f8
g4  f5  f8
g5  f3  f9
g6  f6  f 9
We can reduce the equation system by differencing the
equations, and introducing some new variables, the
relative coordinates instead of the original ones,
X ij  X i  X j , Yij  Yi  Y j , Z ij  Z i  Z j 
  i, j  1,2,3, i  j
xij  xi  x j , yij  yi  y j , z ij  z i  z j 
Then the system of equation to be solved is,
g1   X 13  x13  1  c Y13  c y13  2  b Z 13  b z13  3
g 2   X 23  x23  1  c Y23  c y23  2  b Z 23  b z 23  3
g 3  Y13  c X 13  c x13  1  y13  2  a Z 13  a z13  3
g 4  Y23  c X 23  c x23  1  y23  2  a Z 23  a z 23  3
g 5   Z 13  b X 13  b x13  1  a Y13  a y13  2  z13  3
g6   Z 23  b X 23  b x23  1  a Y23  a y23  2  z 23  3
The Dixon – EDF method
Explicit symbolic solution can be achieved by employing the accelerated
Dixon resultant by the Early Discovery Factors (EDF) algorithm, which was
suggested and implemented in the computer algebra system Fermat by
Lewis.
The basic idea of the Dixon method is to construct a square matrix M
whose determinant D is a multiple of the resultant.
The EDF method is to exploit the observed fact that D has many factors. In
other words, we try to turn the existence of spurious factors to our
advantage. By elementary row and column manipulations (Gaussian
elimination) we discover probable factors of D and pluck them out of M0=M.
Any denominators that form in the matrix are plucked out. This produces a
smaller matrix M1 still with polynomial entries, and list of discovered
numerators and denominators.
Solution with Dixon – EDF method
From our system of equation an univariate polynomial for 1 can be
computed by employing the accelerated Dixon resultant by the Early
Discovery Factors (Dixon - EDF) algorithm as elimination technique.
Using this method one can get the result in the following form:

5
 i  1 
Ki
i 1
where i (1) are irreducible polynomials with low degree, but their
powers, Ki are very big positive integer numbers, so expanding this
expression would result into millions of terms!
Solution with Dixon – EDF method
Consequently, we shall consider Ki = 1, for i = 1;…;5.
We get five irreducible polynomials (i (1)):
1  y13 z23  y23 z13
 2  Z 13 z23  Z 23 z13
 3  x13 y23  1  x23 y13  1  X 13 y23  X 23 y13
 4  x13 2 y23 z23  12  x13 x23 y13 z 23  12  x13 x23 y23 z13  12  x23 2 y13 z13  12
 Z 13 y23 z 23  Y13 y23 z 23  X 13 y23 z 23  Z 13 Z 23 y13 z 23
2
2
2
 Y13 Y23 y13 z 23  X 13 X 23 y13 z 23  Z 13 Z 23 y23 z13  Y13 Y23 y23 z13
 X 13 X 23 y23 z13  Z 23 y13 z13  Y23 y13 z13  X 23 y13 z13
2
2
2
 5  Z 13 x13 x23 z23  12  Z 23 x13 2 z 23  12  Z 13 x23 2 z13  12  Z 23 x13 x23 z13  12
 X 13 Z 13 x23 z 23  1  2 X 13 Z 23 x13 z 23  1  X 23 Z 13 x13 z 23  1
 X 13 Z 23 x23 z13  1  2 X 23 Z 13 x23 z13  1  X 23 Z 23 x13 z13  1
 X 13 Z 23 z 23  X 13 X 23 Z 13 z 23  X 13 X 23 Z 23 z13  X 23 Z 13 z13
2
2
Solution with Dixon – EDF method
The factor polynomial providing proper root has degree of two,
therefore its solution can be expressed in analytical form (only the
positive root is correct, 1). The proper polynomial is:
4  x13 2 y23 z23  12  x13 x23 y13 z23  12  x13 x23 y23 z13  12  x23 2 y13 z13  12
 Z13 y23 z23  Y13 y23 z23  X 13 y23 z23  Z13 Z 23 y13 z23
2
2
2
 Y13 Y23 y13 z23  X 13 X 23 y13 z23  Z13 Z 23 y23 z13  Y13 Y23 y23 z13
 X 13 X 23 y23 z13  Z 23 y13 z13  Y23 y13 z13  X 23 y13 z13
2
2
2
Similarly we can get simple explicit forms for 2 and 3.
Solutions for 1, 2, 3
1 = [X232 y13 z13 + y13 Y232 z13 + X132 y23 z23 + Y132 y23 z23 + y23 Z132 z23 X13 X23 (y23 z13 + y13 z23) - Y13 Y23 (y23 z13 + y13 z23) y23 z13 Z13 Z23 - y13 Z13 z23 Z23 + y13 z13 Z232)]1/2
/ [(x23 y13 - x13 y23) (x23 z13 - x13 z23)]1/2
2 = [-X132 x23 z23 + X13 X23 (x23 z13 + x13 z23) + x23 (Y13 Y23 z13 - Y132 z23 Z132 z23 + z13 Z13 Z23) - x13 (X232 z13 + Y232 z13 - Y13 Y23 z23 Z13 z23 Z23 + z13 Z232))]1/2 / [-(x23 y13 - x13 y23) (-y23 z13 + y13 z23)]1/2
3 = [X132 x23 y23 - X13 X23 (x23 y13 + x13 y23) + x23 (Y132 y23 y13 Y13 Y23 + y23 Z132 - y13 Z13 Z23) + x13 (X232 y13 - Y13 y23 Y23 + y13 Y232 y23 Z13 Z23 + y13 Z232))]1/2 / [(x23 z13 - x13 z23) (y23 z13 - y13 z23)]1/2
Solution for a, b, c
Knowing 1, 2, 3 our system of equation will be linear and a, b, c can be
determined easily.
a  [ X 23Y13  X 13Y23  ( X 23 y13  X 13 y23 ) 2   1 ( x23Y13  x13Y23  ( x23 y13  x13 y23 ) 2 )] /
[ X 23 Z13  X 13 Z 23  ( X 23 z13  X 13 z23 ) 3   1 ( x23 Z13  x13 Z 23  ( x23 z13  x13 z23 ) 3 )]
b  [Y23 Z13  Y13 Z 23  (Y23 z13  Y13 z23 ) 3   2 ( y23 Z13  y13 Z 23  ( y23 z13  y13 z23 ) 3 )] /
[ X 23Y13  X 13Y23  ( X 23 y13  X 13 y23 ) 2   1 ( x23Y13  x13Y23  ( x23 y13  x13 y23 ) 2 )]
c  [ X 23 Z13  X 13 Z 23  ( X 23 z13  X 13 z23 ) 3   1 ( x23 Z13  x13 Z 23  ( x23 z13  x13 z23 ) 3 )] /
[Y23 Z13  Y13 Z 23  (Y23 z13  Y13 z23 ) 3   2 ( y23 Z13  y13 Z 23  ( y23 z13  y13 z23 ) 3 )]
Solution for X0, Y0, Z0
Knowing 1, 2, 3 and a, b, c the translation parameters can be
determined from the original system of equation:
0
 X 0   x1 
 X1 
 x1 
 X1 
 x1   1 /  1
   
 
 





1
1/ 2
 Y0    y1   M  R   Y1    y1     R   Y1    y1    0
Z  z 
Z 
z 
Z 
z   0
0
 0   1
 1
 1
 1
 1 
 1  a 2  b2  c2
2(ab  c)
2(b  ac)   X 1 

  
1
2
2
2

  2(ab  c)
1 a  b  c
2(a  bc)    Y1 
1  a 2  b2  c2 
2
2
2 

2
(

b

ac
)
2
(
a

bc
)
1

a

b

c

  Z1 
0 

0 
1 /  3 
Using the Dixon-EDF method we got fully symbolic solution for all the
parameters and using these explicit formulas the parameters can be
calculated easily just substituting the coordinates to these formulas.
N-point problem
In geodesy normally we have more observations than unknowns,
when the solution is determined by minimizing the sum of the
deviations squared (Least Squares Method). In this case one
possible solution is the before mentioned Gauss-Jacobi
combinatorial method.
In Hungary there are 1138 national stations with known GPS and
local projection coordinates. To determine the parameters of the
affine transformation we should have compute about 250 millions
combinations, which is unrealistic.
Therefore we decided to transform the overdetermined problem
into a determined one.
Definition of the N-point problem
For one 3D control point (which coordinates are given in both systems), 3
nonlinear equations can be written:
vxi   X i  cYi  bZi  xi 1  X 0 1  cyi 2  cY0 2  bzi 3  bZ0 3
vyi  cX i  Yi  aZ i cxi 1  cX 0 1  yi 2  Y0 2  azi 3  aZ0 3
vzi  bX i  aYi  Z i  bxi 1  bX 0 1  ayi 2  aY0 2  z i 3  Z 0 3
Therefore for N control points we can write 3 times N nonlinear equations.
N
u   (vxi2  vyi2  vzi2 )
i 1
u  f (a, b, c, X 0 , Y0 , Z 0 , 1 , 2 , 3 )
The determined model can be evaluated via partial differentiation:









f a  0, f b  0, f c  0, f X 0  0, f Y0  0, f Z0  0, f  1  0, f  2  0, f  3  0
The determined model
Using computer algebra manipulations we get a determined model in
symbolic form consisting of 9 equations. One of them is:
N
Eqa  bNX 0Y0 1 2  aNY0  2  cNX 0 Z 0 1 3  aNZ0  3  bY0 1 2  xi 
2
2
2
2
i 1
N
N
N
i 1
i 1
i 1
N
N
 cZ0 1 3  xi  bY0 2  X i  cZ0 3  X i  aY0 2
2
N
 (2 y ) 
i
i 1
N
 bX 0 1 2  yi  b 1 2  ( xi yi )  b 2  ( X i yi )  a 2
i 1
i 1
2
i 1
N
N
N
N
i 1
i 1
i 1
i 1
N
y
i 1
2
i

 aY0 2  (2Yi ) bX 0 1  Yi Z 0 3  2Yi b 1  ( xiYi ) 
N
N
i 1
i 1
 b ( X iYi ) a 2  2 yiYi aZ0 3
N
2
N
N
N
 (2 z )  cX    z
i
i 1
0
1
3
N
 c 1 3  ( xi zi ) c 3  ( X i zi )   3  (2Yi zi )  a 3
i 1
i 1
i 1
N
N
N
N
i 1
i 1
i 1
i 1
i 1
2
i
N
z
i 1

2
i

 Y0 2  (2 Z i ) aZ0 3  (2 Z i )  cX 0 1  Z i  c 1  ( xi Z i ) 
N
N
N
N
i 1
i 1
i 1
i 1
 c  ( X i Z i )  2  2 yi Z i  a 3  2 zi Z i  a  (Yi  Z i )  0
2
2
Numerical solutions of the
determined model
Unfortunately this multivariate polynomial system of 9 equations with
9 unknowns can not be solved in symbolic form.
Local numerical methods like Newton-Raphson or Newton-Krylov
generally fail to converge, because of the initial guess values
computed from the 3 point solution are wildly different values
depending on the constellation of the selected 3 points.
Global numerical method like linear homotopy is considered as a
good candidate.
Solution with Newton Homotopy Method
The homotopy continuation method deforms the known roots of the start
system into the roots of the target system. It is a very effective tool to find
the roots of polynomial systems.
Employing New tonhomotopy, the start system is,
G(  )  F (  )  F (  0 )
w ith F  ( Fa (  ), Fb (  ), Fc (  ), FX 0 (  ), FY0 (  ), FZ 0 (  ), F 1 (  ), F 2 (  ), F 3 (  )) as target
system,   (a, b, c, X 0 , Y0 , Z 0 ,  1 ,  2 ,  3 ) as unknow nvariables and
 0  (a, b, c, X 0 , Y0 , Z 0 ,  1 ,  2 ,  3 ) as start (initial) values for the homotopy function.
Then the homotopy functionis,
H (  ,  )  F (  )  (1   ) F (  0 ).
The start values were computed by the 3 point solution, because homotopy
solution technique proved to be very robust to ensure convergency.
Solution with different start values
This table illustrates that the values calculated from different triplets can
differ significantly depending on their geometrical configurations.

[″]

[″]

[″]
X0
[m]
Y0
[m]
Z0
[m]
k1
[ppm]
k2
[ppm]
k3
[ppm]
start values - a
8
-8
-8
-243
-227
+337
0
0
0
start values - b
+2
-3
-19
0
0
0
-1
-118
-18
solution
3
-47
14
-2299
527
2143
270
-189
-245
The parameters in this table are transformed into their traditional geodetic
forms, representing the rotation matrix with three rotation angles (, , ) in
seconds, instead of a, b, c, and the deviation of the scale parameters from
one (ki = mi – 1 = 1/I - 1) in ppm (part per million), instead the inverse scale
parameters (1, 2, 3).
Trajectories of the different solutions
Solution of the linear homotopy model means the solution of a system of
nonlinear ordinary differential equations. Figures above show the
trajectories (the homotopy paths) of the variable (parameter) a on the
complex plain, in case of these two different initial values.
We have got the same result independently on the initial values.
Comparing the different methods
Method
Computation time [second]
Start values - a
Start values - b
Newton-Raphson
Fails to converge
Converging to wrong a solution
Newton-Krylov
singularity
3.5
Newton-Homotopy
0.72
0.73
The traditional Newton-Raphson method failed with both initial values, its
modification, the Newton-Krylov method had singularity in the first case,
and it was slowed down by the increasing number of the iteration
ensuring acceptable precision in the second case.
Using homotopy we do not need proper initial value, because this
method is more robust than local methods, enlarges the domain of
convergence and even faster than the traditional Newton’s type methods.
Thank you for your attention!