z - Electronic Journal of Polish Agricultural Universities

z - Electronic Journal of Polish Agricultural Universities

Electronic Journal of Polish Agricultural Universities is the very first Polish scientific journal published exclusively on the Internet, founded
on January 1, 1998 by the following agricultural universities and higher schools of agriculture: University of Technology and Agriculture of
Bydgoszcz, Agricultural University of Cracow, Agricultural University of Lublin, Agricultural University of Poznan, Higher School of
Agriculture and Teacher Training Siedlce, Agricultural University of Szczecin, and Agricultural University of Wroclaw.
ELECTRONIC
JOURNAL
OF POLISH
AGRICULTURAL
UNIVERSITIES
2003
Volume 6
Issue 2
Series
GEODESY AND
CARTOGRAPHY
Copyright © Wydawnictwo Akademii Rolniczej we Wroclawiu, ISSN 1505-0297
Jasińska E., Łacina W., Preweda E., Żygieło P. A MODIFIED ALGORYTHM OF DETERMINING THE SHAPE OF SHELL OBJECTS
USING THE METHOD OF CONICAL INTERSECTION Electronic Journal of Polish Agricultural Universities, Geodesy and Cartography,
Volume 6, Issue 2. Available Online http://www.ejpau.media.pl
A MODIFIED ALGORYTHM OF DETERMINING
THE SHAPE OF SHELL OBJECTS
USING THE METHOD
OF CONICAL INTERSECTION
Elżbieta Jasińska, Wojciech Łacina, Edward Preweda, Piotr Żygieło
Departament of Land Information, AGH University of Science and Technology, Cracow, Poland
ABSTRACT
PLANE INTERSECTIONS
DETERMINATION OF AN AXIS USING PROFESSOR KADAJ’S METHOD
AN EDGE ACCORDING TO A MODIFIED ALGORITHM
MULTIPLE PLANE INTERSECTIONS
APPROXIMATION OF PARAMETERS
CONCLUSIONS
REFERENCES
ABSTRACT
The category of geodetic calculations known as the method of conical intersections could be used for geodetic
stocktaking of lines and surfaces. This study presents modified algorithms of the method of conical intersections
which allow to improve calculations and to carry out a precise assessment of the accuracy of parameters which
are being determined.
PLANE INTERSECTIONS
Conical intersections were formulated and described in the studies [1-3]. Their characteristic feature is the course
of the process of approximation of model parameters in two stages. During the first stage one determines the axis
of an analysed object on the basis of conical intersection. This allows to determine the deflection of a structure.
Then co-ordinates of stations and observation are transformed to a system in which the axis is vertical. Thanks to
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that one finds the parameters of a shell in the canonical form. Finally, on the basis of determined coefficients of the
equation of a mathematical creation one determines deviations of measured points from the model.
The axis of an approximated surface is determined on the basis of plane intersection. A common idea of axis
determination is presented on figure 1.
Fig. 1. The axis formed by crossing planes of mean tangency directions
Description of verticality amounts to determination of parameters of axes of an object. In order to that one shall
find for each station a plane which comprises the axis of the structure and goes through a point of the station. On
the basis of a pencil of planes which is formed in this way one finds averaged parameters of the axis.
Lines which are created by averaged directions to tangency points measured from a given station lie, within the
limits of accuracy of a measurement, in one plane. Parameters of that plane will be determined using the least
square method, however, matching will make sense when the number of mean directions for every plane is
bigger than 2.
An equation of a plane derived from ith station has the following form:
A ⋅ (X − X i ) + B ⋅ (Y − Yi ) + C ⋅ (Z − Z i ) = 0
r
where the normal vector of a plane N = [A, B ,C ] and the vector [X − X i ,Y − Yi , Z − Z i ] are mutually orthogonal
while the point (X ,Y , Z ) is a tangency point and the point S i = [X i ,Yi , Z i ] is a station. Values A, B, C will be
unknown; they will be determined on the basis of known values αśr, ϕśr. Let’s form a vector of mean observation
r
r
direction l which will be unitary in a projection onto a plane 0XY l = [cos α śr , sin α śr , tg ϕ śr ] . Vectors
r
r
N and l should be mutually perpendicular. This condition results in a following equation:
A ⋅ cos α śr + B ⋅ sin α śr + C ⋅ tg ϕ śr = 0 ,
from which results an observation equation of the following type:
A ⋅ cos α śr + B ⋅ sin α śr + C ⋅ tg ϕ śr = v
(1)
r
It should be stated that the searched vector must be normalised, otherwise there are infinitely many vectors N
r
perpendicular to l . The simplest method to normalise is assuming that one of co-ordinates has the value 1,
however only such co-ordinate can be chosen which definitely is different from 0. Because it results from the
very assumption that the searched plane should be approximately vertical, C ≈ 0. Co-ordinates A and B remain,
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definitely one of them will be different from 0. αśr is the factor which determines which of them shall be
assumed as equal to 1.
If αśr ≈ 00 (180o) ⇒ A ≈ 0 ⇔ B ≠ 0 it is assumed that B = 1
If αśr ≈ 900 (270o) ⇒ B ≈ 0 ⇔ A ≠ 0 it is assumed that A = 1
Let’s assume, for example that B ≈ 0, it is assumed that A = 1, then the equation (1) shall have the following
form:
cos α i + B ⋅ sin α i + C ⋅ tg ϕ i = υ i
when
i ∈ { 1, 2, ..., n }
n : number of pairs of observations from a given station
When a matrix is formed:
 1 sin α 1 
 1 sin α 
2
A=
,
...
... 


 1 sin α n 
B
X=  ,
C 
 − cos α 1 
− cos α 
2
L=
 ... 


− cos α n 
one may solve the system Ax = L using the least square method. Using this method one will also determine
parameters of accuracy assessment, when σ A = 0 .
In a situation when B = 1 the whole reasoning is analogous. An equation of intersecting planes for every station
results from calculations:
A1 ⋅ (X − X 1 ) + B1 ⋅ (Y − Y1 ) + C1 ⋅ (Z − Z 1 ) = 0
A2 ⋅ (X − X 2 ) + B 2 ⋅ (Y − Y2 ) + C 2 ⋅ (Z − Z 2 ) = 0
...
As ⋅ (X − X s ) + B s ⋅ (Y − Ys ) + C s ⋅ (Z − Z s ) = 0
(2)
when
i ∈ { 1, 2, ..., s }
s : number of all stations
An equation of the axis is determined on the basis of the following equations:
X = ux ⋅t + X 0 ,
Y = u y ⋅ t + Y0 ,
Z = uz ⋅t + Z0
when the point P0 = (X 0 ,Y0 , Z 0 ) is any point of the line, however, to make its determination unique we will
assume that one of co-ordinates is 0. The axis is approximately vertical, therefore it definitely cuts the horizontal
plane, therefore it could be assumed that Z0 = 0.
[
r
The vector u = u z , u y , u z
] is a directional vector of the line. In order to make its determination unique, the
vector should be normalised. Because the vector defines the axis which is approximately vertical, it may be
assumed that uz = 1.
DETERMINATION OF AN AXIS USING PROFESSOR KADAJ’S METHOD
From among all planes described by the equation (2), one may determine their common edge, that is the axis of
an object. In order to do that for every pair of planes one determines an edge of cutting, however, for such planes
which intersect at a profitable angle. The best results are obtained when planes are approximately perpendicular.
Most often when choosing matching pairs one considers such planes whose dihedral angle belongs to the range
<300,1500>. For example a common axis of planes from stations S1 and S2 will be determined:
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X =
Y=
B (A X + B1Y1 + C1 Z i ) − B1 (A2 X 2 + B 2 Y2 + C 2 Z 2 )
B1C 2 − B 2 C1
Z+ 2 1 1
A1 B 2 − A2 B1
A1 B 2 − A2 B1
A (A X + B 2 Y2 + C 2 Z 2 ) − A2 (A1 X 1 + B1Y1 + C1 Z i )
C1 A2 − C 2 A1
Z+ 1 2 2
A1 B 2 − A2 B1
A1 B 2 − A2 B1
after a comparison with a parametric equation of the axis we obtain the searched parameters:
X0 =
Y0 =
B 2 (A1 X 1 + B1Y1 + C1 Z i ) − B1 (A2 X 2 + B 2 Y2 + C 2 Z 2 )
A1 B 2 − A2 B1
A1 (A2 X 2 + B 2 Y2 + C 2 Z 2 ) − A2 (A1 X 1 + B1Y1 + C1 Z i )
A1 B 2 − A2 B1
ux =
B1C 2 − B 2 C1
A1 B 2 − A2 B1
uy =
C1 A2 − C 2 A1
A1 B 2 − A2 B1
Then the determined parameters of an axis from particular pairs of planes are averaged to obtain in the end a
single equation of the axis. This may be done in two ways. First, one may assume the average of values X0, Y0,
ux, uy, or a weighted average, however, then one should know mean errors for every independently determined
set of values X0, Y0, ux, uy. In this study they will be defined as errors of the function F(Ai,Bi,Ci,Aj,Bj,Cj) having
previously calculated errors of values A, B, C.
σF =
f T Cov( Ai Bi Ci A j B j C j ) f
where
 ∂F
fT =
 ∂Ai
∂F
∂Bi
∂F
∂Ci
∂F
∂A j
∂F
∂B j
∂F 

∂C j 
0
Cov( Ai Bi C i )
Cov( Ai Bi Ci A j B j C j ) = 
Cov
(
A
0
j B jC j


)
Agreeing of results from particular pairs of planes causes that difficulty that every plane is used to determine a
line many limes. Therefore, results depend on each other which is not taken into account by the method.
AN EDGE ACCORDING TO A MODIFIED ALGORITHM
The method proposed by the authors is an exact method which requires a smaller number of calculations and results
in more accurate results. It is not necessary to choose planes because all will be taken into account in the process of
compensation and only once.
The first stage of calculation involves normalisation of planes which will make it possible to compare equations
formed later:
Ai =
Ai
Ai 2 + Bi 2 + C i 2
, Bi =
Bi
Ai 2 + Bi 2 + C i 2
, Ci =
Ci
Ai 2 + Bi 2 + Ci 2
After taking into account unit normal vectors in an equation (2) and moving to equations of corrections you will
get:
A1 ⋅ (X − X 1 ) + B1 ⋅ (Y − Y1 ) + C1 ⋅ (Z − Z 1 ) = v1
A2 ⋅ (X − X 2 ) + B 2 ⋅ (Y − Y2 ) + C 2 ⋅ (Z − Z 2 ) = v 2
...
(3)
As ⋅ (X − X s ) + B s ⋅ (Y − Ys ) + C s ⋅ (Z − Z s ) = v s
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The gist of the method involves cutting of a pencil of planes with an additional plane Z = 0. In this way planes
will be changed into lines and the whole problem will amount to finding a common point for all lines. When you
put Z = 0 to the equation (3) you will get:
A1 ⋅ (X − X 1 ) + B1 ⋅ (Y − Y1 ) − C1 ⋅ Z 1 = v1
A2 ⋅ (X − X 2 ) + B 2 ⋅ (Y − Y2 ) − C 2 ⋅ Z 2 = v 2
...
As ⋅ (X − X s ) + B s ⋅ (Y − Ys ) − C s ⋅ Z s = v s
The above system of equations may be presented as a matrix AX=L, where
 A1

A
A= 2
 ...

 As
B1 

X 
B2 
, X =  0 ,

...
 Y0 

B s 
 C1 Z 1 + A1 X 1 + B1Y1 


C 2 Z 2 + A2 X 2 + B 2 Y2 

L=


...


 C s Z s + As X s + B s Ys 
Solving of the system using the least square method shall allow to determine an estimator of the searched point
of the line X0, Y0 and its assessment of accuracy in a form of a covariance matrix Cov (X0, Y0).
Similarly to co-ordinates X0, Y0 you will find co-ordinates of a directional vector of the line ux, uy. To do this in the
equation (3) the following substitution will be done Z = uz = 1, getting:
A1 ⋅ (X − X 1 ) + B1 ⋅ (Y − Y1 ) + C1 ⋅ (1 − Z 1 ) = v1
A2 ⋅ (X − X 2 ) + B 2 ⋅ (Y − Y2 ) + C 2 ⋅ (1 − Z 2 ) = v 2
...
As ⋅ (X − X s ) + B s ⋅ (Y − Ys ) + C s ⋅ (1 − Z s ) = v s
This system may be written in the following form:
 A1

A
A= 2
 ...

 As
 − C1 ⋅ (1 − Z 1 ) + A1 X 1 + B1Y1 


− C 2 ⋅ (1 − Z 2 ) + A2 X 2 + B 2 Y2 

L=


...


 − C s ⋅ (1 − Z s ) + As X s + B s Ys 
B1 

X 
B2 
, X =  1 ,

...
 Y1 

B s 
A solution made using the least square method will allow to determine the point X1, Y1 and a covariance matrix
Cov (X1, Y1).
r
Finally, co-ordinates of the searched vector u will be calculated in the following way:
(
u x = ( X 1 − X 0 ) ± Cov Xˆ 0 , Yˆ0 ,
(
u y = (Y1 − Y0 ) ± Cov Xˆ 0 , Yˆ0 ,
)
)
1,1
2,2
(
+ Cov Xˆ 1 , Yˆ1 ,
(
+ Cov Xˆ 1 , Yˆ1 ,
)
1,1
)
2,2
MULTIPLE PLANE INTERSECTIONS
This method allows for taking into account all observations and simultaneous approximation of axis parameters.
It is assumed that the axis will be determined as a line. Variants are also possible, e.g. approximation of an axis
with a polynomial, however they will not be discussed in this study.
[
]
Let’s make the point P0 = (xo , y o , z o ) a searched point of the line p and the vector u = u x ,u y ,u z a directional
vector of the line p. Let’s form a vector Po S i = [xi − xo , y i − y o , z i − z o ] and a vector of observation direction j
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[
]
from the station Si eij = u ij , vij , wij . Vectors u , Po S i , eij lie on one plane, therefore a scalar triple product of
those vectors is equal to zero:
(u × P S )⋅ e
o i
ij
=0
After substitution of co-ordinates:
ux
xi − x0
u ij
uy
yi − y0
vij
uz
zi − z0 = 0
wij
On the basis of the above relation an observation equation is formed:
ux
Fij = xi − x 0
u ij
uy
yi − y0
vij
uz
z i − z 0 = δ ij
wij
(4)
where
i ∈ { 1, 2, ..., s }
j ∈ { 1, 2, ..., ri }
s : number of stations
ri : number of observations from the station i
The equation (4) has six unknowns: x0, y0, z0, ux, uy, uz , and four of them are independent. The point
[
]
P0 = (xi , y i , z i ) is any point of the line and the vector U = u x , u y , u z can be normalised in any way. The line p
is approximately vertical, so it crosses any horizontal plane (e.g. a plane with the height z = 0), therefore, it could
be assumed that z0 = 0. Also from the condition of approximated verticality of the axis it is determined that
uz = 1. When these relations are taken into account, the equation (4) shall look in the following way:
(
)
(
)
− y o u ij + xo vij + (u x y 0 − u y xo )wij = δ ij
Fij = u x z i vij − y i wij + u y xi wij − z i u ij + y i u ij − xi vij +
Before the beginning of calculations the function Fij should be linearised
Fij = Fij 0 + dFij = Fij 0 +
∂Fij
∂u x
∂Fij
∂u y
∂Fij
∂u x
du x +
∂Fij
∂u y
du y +
∂Fij
∂x0
dx0 +
∂Fij
∂y 0
dy 0
= wij ( yi − y 0 ) − vij (z i − z 0 )
= u ij (z i − z 0 ) − wij (xi − x0 )
∂Fij
∂x0
∂Fij
∂y 0
= u y wij − u z vij
= u z u ij − u x wij
PRINT
An approximate value of the function will be calculated for approximate unknowns
(
)
(
)
Fij 0 = u~x z i vij − y i wij + u~ y xi wij − z i u ij + y i u ij − ~
xi vij +
~
~
~
~
~
~
− y o u ij + xo vij + u x y 0 − u y x o wij
(
)
Because the axis is approximately vertical:
u~x = u~ y = 0
x0 , ~y 0 will be determined from a forward intersection:
Approximated co-ordinates ~
y − y1 + x1 tg α1 − x 2 tg α 2
~
x0 = 2
,
tg α1 − tg α 2
~y = x 2 − x1 + y1 ctg α1 − y 2 ctg α 2
0
ctg α1 − ctg α 2
Finally, a matrix system of linear equations Ax = L can be built, where:
 ∂F11
 ∂u
 x
 ∂F12
A =  ∂u x

 ...
 ∂Fsr
 ∂u
 x
∂F11
∂u y
∂F12
∂u y
...
∂Fsr
∂u y
∂F11
∂x 0
∂F12
∂x 0
...
∂Fsr
∂x 0
∂F11 
∂y 0 
∂F12 
∂y 0  ,

... 
∂Fsr 
∂y 0 
 du x 
− F110 
du 
− F 
X =  y  , L =  120 
 dx 0 
 ... 




 dy 0 
− Fsr 0 
In result of solving of this system using the least square method, estimators of unknowns and a covariance matrix
are obtained for estimated parameters.
Deflection κ and the azimuth of deflection ϑ are determined according to the relation:
 u 2 +u 2
 x
y
κ = arctg 
u
z



 uy

 , ϑ = arctg 

 ux


.


An analysis of accuracy of deflection and of the azimuth of deflection can be carried out on the basis of the right
of determination of variations for correlated values. Finally
σκ 2 =
1
( u x 2 + u y 2 )( 1 + u x 2
σϑ =
( ux
2
T
(
− u y   V (u x )
2 2  u  cov u u
x y
+ uy )  x  
1
2
u x   V (u x )
  
+ u y 2 ) 2 u y  cov u x u y
T
(
)
)
( )
( )
cov u x u y  u x 
V u y  u y 
( )
( )
cov u x u y  − u y 
V u y   u x 
PRINT
APPROXIMATION OF PARAMETERS
In effect of realisation of algorithms presented here one can determine deflection of the axis κ and an azimuth of
deflection ϑ and the point in which the axis cuts through the horizontal axis.
Transformation
The next stage of calculation is determination of such a co-ordinate system where the axis of an object is
vertical. In order to do that one shall form a transformation matrix and transform co-ordinates of stations and
observations.
Forming of a transformation matrix K
K = Aυ A-κ A-υ
where
Aυ : a matrix of a revolution around an axis by angle υ
A-κ : a matrix of a revolution around an axis by angle -κ
A-υ : a matrix of a revolution around an axis by angle -υ
cos ϑ
K =  sin ϑ
 0
− sin ϑ 0 cos κ
cos ϑ 0  0
0
1  sin κ
0 − sin κ   cos ϑ sin ϑ 0
1
0  − sin ϑ cos ϑ 0
0 cos κ   0
0
1
 cos 2 ϑ (cos κ − 1) + 1 sin ϑ cos ϑ(cos κ − 1 ) − sin κ cos ϑ 


K =  sin ϑ cos ϑ(cos κ − 1 ) sin 2 ϑ (cos κ − 1) + 1 − sin κ sin ϑ 


sin κ cos ϑ
sin κ sin ϑ
cos κ


Transformation of co-ordinates of stations Si
  xi   x 0  
 xi 
y  = K y  − y 
 i   0 
 i
z   0 
 z i 
 i   
When the axis of an object is vertical, the matrix K is unitary, therefore transformation of co-ordinates amounts
to translation by a vector - [x0 y 0 0] .
A vector transformation of observation eij
[
eij = cos α ij , sin α ij ,tg ϕ ij
]
 Lij 
cos α ij 




 M ij  = K  sin α ji 
 N ij 
 tg ϕ ij 




aij = arctg
ϕ ij = arctg
M ij
Lij
N ij
2
Lij + M ij 2
PRINT
When the axis of an object is vertical, observations do not change. Transformed values form observation sets
Oij = xi , y i , z i ,α ij ,ϕ ij where: i ∈ { 1, 2, ..., s }, j∈{1,2,..., ri}.
{
}
Approximation of parameters can be done using two methods described below.
Method 1
After a transformation of a system of equations the axis of a hyperboloid in a new system is vertical and the
surface can be described by a canonical equation
f ( x, y , z ) =
x2
a2
+
y2
b2
−
(z − z 0 )2
c2
−1 = 0
where a, b, c are geometric parameters of a shell and z0 determines the height of the centre of symmetry in a coordinate system.
Let’s make the point P = (x , y , z ) a point of tangency of the axis of an aiming line from a station
S i = (xi , y i , z i ) and a model surface of a hyperboloid.
Let’s form a vector L = [x − xi , y − yi , z − z i ] which lies on the axis of the aiming line and a normal vector to the
r
surface N = f x , f y , f z in the point P:
[
]
fx =
∂f
2x
,
=
∂x a 2
fy =
∂f 2 y
,
=
∂y b 2
fz =
2(z − z 0 )
∂f
=−
∂z
c2
r r
Vectors L , N are perpendicular, therefore:
r r
z−z
x
y
L ⋅ N = (x − xi ) 2 + (y − y i ) 2 − (z − z i ) 2 0
a
b
c
x2
a
2
+
y2
b
2
(z − z 0 )2
−
c
2
−
xxi
a
2
−
yy i
b
2
+
(z − z 0 )(zi − z 0 ) = 0
c2
xxi yy i (z − z 0 )(z i − z 0 )
+ 2 −
=1
a2
b
c2
The condition which describes aiming axes in combination with observations and known co-ordinates of stations
is as follows:
x = l cos α + xi , y = l sin α + y i ,
z = l tg α + z i
After putting the above equations in a canonical equation of a hyperboloid and considering the condition for a
rotary hyperboloid, that is the condition a =b, the following will be obtained:
(l cos α + xi )2 + (l sin α + yi )2 − (l tg ϕ + z i − z 0 )2
a2
(l cos α + xi ) xi + (l sin α + yi ) yi
a2
−
=1
c2
(l tg ϕ + z i − z 0 )(z i − z 0 )
c2
=1
From the above equations, when l is eliminated, the equation which describes the surface of a hyperboloid and
observations will be obtained.
a2
c
2
z0 2 + a 2 +
a2
{
2a 2 z 0
c
2
[ z i − tg ϕ ( xi cos α − y i sin α )] −
a4
c
2
tg 2 ϕ +
}
[ z i − tg ϕ ( xi cos α − y i sin α )] 2 + tg 2 ϕ ( xi cos α − y i sin α ) 2 +
c2
− ( xi cos α − y i sin α ) 2 = 0
+
(5)
PRINT
Let’s introduce auxiliary variables:
ξ=
a2
c
2
zo 2 + a 2 ,
η=
2a 2 z 0
c
2
,
ζ =
Coefficients which are a function of observed elements
(
Fij = z i − tg ϕ ij xi cos α ij + y ij sin α ij
a2
(6)
c2
)
2
Gij = tg ϕ ij
(7)
(
H ij = xi cos α ij − y ij sin α ij
)2
After taking into account (6) and (7), you will get a simple notation of the equation (5)
(
)
η2

Φ = ξ + ηFij + ζ Fij 2 + Gij H ij + 
− ξζ Gij − H ij = 0
 4

Observation equations will have the following form:
(
)
η2

ξ + ηFij + ζ Fij 2 + Gij H ij + 
− ξζ Gij − H ij = δ ij
 4

The function Φ is not a linear function of searched parameters – before beginning of further calculations one stall
carry out linearisation
Φ = Φ 0 + dΦ = Φ 0 +
(
∂Φ
∂Φ
∂Φ
ζdζ
dξ +
dη y +
∂ξ
∂η
∂ξ
)

η 2
ξ 0 + η 0 Fij + ζ 0 Fij 2 + Gij H ij +  0 − ξ 0ζ 0 Gij − H ij +

 4


η


+ 1 − ζ 0 Gij dξ +  Fij + 0 Gij  dη + Fij 2 + Gij H ij − ξ 0 Gij dζ = δ ij
2


(
(
)
)
(8)
Approximate parameters can be determined by introduction to the equation (8) of an auxiliary unknown
λ0 =
η0 2
− ξ 0ζ 0
4
Approximate unknowns will be obtained from a solution of a system of linear equations
(
)
ξ 0 + η 0 Fij + ζ 0 Fij 2 + Gij H ij + λ0 Gij − H ij = 0
which can be written in a form of a matrix:
 ∂Φ 11
 ∂ξ

 ∂Φ 12
A =  ∂ξ
 ..
 ∂Φ
 sri
 ∂ξ
∂Φ 11
∂η
∂Φ 12
∂η
...
∂Φ sri
∂η
∂Φ 11
∂ζ
∂Φ 12
∂ζ
...
∂Φ sri




,



∂ζ 
 dξ 
X =  dη  ,
dζ 
− Φ 110 
− Φ 
120 
L=
 ... 


− Φ sr 0 
PRINT
Finally:
( )
Cov ( Xˆ )
Cov ( Xˆ )
ξ = (ξ 0 + d ξ ) ± Cov Xˆ
1,1
η = (η 0 + dη ) ±
2,2
ζ = (ζ 0 + d ζ ) ±
a= ξ −
3,3
η2
ξ
η2
η
, c=
−
, z0 =
2
4ζ
2ζ
ζ 4ζ
Method 2
The equation (5) without introduction of auxiliaries. Unknown parameters a, c, z 0 will be determined after the
following transformations:
Φ = a 2 [z 0 − z i + tg( xi cos α + y i sin α )]2 +
[
](
)
+ a 2 − ( xi cos α − y i sin α ) 2 ⋅ c 2 − a 2 tg 2 ϕ = 0
a 2 [z 0 − z i + tg( xi cos α + y i sin α )]2 +
[
](
)
+ a 2 − ( xi cos α − y i sin α ) 2 ⋅ c 2 − a 2 tg 2 ϕ = δ ij
Φ = Φ 0 + dΦ = Φ 0 +
∂Φ ij
∂a
{[
c
+ (~
∂Φ
∂Φ
∂Φ
da +
dc +
dz 0
∂a
∂c
∂z 0
]
= 2a~ z 0 − z i + tg( xi cos α ij + y i sin α ij ) 2 +
2
[
)
− a~ 2 tg 2 ϕ ij − tg 2 ϕ ij ⋅ a~ 2 − ( xi cos α ij − y i sin α ij ) 2
∂Φ ij
]}
= 2c~ [a~ 2 − ( xi cos α ij − y i sin α ij ) 2 ]
∂c
∂Φ ij
[
]
= 2a~ 2 ~z 0 − z i + tg( xi cos α ij + yi sin α ij )
∂z 0
[
]
Φ ij 0 = a~ 2 ~z 0 − z i + tg( xi cos α ij − y i sin α ij ) 2 +
+ a~ 2 − ( xi cos α ij − y i sin α ij ) 2 ⋅ c~ 2 − a~ 2 tg 2 ϕ ij
[
](
)
During calculations it is useful to adopt auxiliary marking
(
Fij = z i − tg ϕ ij xi cos α ij + yij sin α ij
)
2
Gij = tg ϕ ij
(
H ij = xi cos α ij − yij sin α ij
)2
PRINT
∂Φ ij
∂a
)2 + (~c 2 − a~ 2 Gij )− Gij ⋅ (a~ 2 − H ij )
(
= 2a~ z 0 − Fij
∂Φ ij
∂c
∂Φ ij
∂z 0
[
Φ ij 0 = a~ 2 ~z 0 − Fij
= 2c~ a~ 2 − H ij
[
]
[
]
= 2a~ 2 ~z 0 − Fij
]2 + [a~ 2 − H ij ]⋅ (~c 2 − a~ 2Gij )
Finally, a system of equations Ax = L will be built, where
 ∂Φ 11
 ∂a

 ∂Φ 12
A =  ∂a

 ∂Φ..
 sri
 ∂a

∂Φ 11
∂c
∂Φ 12
∂c
...
∂Φ sri
∂Φ 11
∂z 0
∂Φ 12
∂z 0
...
∂Φ sri
∂c
∂z 0




,





 dξ 
X =  dη  ,
dζ 
− Φ 110 
− Φ 
120 
L=
 ... 


− Φ sr 0 
In result of a solution of this system one will get estimators of parameters with a full assessment of accuracy.
Deviations from the model surface can be determined according to the formulas
d ij = ∆α ij l ij
d ij =
(
δ ij
2 H ij c 2 − a 2 Gij
)
CONCLUSION
The problem with determination of the shape and situation of second degree surfaces stall exists. This study is
devoted to the conical intersection method which, although little known, offers interesting solutions. Conical
intersections can be characterised by the fact that it is easy to measure them and to program calculations, it is
easy to measure the accuracy of calculated parameters and the fact that there are many stages allows to control
intermediate results. The algorithms presented here have features of universality. They could be used both to
analyse shapes of shell objects and, for example, to determine deviations of such objects as industrial chimneys,
masts and piles.
The algorithms that are presented in the study have been verified, checked on the basis of test objects and
on results from real measurements of chimney cooler situated in ‘Łaziska’ power plant.
REFERENCES
1.
2.
3.
4.
5.
Kadaj R.(1973), Methods of Geodetic Stocktaking of Structures with a Shape of One-Shell Rotary Hyperboloid.
Geodezja book 23, AGH, Kraków.
Kadaj R. (1972, Determination of the Situation and Shape of Some Spatial Geometric Forms Using the Methods of
Geodetic Conical Intersections. A dissertation, AGH, Kraków.
Kadaj R. (1975), Geodetic Methods of Measurement of Curvilinear Spatial Forms with Examples of Application in
Engineering, Rozprawy 3, Wydawnictwo Naukowe Politechniki Rzeszowskiej, Rzeszów.
Łacina P., Żygieło W (2001), An Analysis of the Method of Conical Intersections of Determination of the Surface of a
Hyperboloid Cooler. A mater degree dissertation supervised by E. Preweda. AGH, Kraków.
Preweda E. (1994), A System of Measurement, Calculation and Visualisation of Geometric Changes of Shelled Objects
with Quadratic Surface. A dissertation AGH, Kraków.
PRINT
Elżbieta Jasińska, Wojciech Łacina, Edward Preweda, Piotr Żygieło
Departament of Land Information
AGH University of Science and Technology
Al. Mickiewicza 30, 30-059 Cracow, Poland
e-mail: [email protected]
Responses to this article, comments are invited and should be submitted within three months of the publication
of the article. If accepted for publication, they will be published in the chapter headed ‘Discussions’ in each
series and hyperlinked to the article.
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