COMPLEX ROOF STRUCTU RES GENERALIZED REPRESEN TATION [614809]

COMPLEX ROOF STRUCTU RES GENERALIZED REPRESEN TATION
ESTABLISHED BY APPRO XIMATING THE MATHEMA TICAL SHAPE
USING THE LEAST -SQUARES METHOD
ERSILIA ONIGA –Teaching assistant, Eng. , Technical University „Gheorghe Asachi” of Iasi, Faculty of
Hydrotechnical Engineering , Geodesy and Environmental Engineering, Department of Terrestrial Measurements and
Cadastre , PhD candidate , Technical University of Civil Engineering, Faculty of Geodesy, Bucharest
e-mail: [anonimizat]
CONSTANTIN CHIRILA –Lecturer, PhD Eng. , Technical University „Gheorghe Asachi” of Iasi, Faculty of
Hydrotechnical Engineering , Geodesy and Environmental Engineering, Department of Terrestrial Measurements and
Cadastre
e-mail: [anonimizat]
Abstract : Representing a complex roof structure is useful both for creating the 3D building model
and also for the evaluation process of the building setting out accuracy , in a rigoro us analytical
framework regarding the spatial position of the characteristic points. This article presents the
methodology for determining the most probable mathematical form which approximates the roof
structure, based on the topographic measurements of a large number of points, evenly distributed
over the roof ‘surface . Applying the least squares method still represents an effective tool used to
evaluate the precision of the obtained results, in accordance with the principles of mathematical
statistics. In the chosen example we are describing the steps for modelling the roof surface of the
Dean's Office building from the Faculty of “Hydrotechnics, Geodesy and Environme ntal
Engineering” from Iasi, starting with the field measurements and aiming to obtain the appropriate
geometric shape , in this case a hyperbolic paraboloid.
Keywords: least-squares, modelling , hyperbolic paraboloid, evaluation
1. Introduction
For this case study the dean's office building from the “Faculty of Hydrotechnics, Geodesy and
Environmental Engineering” – „Gheorghe Asachi” University of Iasi, Romania, has been chosen,
which is a complex shape building, the roof structure having a shape of a hyperbolic parabo loid
(Figure 1).

(a) (b)

Fig. 1 –The study zone (a) Digital image representing the principal facade (b) Digital image representing the
secondary facade
The topographic measurements have been made with the Leica TCR 405 total station, having a
2mm+2ppm precision for distance measuring in Fine -Mode (IR). The topographic data
processing made use of specialized software, while the spatial model ling was based on
mathematical computations using the MATLAB software and graphical representations in the
AutoCAD design environment.

2. The Design, Execution and Compensation for the Topographic Basic Network, in a Local
Coordinate System.
2.1. Total S tation Surveying
On the roofs of three buildings placed in the immediate vicinity of the case study building there
were chosen three station points defining the topographic basic network, required for the detailed
levelling survey of the characteristic points for the build ing's roof . The chosen coordinate system
for defining the basic network points is a three -dimensional local system with its origin (O)
having a given set of coordinates (1000m, 1000m, 100m), the OZ axis oriented towards the
zenith, the OY axis in the horiz ontal plane on the 1 -2 segment direction, and OX axis
perpendicularly oriented on the OY axis and rotated anti -clockwise (Figure 2).

Fig. 2 – Topographic basic network in the local three -dimensional coordinate system
In the newly formed triangle, measurements were made for slant distances, vertical angles and
azimuthal directions for every station point. Following up, same measurements were made for a
number of 118 detail points, uniformly distributed on the considered roof surface (Figure 3).

Fig. 3 – Surveying measurements to acquire the detail points

2.2. Adjustm ent Computations for the Topographic Basic Network Using the Conditional
Observations Method.
Due the inherent measuring errors, after the station processing step for the acquired observations,
the topographic network defined by the three station points is geometrically adjust contains by
imposing several internal conditions, using both a horizontal and vertical point of view. As a
consequenc e, four equations are obtained for the geometric shape of the spatial triangle and for
the pole or line segment, which in their linearized form are:
1 2 3 1 0    v v v w ,
where:
1 200o o o gw ( )      (1)
4 5 6 7 1 2 2 3 3 1 1 2 1 2
0 8 9 2 2 3 2 3 3 1 3 1o o o o ocos Z v cos Z v cos Z v L sin Z v
o o o oL sin Z v L sin Z v w ,       
      
where:
3
2
1oo
ii
iw L cos Z

(2)
121 3 4 1 2 1 2 1 2 1 2 2
0 5 7 8 3 2 3 1 2 1 2 2 3 2 3o oo sin Z cos sin o o o o o oL cos v L sin Z sin v sin Z vo o osin (sin ) sin
osin o o o o osin Z v L cos Z v L cos Z v w ,osin
  

      
        

where:
3 2 3 2 3 1 2 1 2osin o o o ow L sin Z L sin Zosin
    (3)
122 3 4 1 2 1 2 1 2 1 2 2
0 6 7 9 43 1 1 2 1 2 3 1 3 1o oo sin Z cos sin o o o o o oL cos v L sin Z sin v sin Z vo o osin (sin ) sin
osin o o o o osin Z v L cos Z v cos Z L v w ,osin
  

   
        

where:
4 3 1 3 1 1 2 1 2osin o o o ow L sin Z L sin Zosin
    (4)
The conditional equation system is solved using matrix computation, by imposing a minimum
condition on the sum of the square values for the corrections, resulting the correction vector [1]:
1
1 1 1 1TT
n, n,r r, r, r,n n,r r,V A K where K ( A A ) W  
(5)
Based on the previously obtained corrections, the compensated values for the horizontal angles in
the triangle are calculate d, and also the vertical angles and the slant distances, and by making use
of these, the rectangular spatial coordinates ( X, Y, Z) for the topographic basic network points are
calculated, in the local system (Table 1).
Table 1
Compensating the topographic basic network
Point
station
Point
no. Triangle
angles
(g c cc) Slant
distances Vertical
angle s Elevation
differences Rectangular coordinates of the
triangle vertex
oL / Lii
(m)
oZ / Zii (m)
oZ / Z ( m )i j i j X (m) Y (m) Z (m)
1 2 αo=38.4364 117.328 103.1592 -5.820 1000.000 1000.000 100.000 3 α=38.4436 117.328 103.1597 -5.821
2 3 βo=38.2029 71.653 93.5282 7.272 1000.000 1117.183 94.179 1 β=38.2031 71.653 93.5274 7.273
3 1 γo=123.3637 70.907 101.3015 -1.449 1040.255 1058.354 101.452 2 γ=123.3532 70.907 101.3034 -1.452

3. Determining the Spatial Coordinates for the Detail Points on the Building's Roof Based
on the Local Basic Network.
The new detail points, situated on the building's roof in the considered case study, have been
surveyed from all the three station points in the topographic basic network. In a first phase,
intermediate sets of coordinates ( Xo, Yo, Zo) have been computed for the detail points, as the
arithmetic mean of the spatial coordinates received from each station point. In the second phase,
the compensated spatial coordinates were determined by applying the functional model for
weighted indirect measurements.
For eve ry new point, the correction equations system has been built, written for every azimuthal
direction, zenithal angle and measured slant distance. The number of the system's unknown
variables results from the three corrections (dx P, dy P and dz P) for the new point coordinates and
from the three corrections (dz 1, dz 2 and dz 3) for the medium orientation angle corresponding to
each station point:
1 1 1 1 1
1 12 12
1 13 13P P P P P P dz a dx b dy l v
dz l v
dz l v    
   
   

2 2 2 2 2
2 21 21
2 23 23P P P P P P dz a dx b dy l v
dz l v
dz l v    
   
   

3 3 3 3 3
3 31 31
3 32 32P P P P P P dz a dx b dy l v
dz l v
dz l v    
   
   

(6)
1 1 1 1
2 2 2 2
3 3 3 3P P P P P P
P P P P P P
P P P P P Pa dx b dy l v
a dx b dy l v
a dx b dy l v  
  
  
(7)
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3P P P P P P P P
P P P P P P P P
P P P P P P P Pc dx d dy e dz l v
c dx d dy e dz l v
c dx d dy e dz l v   
   
   
(8)

The weights for each equation are determined using the mean square errors for the measured
values on the terrain: directions, zenithal angles and slant distances [2]:
2 2 21 1 1
dZ
Z dp ; p ; p
s ss
  

(9)
The free terms are determined by subtracting the provisional values for these elements calculated
from provisional coordinates, and the corresponding values measured on terrain:
o o o * o * o
i P i P i P i i P i P i P i P i P i Pl ( ) Z ; l L L ;l Z Z              
(10)
The corrections equations system can be expressed using the matrix syntax, as shown next [1]:
1 1 1 r,n n, r, r,B X L V ,
rr with P
(11) the correction
equations for the
slant distances
the correction
equations for the
azimuthal directions

the correction
equations for the
zenithal angles

or detailed:
11
1 1 1 1
22
2 2 2 2
33
3 3 3 3
11 0 0 0
1 0 0 0 0 0
1 0 0 0 0 0
0 1 0 0
0 1 0 0 0 0
0 1 0 0 0 0
0 0 1 0
0 0 1 0 0 0
0 0 1 0 0 0
000oo
PP
o o o o
P P P P
oo
PP
o o o o
P P P P
oo
PP
o o o o
P P P P
o
Psin cos
L sin Z L sin Z
sin cos
L sin Z L sin Z
sin cos
L sin Z L sin Z
sin Z cos


   

   

   









1 1 1 1
2 2 2 2 2
3 3 3 3 3
1 1 1 1 1
1 1 1
22
2000
000
000
000o o o o
P P P P
o o o o o
P P P P P
o o o o o
P P P P P
o o o o o
P P P P P
o o o
P P P
oo
PP
o
Psin Z sin cos Z
sin Z cos sin Z sin cos Z
sin Z cos sin Z sin cos Z
cos cos Z sin cos Z sin Z
L L L
cos cos Z si
L



   
    
    
    
  

1
1
2
3
2 2 2
22
3 3 3 3 3
3 3 3000P
P
P
o o o
P P P
oo
PP
o o o o o
P P P P P
o o o
P P Pl
dz
dz
dz
dx
dy
dz
n cos Z sin Z
LL
cos cos Z sin cos Z sin Z
L L L
  

    
  











 
 
 
  
 
 
  






 


 


1
12 12
13 13
22
22
33PP
PP
PP
PPv
lv
lv
lv
… …
… …
… …
lv
lv   
   
   
   
   
   
      
   
   
   
   
   
      

The system's unknowns result from solving the normal equations, having the structure [1]:
11
1 1 1
nnT T T
n nr rr rn nr rr r nn nr rr r
NX ( B P B ) B P L N B P L 
(13)
The determination precision of each point coordinates from the unknown matrix is expressed
with a mean square error [1]:
    
P P P P P P x x x y y y z z zs Q ; s Q ; s Q
[ pvv]in which :rn
(14)
The maximum positioning error for the new points has been encountered for the number 73 point,
being of 0.040m in plane, 0.018m on the vertical axis, respectively 0.044m from the spatial point
of view. The minimum positioning error for the new points occurred for the number 102 point,
being of 0.006m in plane, 0.003m on the ver tical axis, respectively 0.007m from the spatial point
of view. These precision elements serve on the next step to establish the weights for the indirect
measurements functional model to approximate the mathematical form for the building's roof,
using the on terrain measured characteristic points.
(12)

4. Approximating the Roof's Structure with the Most Probable Geometric Form of a
Hyperbolic Paraboloid.
In order to approximate the roof's structure with the most probable geometric form of a
hyperbolic paraboloi d, the following computation steps were made :
4.1. Approximating the Main Axis for the Hyperbolic Paraboloid Using the Least Squares
Method
To approximate the most probable line represented by the plane image of the parabola in the XOZ
plane of the hyperbo lic paraboloid, the starting point is the general line equation, with the form:
y mx n
(15)
The equation can be written for the v x and v y deviations of the point coordinates obtained by
measurements, related to the most probable line, by using the default function of unknown
variables (m,n), like:
( , , , ) ( ) 0xi yi i yi i xi F m n v v y v m x v n      
(16)
By linearizing the function and conveniently grouping the terms, the coordinate corrections
equations system results, with the form [2]:
11
221 1 1
2 2 2()
()
…………………………………. ………………………..
()
r r ro x y o o
o x y o o
o x y o r o rm v v x dm dn m x n y
m v v x dm dn m x n y
m v v x dm dn m x n y      
      

      

(17)
while using matrices:
1 2 2 1 1 r,n n, r, , r, n,nB V J X K , with cofactor matrix Q (18)
where:
1 0 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 0 0 1o
o
o r,n
om …
m …
m … B,
… … … … … … … … …
m

  


 
1
1
2
2 1
r
rx
y
x
n, y
x
yv
v
v
V, v
…
v
v









1
2
2 31
1
1
1r,
rx
x
J, x
… …
x





21,dmX,dn
11
22
33 1oo
oo
oo r,
o r o r( m x n y )
( m x n y )
( m x n y ) K,
…
( m x n y )  
  
  


  
1
12
2
2
20 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0r
rx
y
n,n
x
ys …
s …
Q … … … … … …
… s
… s









The weights in the corrections equations were considered to be inversely proportional with the
mean square errors for the compensated values of the dxPi, dy Pi unknowns, from the phase of
determining the coordinates for the new points using the indirect measurements method.
By solving the equation system, the compensated coordinates of the points situated on the line are
resulting, while the line parameters are also obtained by com pensation:
0.520; 461.610      oo m m dm n n dn m
(19)
The most probable line (the image in plane of the parabola from the XOZ plane of the hyperbolic
paraboloid) which approximates the 12 points previously computed at step 3, has the following
shape (Figure 4).

Fig. 4 – The most probable line – the image in plane of the parabola from the XOZ plane of the hyperbolic
paraboloid
4.2. Approximating the Parabola from the XOZ Plane of the Paraboloid Using the Least
Squares Method
To approximate the parabola (the “saddl e”) from the XOZ plane of the hyperbolic paraboloid, the
starting point is the parabola equation, of the form:
2() Z F x AX BX C   
(20)
The corrections equations in linear form are [2]:
2
1 1 1
2
2 2 2
20
0
………………………………
0r r rAX BX C Z
AX BX C Z
AX BX C Z   
   

   
(21)
and as matrices:
1 1 1 r,n n, r, r, r,rB X L V , with P
where :
(22)
1
12211 1
22 2 2211
2
21001
100
100
rz
z r,n n, r, r,r
rrr
z…XX Z ( s )AZ XX( s ) … B ; X B ; L ; P… … … …C … … … …ZXX… ( s )
              

(23)

The weights were considered to be inversely proportional with the mean square errors
corresponding to the compensated values for the unknowns dzPi from the phase of determining
the coordinates of the new points through indirect measurements method.

The unknowns of the (A, B, C) system represent the defining parameters for the parabola, which
result by solving with matrix resolution the normal equation system:
A=0.0190; B= -55.3474; C= -40478.2029
4.3. Computing the Necessary Parameters to Transform the Detail Points Coordinates
From the Local System in the Paraboloid System
To establish the paraboloid system origin, the minimum value for the vertical coordinate of the
points belonging to the approximated parabola is being chosen, which means enforcing the first
order partial derivative of the following function to be equal to zero:
2min 2 02BZ AX BX C im AX B XA        
(24)
The local system coordinates for the parabola's vertex, after es tablishing its position on the
previously determined line by calculating the Y coordinate, are:
1029.132m; Y=1092.216m; Z=90.095mX

The most probable parabola which approximates the 12 points, with their coordinates on the X
axis being previously calculated at step 4.1, has the following shape (Figure 5).

Fig. 5 – The most probable parabola (the “saddle”) from the XOZ plane of the hyperbolic paraboloid and its vertex P
To transform the detail points coordinates from the local system in the paraboloid system, two
operations are made: of translation and rotation [3].
The previously determined point P in the local coordinate system, representing the vertex of the
hyperbolic paraboloid, will be translated in the origin of the system of axes (0,0 ,0), together with
the detail points on the roof surface. The rotation takes place around the OZ axis, anti -clockwise,
while the rotation angle is the complement for the α angle.
1 1 1 1 1 1
2 2 2 2 2 2
3 3 3 3 3 30
0
0 0 1L L L P P P
L L L P P P
L L L P P P
rL rL rL r P r P r PX Y Z X X Y Y Z Z
cos ' sin ' X Y Z X X Y Y Z Z
sin ' cos ' ( m ), X Y Z X X Y Y Z Z
… … … … … …
X Y Z X X Y Y Z Z
      
                        
         
: ( )
' 100gwhere arctg m


(25)
4.4. Approximating the Geometric Shape of the Hyperbolic Paraboloid Using the Least
Squares Method
To approximate the hyperbolic paraboloid by applying the least squares principle, the starting
point is the general equation for the hyperbolic paraboloid:

22
222,xyzab (26)
where: a,b are the parameters defining the hyperbolic paraboloid, and (x,y,z) are the coordinates
for the paraboloid surface in its own coordinate system.
The function with the “a” and “b” unknowns, with the form:
22
22( , ) 2 0xyF a b zab   
(27)
is linearized by a Taylor series expansio n around two provisional values (a0, b0):
22
22
3 3 2 222( , ) ( , ) 0, : ( , ) 2o o o o
ooxyF a b F a b x da y db where F a b za b a b      
(28)
The correction equations, linearly expressed, are:
22
22 11
1 1 13 3 2 2
00
22
22
3 3 2 2
002220
…………………………………. ………………………..
2220oo
rr
r r r
ooxyx da y db za b a b
xyx da y db za b a b     


     

(29)
and as matrix representation:
1 1 1 r,n n, r, r,B X L V , with Pr,r
(30)
where:
22
1133
00
22
2233
00
22
33
0022
22
22r,n
rrxy
ab
xy
B, ab
… …
xy
ab







22
11122
00
22
222 221 00
22
22
002
2
2r,
rrrxyz
ab
xyzL, ab
…
xyz
ab







21,daX,db
1
2
1r,
rv
vV…
v





The weight matrix has on its main diagonal the inverses of the squares of the spatial errors
corresponding to the newly determined points, and all the other elements equal to zero:

1
22
2
21 0 0 0
0 1 0 0
0 0 0 1
rt
t
r,r
t/ s …
/ s …
P… … … … …
… / s








At the end of the compensation with the weighted indirect measurements method, the
compensated values for the hyperbolic para boloid parameters are obtained:
0
0a a da 5.116(m)b b db 5.594                 

Using these parameters and the intervals for the ( X, Y, Z) coordinates, the hyperbolic paraboloid
is automatically generated, with the needed coordinates computed in the MATLAB environment.
The resulting paraboloid represents the roof surface of the Dean's Office building from the
Faculty of “ Hydrotechnical Engineering , Geodesy and Environmental Engineering” from Iași
(Figure 6).

(a) (b)
Fig. 6 – (a) The best fitting hyperbolic paraboloid shape, modelled in 3D space , (b) The roof surface modelled in 3D
using the detail points (red colour ) and the best fitting hyperbolic paraboloid shape (green colour )
5. Conclusions
In this paper, a technique for least squares fitting of a complex roof structure with a hyperbolic
paraboloid shape, from 3D data, has been presented. Also the steps for modelling a roof surface
have been described , starting with the field measurements, continuing with the calculation of the
best fitting hyperbolic paraboloid parameters and ending with the modelling part using the
MATAB program and the AutoCAD softwar e.
The compensations for the measurements using the functional model of indirect or conditional
observations, based on the least squares principle, accurately reflect the real world situation, by
considering all the measured elements in the 3D space and al so by correctly establishing the
weights to fit the obtained results as close as possible to the on -terrain positioning.
The linear, curve and spatial elements approximations were implemented also using the least
squares method, in this case having to adap t the local coordinate system to the coordinate system
of the studied geometric figure.
Finally, the mathematical modelling of the geometric shape of the roof as a hyperbolic paraboloid
can be successfully integrated in the 3D spatial model of the Dean's O ffice building from the
Faculty of “ Hydrotechnical Engineering , Geodesy and Environmental Engineering” from Iasi.
References
[1] Nistor Gh., Teoria prelucrarii masuratorilor geodezice, Universitatea Tehnică „Gheorghe Asachi”, Iasi, 1996
[2] Charles D. Ghilani, Paul R. Wolf – Adjustment computations. Spatial Data analysis , Fourth edition, John
Wiley&Sons Inc., U.S.A., 2006.
[3] Bofu C., Chirilă C. – Sisteme Informaționale Geografice. Cartografierea și editarea hărților , Ed. Tehnopress,
Iași, 2007
[4] Min D., Timothy S. N., Chunguang Cao -Least -squares -based fitting of paraboloids, Science Direct, Pattern
Recognition 40 (2007) 504 -515