Construction ruled surfaces from scattered data by Kohonen network

glibdoadingAI and Robotics

Oct 20, 2013 (3 years and 9 months ago)

71 views

Construction ruled
surface
s

from scattered data by Kohonen network

Emőd Kovács
, emod@ektf.hu


Institute of Mathematics and Informatics, Eszterházy Károly College


Abstract
.

In this paper a new approach is presented, where first an artificial neural
networ
k is used to order the data and form a grid of control vertices with quadrilateral
topology, hence after this step the original, well
-
known free
-
form methods can be applied to
construct the
ruled surface.
Ruled surfaces and their special subclass, developa
ble surfaces are
well
-
known in classical geometry, but also have extended applications in computer geometry
and graphics as well as in computer aided manufacture. In these latter fields, however, the
standard way of descripting surfaces is the different ty
pes of spline
-
surfaces, like B
é
zier,
B
-
spline or NURB surface. Hence it is highly desired to describe and construct ruled and
developable surfaces as spline surface. Several methods have been developed to approximate
or interpolate such kind of surfaces
.

T
he standard definition of any type of spline surfaces
uses a quadrilateral control grid as input data. The main purpose of most of the known
methods is to construct this control grid. In our method the Kohonen neural network has been
applied in a preproces
sing step to produce a control grid from the original input data.

In our paper
original

input data structure consists of a set of given lines, called rulings.
In this case the description of these lines and the neural network by homogeneous coordinates
wil
l be
advantageous
, which is a well
-
known technique in projective geometry. The Plücker
coordinates of the projective lines will be also applied
.


References

1.

H
OSCHEK
,

J.,
S
CHWANECKE
,

U., Interpolation and approximation with ruled surfaces,
in: The Mathemat
ics of Surfaces VII. (ed.:Robert Cripps), Elsevier, pp. 213

231,
1988.

2.

B
ODDULURI
,

R.,
R
AVANI
,

B., Geometric Design and Fabrication of Developable
Surfaces, ASME Adv. Design Autom. 2, pp. 243

250, 1992.

3.

C
HEN
,

H.,
P
OTTMANN
,

H., Approximation by Ruled Surface
s, Journal of
Computational and Applied Mathematics, 102, pp. 143

156, 1999.

4.

F
ARIN
,

G., Curves and Surfaces for Computer Aided Geometric Design A Practical
Guide, Academic Press, 1996.

5.

F
REEMAN
,

J.,
S
KAPURA
,

D., Neural Networks; Algorithms, Applications an
d
Programming Techniques, Addison
-
Wesley, 1991.

6.

H
LAVATY
,

V., Differential line geometry, Nordhoff Ltd, Groningen, 1953.

7.

H
OFFMANN
,

M.,
K
OVÁCS
,

E., Constructing ruled B
-
spline surfaces by Kohonen
neutral network, Acta Acad. Paed. Agriensis, TOM. XXVII., Sect
io Matematicae, pp.
63

68, 2000.

8.

H
OFFMANN
,

M.,
V
ÁRADY
,

L., Free
-
form Surfaces for Scattered Data by Neural
Networks, Journal for Geometry and Graphics, Vol.2, No.1, pp. 1

6, 1998.

9.

K
OHONEN
,

T., Self
-
O
rganization
Maps
, Springer Verlag,
2001
.

10.

V
ÁRADY
,

L.,
H
OFFMANN
,

M.,

K
OVÁCS
,

E., Improved Free
-
form
Model
l
ing

of
Scattered Data by Dynamic Neural Networks, Journal for Geometry and Graphics,
Vol.3,No.2, pp.177

181, 1999.