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where P i , j forms a bi-directional network of control points, N i , p and N j , q are the basic
B-spline functions defined on knot vectors:
U
={
a , . .. , a
p
, u p + 1 , ...u r p + 1 , b , . .. , b
p
}
+
1
+
1
(14.11)
V
={
c , . .. , c
q
, u q + 1 , ...u s q + 1 , d , . .. , d
q
}
+
1
+
1
with r
=
n
+
p
+
1, s
=
m
+
q
+
1, the limits [ a , b ] and [ c , d ] being generally fixed at
[0, 1].
After this theoretical interlude, I think it is important to discuss the advantages and
disadvantages of using NURBS as modelling surfaces. The undeniable advantages are:
A wide range of objects can be modelled in an accurate manner; interactive modi-
fication is easy thanks to the use of control points and knots. NURBS can be used
for modelling conical sections, which is not possible if Bezier surfaces are used;
The calculation cost for the representation is not huge, and most of the calculations
are digitally stable;
The NURBS do not vary with the common geometric transformations (rotation,
translation, perspective projection). It is sufficient to apply these transformations
to the control points;
The geometric interpretations are easy as the NURBS are simple generalisations
of Bezier curves and surfaces;
The representation is particularly compact for complex forms, which cannot even
be represented by other methods.
However, there also are some disadvantages:
The NURBS contain huge geometric data; the representation of simple regular
curves and surfaces can be inconvenient in terms of storage. Thus, the represen-
tation of a circle in the space normally requires the radius, the centre of the circle
and a perpendicular to the plane of the circle, i.e. 7 real numbers. However, if we
use NURBS, 7 homogenous control points and 10 knots, i.e. 38 real numbers are
required. This largely depends on the type of surface to be represented;
The manipulation of the weights can lead to aberrant constructions;
Common operations (intersection of surfaces) are extremely difficult to carry out,
which can prove to be particularly troublesome in virtual reality;
Certain fundamental algorithms are digitally unstable (calculation of parameters
( u , v ) of a point ( x , y , z ) belonging to a surface).
14.4 ALGORITHMIC GEOMETRY
In this section, we will discuss certain concepts of algorithmic geometry for the conver-
sion of models. For questions of speed, the models used in virtual reality are polygonal
 
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