Game Development Reference

In-Depth Information

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

Search Nedrilad ::

Custom Search