Graphics Reference
In-Depth Information
9.6.4 Non-Uniform Rational B-Splines
Non-uniform rational B-splines
(NURBS) combine the advantages of non-
uniform B-splines and rational polynomials: they support periodic shapes such
as circles, and they accurately describe curves associated with the conic sec-
tions. They also play a very important role in describing geometry used in the
modelling of computer animation characters.
NURBS surfaces also have a patch formulation and play a very important
role in surface modelling in computer animation and CAD. However, tempting
though it is to give a description of NURBS surfaces here, they have been
omitted because their inclusion would unbalance the introductory nature of
this text.
9.7 Surface Patches
9.7.1 Planar Surface Patch
The simplest form of surface geometry consists of a patchwork of polygons
or triangles, where three or more vertices provide the basis for describing the
associated planar surface. For example, given four vertices
P
00
,P
10
,P
01
,P
11
,
as shown in Figure 9.17, a point
P
uv
can be defined as follows. To begin with,
a point along the edge
P
00
−
P
10
is defined as
P
u
1
=(1
−
u
)
P
00
+
uP
10
(9.54)
and a point along the edge
P
01
−
P
11
is defined as
P
u
2
=(1
−
u
)
P
01
+
uP
11
(9.55)
Therefore, any point
P
uv
is defined as
P
uv
=(1
−
v
)
P
u
1
+
vP
u
2
P
uv
=(1
u
)
P
01
+
uP
11
]
P
uv
=(1
− u
)(1
− v
)
P
00
+
u
(1
− v
)
P
10
+
v
(1
− u
)
P
01
+
uvP
11
−
v
)[(1
−
u
)
P
00
+
uP
10
]+
v
[(1
−
(9.56)
P
01
P
11
P
uv
v
P
00
P
10
u
Fig. 9.17.
A flat patch defined by
u
and
v
parameters.
