Graphics Reference
In-Depth Information
x
1
w
1
,y
=
w
1
,z
=
z
1
y
1
w
1
.
On the other hand if
P
(
x, y, z
) is a Cartesian point, it corresponds to the
projective point
P
(
x, y, z,
1). Hence, the homogeneous representation of an
object in n-space can be viewed as an object in (n+1)-space. The coordinates
in n-space are called ordinary coordinates and those in (n+1)-space are called
homogeneous coordinates. The mapping from n-space to (n+1)-space is one-
to-many, i.e., there is an infinite number of equivalent representations of n-
space object in (n+1)-space. The inverse mapping from (n+1)-space to n-space
is many-to-one. The homogeneous representation of (
x, y, z
)is(
wx, wy, wz, w
)
for any
w
x
=
= 0 and a homogeneous point (
a, b, c, d
) has a three dimensional
image (
a/d, b/d, c/d
).
5.5.2 Essentials of Rational B-Spline Curves
With the concept of homogeneous coordinates discussed above, a rational B-
spline curve is defined in 3-d Cartesian space as a projection of a nonrational
B-spline in 4-d homogeneous coordinate space by
n
P
r
(
u
)=
R
i,m
(
u
)
V
i
,
(5.25)
i
=
o
where
V
i
s
are the 3-d control polygon vertices and
R
i,m
is the rational B-spline
basis functions, and are connected to nonrational B-spline basis functions in
the way as
w
i
B
i,m
(
u
)
n
R
i,m
(
u
)=
,
(5.26)
w
i
B
i,m
(
u
)
i
=0
where
w
i
≥
i
.Thus,
R
i,m
(
u
)
V
i
is the projection in 3-space from
B
i,m
(
u
)
V
i
in homogeneous 4-space. Hence, the rational B-spline basis func-
tions and curves are generalizations of nonrational B-spline basis functions
and curves.
For rational B-spline basis functions, it is also true that
n
0for
∀
R
i,m
(
t
)=1
,
(5.27)
i
=0
where
t
is any parameter. About the properties of rational B-spline curve, we
can say that:
(1) It is also variation diminishing like the B-spline curve.
(2) It also lies within the union of convex hulls formed by
m
successive defining
polygon vertices like the B-spline curve.
(3) Like B-spline, it also follows the shape of the defining polygon.
(4) The curve is invariant with projective transformation. Thus, it follows a
stronger condition compared to B-spline curves, which are a
ne invariant.
Search WWH ::
Custom Search