Graphics Reference
In-Depth Information
Figure 11.28.
Projectively equivalent
ellipse and parabola.
()
()
()
()
xu
zu
yu
zu
Ê
Ë
ˆ
¯
()
=
pu
,
because the central projection is gotten simply by dividing by the z-coordinate. But
we can think of equation (11.101) as defining a conic with homogeneous coordinates
in projective space
P
2
. The important observation is then that conics do have
polynomial representations if we use homogeneous coordinates.
In summary, we have shown that we can handle a larger class of curves if we use
homogeneous coordinates. In that setting, the analog of equation (11.67) is
n
Â
()
=
()
Pu
b u
i
P
,
(11.102)
i
i
=
0
where the b
i
(u) are suitable basis or blending functions and the
P
i
are points described
with homogeneous coordinates. Everything we did earlier for polynomial curves
applies to the curves defined by equation (11.102) since the nature of the coordinates
did not play a role. In particular, we have the obvious notions of Bézier and B-spline
curves for homogeneous coordinates. Furthermore, if we write
P
i
in the form
P
i
=
(x
i
w
i
,y
i
w
i
,z
i
w
i
,w
i
), then the projective space curve defined by P(u) projects to the curve
n
Â
()
wb u
p
ii
i
i
=
0
pu
()
=
(11.103)
n
Â
()
wb u
ii
i
=
0
where
p
i
= (x
i
,y
i
,z
i
). There are several important special cases of such curves.
Definition.
The curve p(u) defined by equation (11.103) is called a
rational Bézier
curve
if its domain is [0,1] and b
i
(u) = B
i,n
(u). (The B
s,t
(u) are the functions defined
by equation (11.50).) The curve p(u) is called a
rational B-spline curve of order k
if the
