Graphics Reference
In-Depth Information
3
3
Â
Â
(
)
=
()
(
)
(
)
=
()
(
)
pu
,
03
Bu
qq
-
,
pu
,
13
Bu
qq
-
,
(12.45a)
v
i
,
3
i
10
i
00
v
i
,
3
i
30
i
20
i
=
0
i
=
0
3
3
Â
Â
(
)
=
()
(
)
(
)
=
()
(
)
p v
0
,
3
Bu
qq
-
,
p v
1
,
3
Bu
q q
-
.
(12.45b)
u
i
,
3
1 1
i
0 0
i
u
i
,
3
3 0
i
2 1
i
i
=
0
i
=
0
(5) It satisfies the compatibility condition for twist vectors and those do not have to
be specified.
Because of property (4), it is easy to use Gregory patches to define composite sur-
faces in which all the patches meet smoothly along common edges. They are also good
for blending since one can define nonrectangular Gregory patches. See [Chiy88] for
how one works with these patches in practice and how the control points are used.
12.12
B-spline Surfaces
12.12.1
The Basic B-spline Surface
Let N
i,k
(u) and N
j,h
(v) be the functions defined by equations (11.69) with respect to
given nondecreasing knot vectors (u
0
,u
1
,..., u
m+k
) and (v
0
,v
1
,...,v
n+h
), respectively.
Let
p
ij
be a given set of points. Define a function p(u,v) by
m
n
Â
Â
(
)
=
()
()
puv
,
N
uN
v
p
.
(12.46)
ik
,
jh
,
ij
i
=
0
j
=
0
Definition.
The parametric surface p(u,v) defined by equation (12.46) is called a
B-
spline surface of order
(k,h) and
degree
(k - 1,h - 1) with
control points
p
ij
and
u-knots
u
i
and
v-knots
v
j
. The
domain
of the surface is defined to be the rectangle
[u
k-1
,u
m+1
] ¥ [v
h-1
,v
n+1
]. If k = h = 3, then the surface is called a
bicubic
B-spline surface.
The B-spline tensor product surface defined by equation (12.46) satisfies some
important properties that follow from those of the corresponding curves. For example,
(1) (Local control) If a point
p
ij
is moved, then the only change to the function p(u,v)
occurs in the rectangle [u
i
,u
i+k
) ¥ [v
j
,v
j+h
).
(2) At a u-knot of multiplicity r, the partial derivatives ∂
i
p/∂u
i
exist and are continuous
for 0 £ i £ k - 1 - r. At a v-knot of multiplicity s, the partial derivatives ∂
j
p/∂v
j
exist
are continuous for 0 £ j £ h - 1 - s.
(3) (Local convex hull property) The surfaces satisfy the convex hull property, that is,
they lie in the convex hulls of their control points. In fact, like in the case of B-spline
curves, a stronger property holds: if (u,v) Π[u
i
,u
i+1
) ¥ [v
j
,v
j+1
), then p(u,v) lies in the
convex hull of the points
p
st
, i - k + 1 £ s £ i , j - h + 1 £ t £ j.
(4) If the B-splines N
i,k
(u) and N
j,h
(v) have clamped knot vectors, then p(u,v)
interpolates the four corner points, that is,
