Bspline Surfaces: Important Properties
Several important properties of Bspline surfaces are listed here.
These properties can be proved easily by applying the same techniques used
for Bézier curves. Please compare these important properties with
those of Bspline curves'. Please recall that the equation of a Bspline
surface is the following
where the degrees in the u and vdirections are p and
q, respectively, and there are m+1 rows and n+1 columns
of control points.

Nonnegativity: N_{i,p}(u)
N_{j,q}(v) is nonnegative for all
p, q, i, j and u and v
in the range of 0 and 1.
This is obvious.

Partition of Unity: The sum of all
N_{i,p}(u)
N_{j,q}(v) is 1 for all
u and v in the range of 0 and 1.
More precisely, this means for any pair of u and v
in the range of 0 and 1, the following holds:

Strong Convex Hull Property: if (u,v) is in
[u_{i},u_{i+1}) x
[v_{j},v_{j+1}),
then p(u,v) lies in the convex hull defined
by control points p_{h,k}, where
ip <= h <= i and
jq <= k <= j.
This strong convex hull property for Bspline surfaces follows
directly from the
strong convex hull property for Bspline curves.
For the udirection, if u is in
[u_{i},u_{i+1}),
then there are at most p+1 nonzero basis functions, namely,
N_{i,p}(u),
N_{i1,p}(u), ..., and
N_{ip,p}(u). Thus, only the control
points on row ip to row i have nonzero basis
functions in the udirection. Similarly, if v is in
[v_{j},v_{j+1}),
there are at most q+1 nonzero basis functions on this
knot span, namely
N_{j,q}(v),
N_{j1,q}(v), ..., and
N_{jq,q}(v). Thus, only the control
points on column jq to column j have nonzero basis
functions in the vdirection. Combining these two facts
together, only the control points in the range of row ip to
row i and column jq to q have nonzero
basis functions. Since these basis functions are nonnegative and
their sum is one (i.e., the partition of unity property),
p(u,v) lies in the convex hull defined by
these control points.
As a result, the surface patch defined on rectangle
[u_{i},u_{i+1}) x
[v_{j},v_{j+1}) lies
completely in the same convex hull.

Local Modification Scheme:
N_{i,p}(u)N_{j,q}(v)
is zero if (u,v) is outside of the rectangle
[u_{i},u_{i+p+1}) x
[v_{j},v_{j+q+1})
From the
local modification scheme property, we know that in the
udirection N_{i,p}(u)
is nonzero on
[u_{i},u_{i+p+1})
and zero elsewhere. The local modification scheme property of
Bspline surfaces follows directly from the curve case.
If control point p_{3,2} is moved to a new location,
the following figures show that only the neighboring area on the
surface of the moved control point changes shape and elsewhere is
unchanged.

p(u,v) is C^{ps}
(resp., C^{qt}) continuous in the
u (resp., v) direction if u
(resp., v) is a knot of multiplicity s
(resp., t).

Affine Invariance
This means that to apply an affine transformation to a
Bspline surface one can apply the transformation to all
control points and the surface defined by the transformed control
points is identical to the one obtained by applying the same
transformation to the surface's equation.

Variation Diminishing Property:
No such thing exists for surfaces.

If m = p, n = q, and U =
{ 0, 0, ..., 0, 1, 1, ...., 1 }, then a Bspline
surface becomes a Bézier surface.