Graphics Reference
In-Depth Information
(2) They provide local control.
(3) The nonperiodic B-splines interpolate the first and last control points, the peri-
odic ones do not.
(4) The degree of a Bézier curve increases with the number n of points, whereas with
B-splines one can vary n and the differentiability via k independently.
(5) Mathematically, a Bézier parameterization is a special case of a B-spline
parameterizations.
(6) The higher the multiplicity of a knot, the lower the differentiability at that point.
Periodic B-splines have a simple matrix formulation (which converts the curve to
the power form). The main advantage of this is realized if the matrix multiplication
is handled by hardware.
[FarR87], [FarR88], and [Faro91] showed that the Bernstein form of Bézier curves
is numerically more stable than the power form, the caveat being however that one
should not convert between the two. One would have to do
everything
, including algo-
rithms and how the data is stored, in the barycentric form. See [DanD89].
Differences between Hermite and B-spline reprentations
(see [Fari97])
:
(1) B-splines are numerically more stable.
(2) Hermite basis functions are not invariant under affine parameter changes.
(3) B-splines use less storage. For the interpolation of n points, the B-spline curve
needs n + 2 control points plus n + 1 knots (assuming that multiple knots
are stored just once), whereas the Hermite curve needs 2n points plus n + 1
knots.
Next, some comments about NURBS curves.
Advantages to using NURBS curves:
(1) They can represent a wide variety of objects and enable a uniform approach. Using
them one only needs to support
one
type of object. Their ability to represent conics
is extremely important for CAD. They are supported by many standards such as
IGES and STEP.
(2) Many tools exist for manipulating their geometry, such as, insertion of knots, etc.
(3) They support modification of local geometry.
(4) They are easy to transform under scaling, rotation, translation, shear, parallel and
perspective projection.
(5) In general, they have pretty much all the same nice properties as ordinary B-
splines.
Disadvantages to using NURBS curves:
(1) They are more complicated to compute in comparison to special types such as
circles, spheres, etc. On the other hand, current algorithms are fast and numeri-
cally stable.
