9

NURBS Concepts 1093
one or two CVs combine their influence in that
vicinity of the curve.
Effects of multiplicity: there are three CVs at the apex on the
left, two CVs at the apex on the right.
By moving one CV away from the other, you
increase the curve’s continuity lev el again.
Multiplicity also applies w hen you fuse CVs. Fused
CVs create a sharper curvature or a cusp in the
curve. Again, the effect goes away if you u nfuse the
CVs and move one away from the other.
Degree, continuity, and multiplicity apply to
NURBSsurfacesaswellastocurves.
Refining Curves and Surfaces
Refining a NURBS curve means adding more CVs.
Refining gives you finer control over the shape of
the curve. When you refine a NURBS curve, the
software preserves the original curvature. In other
words, the shape of the curve doesn’t change,
but the neighboring CVs move away from the
CV you add. This is because of multiplicity: if
the neighboring CVs didn’t move, the increased
presence of CVs would sharpen the curve. To
avoid this effect, first refine the curve, and then
change it by transforming the newly added CVs,
or adjusting their weights.
Refining a NURBS cur ve.
NURBS surfaces have essentially t he same
properties as NURBS curves, extended from
a one-dimensional parameter space to two
dimensions.
Reparameterizing CV Curves and
Surfaces
When you refine a NU RBS curve or surface, it is a
good idea to reparameterize it. Reparameterizing
adjusts the parameter space so the curve or surface
willbehavewellwhenyouedititinviewports.
There are two ways to reparameterize:
•Chord-length
Chord-length reparameterization spaces knots
in parameter space based on the square root of
the length of each curve se gment.
•Uniform
Uniform reparameterization spaces knots
uniformly. A uniform knot vector has the
advantage that t he curve or surface changes
only locally when you edit it.
CVcurveandsurfacesub-objectsgiveyou
the option of reparameterizing automatically
whenever you edit the curve or surface.
Point Cur ve an d Surfa ce Concepts
You can work with point curves and point surfaces
as well as with CV cur ves and surfaces. The
points that control these objects are constrained
to lie on the curve or surface. There is no control
lattice, and no weight control. This is a simpler
interface that you might find easier to work
w ith. Also, point-based objects give you the
ability to constr uct curves based on dependent
(constrained) points, and then use these to
construct dependent surfaces.
You can think of point curves and surfaces as an
interface to CV curves and surfaces, w hich are
thefullydefinedNURBSobjects.Theunderlying
representation of the curve or surface is still
constructed using CVs.