8
NURBS Concepts 1093
This operation removes the animation of
anything di rectly dependent on the object.
• Break,Extend,JoinandZip,Refine,Delete,
Rebuild, Reparameterize, Close, Make Loft,
Convert Cur ve, and Convert Surface
Any operation that changes the number of
points or CVs in a curve or surface removes the
animation of all points or CVs that are lost.
•Fuse
The animation of the point or CV being fused
to the other point or CV (the second one
chosen) is lost. The first point or CV acquires
the animation of the second.
NUR B S Concepts
NURBS curves and surfaces did not exist in the
traditional drafting world. They were created
specifically for 3D modeling using computers.
Curves and surfaces represent contours or shapes
within a 3D modeling space. They are constructed
mathematically. NURBS mathematics is complex,
and this section is simply an introduc t ion to some
NURBS concepts that might help you understand
what you are creating, and why NURBS objects
behave as they do. For a comprehensive
descript ion of the mathematics and algorithms
involved in NURBS modeling, see
The NURBS
Book
by Les Piegl and Wayne Tiller (New York:
Springer, second edit ion 1997).
Definition and Parameter Space
The term NURBS stands for
Non-Uniform
Rational B-Splines
. Specifically:
•
Non-Uniform
means that the extent of a control
vertex’s influence can vary. This is useful when
modeling irregular surfaces.
•
Rational
means that the equation u sed to
represent the curve or surface is expressed as a
ratio of two polynomials, rather than a single
summed polynomial. The rational equation
provides a better model of some important
cur ves and surfaces, especially conic sections,
cones, spheres, and so on.
•A
B-spline
(for
basis spline)
is a way to
constructacurvethatisinterpolatedbetween
threeormorepoints.
ShapecurvessuchastheLinetoolandother
ShapetoolsareBeziercurves,whicharea
special case of B-splines.
The non-uniform property of NURBS brings up
an important point. B ecause they are generated
mathematically, NURBS objects have a
parameter
space (page 3–1082)
in addition to the 3D
geometric space in which they are displayed.
Specifically, an array of values called
knots (page
3–1055)
specifies the extent of influence of each
control vertex (CV) on the curve or surface. Knots
areinvisiblein3Dspaceandyoucan’tmanipulate
them directly, but occasionally their behavior
affectsthevisibleappearanceoftheNURBSobject.
This topic mentions those situations. Parameter
space is one-dimensional for curves, which have
only a single U dimension topologically, even
though they exist geometrically in 3D space.
Surfaces have two dimensions in p arameter space,
called U and V.
NURBS curves and surfaces have the importa nt
properties of not changing under the standard
geometric affine transformations (Transforms), or
under perspect ive projections. The CVs have local
control of the object: moving a CV or changing
itsweightdoesnotaffectanypartoftheobject
beyond the neighboring CVs. (You can override
this property by using the
Soft Selection (page
1–1148)
controls.) Also, the control lattice that
connects CVs surrounds the sur face. This is
known as the
convex hull (page 3–1018)
property.