9

1092 Chapter 9: Surface Modeling
Shape curves such as the Line tool and other
ShapetoolsareBeziercurves,whicharea
special case of B -splines.
The non-uniform proper ty of NURBS brings up
an important point. Because they are generated
mathematically, NURBS objects h ave a parameter
space (page 3–988) in addition to the 3D geometric
space in which they are displayed. Specifically, an
array of values called knots (page 3–961) specifies
the extent of influence of each control vertex (CV)
on the curve or surface. Knots are invisible in 3D
space and you can’t manipulate them directly,
but occasionally their behavior affects the visible
appearance of the NURBS object. This topic
mentions those situations. Parameter sp ace is
one-dimensional for curves, which have only a
single U dimension topologically, even though
they exist geometrically in 3D sp ace. Surfaces have
two dimensions in parameter sp ace, called U and
V.
NURBS curves and surfaces have the important
properties of not changing under the standard
geometric affine transformations (Transforms), or
under perspective projections. The CVs have loca l
control of the object: moving a CV or changing
its weight does not affect any part of the object
beyond the neighboring CVs. (You can override
this property by using the Soft Selection (page
1–1147) controls.) Also, the control lattice that
connects CVs surrounds the surface. This is
known as the convex hull (page 3–924) property.
Degree and Continuity
All curves have a degree (page 3–927).Thedegree
of a curve is the highest exponent in the equation
used to represent it. A linear equation is degree 1;
a quadratic e quation is degree 2 . NURB S curves
typically are represented by cubic equations and
have a degree of 3. Higher degrees are possible,
but usually unnecessary.
Curves also have continuit y (page 3–923).A
continuouscurveisunbroken.Therearedifferent
levels of continuity (page 3–923).Acurvewithan
angle or cusp is C
0
continuous: that is, the curve
is continuous but has no derivative at the cusp.
A curve with no such cusp but whose curvature
changes is C
1
continuous. Its derivative is als o
continuous, but its second derivative is not. A
curve with uninterrupted, unchanging curvature
is C
2
continuous. Both its first and second
derivatives are also cont inuous.
Levels of curve continuity:
Left: C
0
, because of the angle at the top
Middle: C
1
, at the top a semicircle joins a semicircle of smaller
radius
Right: C
2
, the difference is subtle but the right side is not
semicircular and blends with the left
A cur ve can have still hig her levels of continuity,
but for computer modeling these three are
adequate. Usually the eye can’t distinguish
betweenaC
2
continuous curve and one with
higher continuity.
Continuity and degree are related. A degree 3
equat ion can generate a C
2
continuous curve.
This is why higher-degree curves aren’t generally
needed in NURBS modeling. Higher-degree
cur ves are also less stable numerically, so using
them isn’t recommended.
Different segments of a N URBS c urve can have
different levels of continuity. In par ticular, by
placing CVs at the same location or very close
together, you reduce the cont inuity level. Two
coincident CVs sharpen the cu rvature. Three
coincident CVs create an angular cusp in the
curve. This property of NURBS curves is known as
multiplicity (page 3–977).Ineffect,theadditional