8

1094 Chapter 9: Sur face Modeling
Degree and Continuity
All cur ves have a degree (page 3–1021) .The
degree of a curve is the highest exponent in the
equation used to represent it. A linear equation
is degree 1; a quadratic equation is degree 2.
NURBS curves typically are represented by cubic
equations and have a degree of 3. Higher degrees
are p ossible, but usua l ly unnecessary.
Curves also have
continuity (page 3–1017) .A
cont inuous curve is unbroken. There are different
levels of continuity (page 3–1017) .Acurvewith
an angle or cusp is C
0
continuous: that is, the
curveiscontinuousbuthasnoderivativeatthe
cusp. A curve with no such cusp but w hose
cur vature changes is C
1
continuous. Its derivative
is also continuous, but its second derivative is
not. A curve with uninterrupted, unchanging
cur vature 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 semicirc le joins a semicircle of
smaller radius
Right: C
2
, the difference is subtle bu t the right side is not
semicircular and blends with the left
A curve can have sti ll higher levels of continuity,
but for computer modeling these three are
adequate. Usua l ly the eye can’t distinguish
between a C
2
continuous curve and one with
higher continuity.
Continuity and degree are related. A degree 3
equation 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, s o using
them isn’t recommended.
Different segments of a NURBS curve can have
different levels of cont inuity. In particular, by
placing CVs at the same location or ver y close
together, you reduce the cont inuity level. Two
coincident CVs sharpen the curvature. Three
coincident CVs create an angular cusp in the
curve. This property of NURBS curves is known
as
multiplicity (page 3–1071) . In effect, the
additional one or two CVs combine their influence
in that v ic inity 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 cur ve’s continuity level again.
Multiplicity also applies when you fuse CVs. Fused
CVs create a sharper curvature or a cusp in the
cur ve. Again, the effect goes away if you unfuse the
CVs and move one away from the other.
Degree, continuity, and multiplicity apply to
NURBSsurfacesaswellastocurves.
R efi ning Cur ves a nd S ur f aces
Refining a NURBS curve means adding more CVs.
Refining gives you finer control over the shape of
thecurve.WhenyourefineaNURBScurve,the
softw are preserves the original curvature. In other
words,theshapeofthecurvedoesn’tchange,
but the neighboring CVs
move away from the
CVyouadd. Thisisbecauseofmultiplicity: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