Graphics Reference
In-Depth Information
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5.4.1 Properties of B-Spline Curves
A B-spline curve,
P
(
t
) is a polynomial spline function of degree
m
−
1 such
that in each interval
t
i
≤
t<t
i
+1
,
P
(
t
) is a polynomial of degree
m
−
1,
and
P
(
t
) and its derivatives of order 1
,
2
,
3
2 are all continuous over the
entire curve. When
m
= 4, we get a cubic B-spline curve. This means in each
interval, the curve is a cubic polynomial. Since for any parameter u, the sum
of all the basis functions is one, i.e.,
·
m
−
n
B
i,m
(
u
)=1
,
i
=0
the B-spline curve lies within the convex hull defined by the control points.
We can note in this context that the B-spline convex hull is different from the
Bezier convex hull. Any point on a B-spline curve lies within a convex hull of
m
neighboring points. Hence,
(1) The entire curve lies within the union of all such convex hulls formed by
taking
m
successive defining polygon vertices.
(2) The curve has variation diminishing property, i.e., the curve does not
oscillate about any straight line more often than its defining polygon.
(3) The curve is ane invariant.
(4) The curve follows the shape of the defining polygon.
5.4.2 Effect of Multiplicity
Sometimes we need to insert corner points on a curve to depict a realistic
shape. Corner points are the locations of high curvature regions. This may be
effected by increasing the multiplicity of one or more control points. Multi-
plicity of a control point means counting it more than once. Thus, multiplicity
of a control point by 2 means the same control point is considered twice, while
multiplicity of 4 means it is considered 4 times, and so on. The effect is that
the curve is pulled on and on, and finally passes through it. Readers interested
in details can consult the topic [142]
5.4.3 End Condition
Sometimes, we have diculty in designing closed or periodic B-spline curves.
The curve does not pass through the extreme end or control points of the
guiding polygons. Barsky [16] has examined the conditions of the end control
points for cubic B-splines. David and Rogers provided a generalized treatment
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