Graphics Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0.5
1
1.5
2
Default continuity,
c
0
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.5
1
1.5
2
0.5
1
1.5
2
Low continuity,
c
3/4
High continuity,
c
3/4
FIGURE B.48
The effect of varying the continuity parameter (with default tension).
varies between the first and last values of the knot vector. The knot vector establishes a relationship
between the parametric value and the control points. With replication of values in the knot vector, the
curve can be drawn closer to a particular control point up to the point where the curve actually passes
through the control point.
A particularly simple, yet useful, type of B-spline curve is a uniform cubic B-spline curve. It is
defined using four control points over the interval zero to one (
Eq. B.89
)
. A compound curve is gen-
erated from an arbitrary number of control points by constructing a curve segment from each four-tuple
of adjacent control points: (
P
i
,P
i þ
1
,
P
i þ
2
,
P
i þ
3
) for
i ¼
3, where
n
is the total number of
control points (
Figure B.50
). Each section of the curve is generated by multiplying the same 4
1, 2, . . . ,
n
4
matrix by four adjacent control points with an interpolating parameter between zero and one. In this
case, none of the control points is interpolated.
2
3
2
3
13
31
P
i
P
iþ
1
P
iþ
2
P
iþ
3
4
5
4
5
1
6
u
3
630
3
2
PðuÞ¼
u
u
1
(B.89)
3030
1410
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