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# Curves and surfaces in computer aided geometric design - Yamaguchi F.

Yamaguchi F. Curves and surfaces in computer aided geometric design - Tokyo, 1988. - 390 p.
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*) Note that in t,^t<tl+1, both N,+ liM_1(t) and are zero (refer to Fig. 6.37)
6.12 Geometrical Properties of B-Spline Curves
311
From this equation we can find Q{p and substitute into Eq. (6.122) to obtain:
6.12 Geometrical Properties of 5-Spline Curves
The general geometrical properties of Á-spline curves are summarized below.
(1) Locality
A point on the curve is determined by only the M curve defining vectors Q} in the immediate neighborhood of that point. Consequently, if a certain polygon vertex is varied, only the part of the curve in the immediate neighborhood of that vertex is affected (refer to Fig. 6.47).
(2) Continuity
In general CM_2 continuity between curve segments is maintained.
(3) Convex Hull Property
Equations (6.84), (6.85) and (6.86) hold for a Á-spline function, so the curve segments that comprise the Á-spline curve are convex combinations of the nearest M vectors Qr That is to say, each curve segment is contained inside a convex hull formed by the M points. Consequently, compared to a Bezier curve, a Á-spline curve is more faithful to variations in the polygon shape.
As a special case of this property, if Qj=Qj+i= ... =á/+ì-1> these convex hulls reduce to the points Qp the curve segments defined by this sequence of points degenerate to the points Qj, and the curve passes through the Qj.
Also, if Qj, Qj+i,..., Qj+M-i he colinear straight line, from the convex hull property of a Á-spline curve it follows that the Á-spline curve segments are
p^(t) = h-'YJNj,M-rWQj-
(6.124)
j
Fig. 6.47. Local control of a Â-spline curve
Qe
312
6. The Á-Spline Approximation
straight line segments. Since a Á-spline curve generally maintains CM~2 continuity at the connection points between curve segments, it can include straight line segments that are connected to curved segments with CM1 continuity.
(4) Variation Diminishing Property
The number of intersections between an arbitrary straight line and à Á-spline curve is not more than the number of intersections between that straight line and the curve defining polygon. Therefore, à Á-spline curve assumes a shape that is a smoothed form of the curve defining polygon shape (refer to Sect. 6.9®).
6.13 Determination of a Point on a Curve by Linear Operations2 9)
DeBoor proposed an algorithm for finding a point on a curve recursively by
repeated application of linear operations, without calculating the Á-spline
function values (refer to Appendix B.2).
De Boor’s algorithm
We are to find a point P(ts) on a curve at the parameter value t = ts.
Step 1: Find an i for which ti^ts<ti + i, then set:
r = i-M+1. (6.125)
Step 2: Let:
Q?KQ = Qj (j=r, r+i,..., r+M-i). (6.126)
Step 3: Repeatedly apply the formulas:
Qf\ts) = (1 - A) Qf-~il](ts) + X Qf-40 (6.127)
t — t ¦
X =------—J-— (6.128)
tj+M-k — tj
to find:
In this algorithm, in the case of a closed curve, the j in Qj is replaced by jmod(n+1).
If a uniform knot vector having a span of 1 is given, then:
6.13 Determination of a Point on a Curve by Linear Operations
313
Qr+2=Q'r°Mts)
Qr+1 (0 = Qr+i
Qlol(ts) = Qr
Qr+4 = Q[r°Mts)
Fig. 6.48. Geometrical interpretation of De Boor’s algorithm (case of M = 5)
In the above algorithm, in step 1 we are finding the r in the expression Qj (j = r, r + 1, ..., r + M—1) for the curve defining vectors related to the curve segment in which ts is located. Step 2 is the initial value setting. Steps 2 and 3 can be geometrically interpreted as follows (refer to Fig. 6.48). The case M = 5 will be used as an example in this explanation. In Step 1, suppose that the curve defining vectors Qr, Qr + l5 Qr+2, Qr+ç and Qr + Ar of the segment in which ts is located are known. First, according to Step 2, these are the initial value vectors for the algorithm. That is,
Q l-Af+l (ts)
\
Fig. 6.49.
Relations in De Boor’s
Ql 0l(O—------------------------algorithm
314
6. The 5-Spline Approximation
are produced on the sides of the polygon. A new polygon can be formed from these 4 points. Again linearly interpolate between the points of this new polygon to obtain another new polygon consisting of the points
Again linearly interpolate between these points to obtain the 2 points +
Qlr3+4(ts), which are the ends of a line segment. Finally, linearly interpolate
between these 2 points to determine a single point QlV^ih)- This point is the point P(ts) on the curve that corresponds to t = ts. In general, the linear interpolation parameter value X is different in each interpolation.
The relationships in De Boor’s algorithm are shown in Fig. 6.49.
Example 6.8. Find the point on the open Á-spline curve found in Example 6.4 at t = 4.75, using De Boor’s algorithm.
Solution. From the knot vector of Example 6.3:
T= [to ^1 t-2 13 t4 t5 t6 t-j t8 tg tl0 tn tl2]
= [0 00012345666 6]
we see that, since f7^4.75<f8, i = l. Therefore r = 7 — 4+1=4.
012Ø, Ql+M, QllUts).
Q\ç11 (4.75) = (1 — A) Q,™ (4.75) + X Q™ (4.75)
= (1 ~X)Q4 + XQ5 = 0.083 Q4 + 0.917 Q5
— (I-05+^06 = 0.417 Q5 + 0.583 Q6
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