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

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In this book, these functions will be expressed by the notation NlA(t):

As can be seen from the above derivation, the continuity of the segments at the connection points is assured for all Qj(j = 0, 1, ..., n), so the difficult connection problems encountered with Ferguson curves and Bezier curves are absent. An example of a curve is shown in Fig. 6.2.

Similarly, a curve can be formed from uniform B-spline functions of order 3 degree 2 by assuming quadratic functions and forming linear combinations of 3 points; more generally, a curve can be formed from uniform B-spline functions of Order n+1, degree n as linear combinations of (n+1) points. For example, the uniform B-spline functions of order 3, degree 2 ^i(0> Y2(t) when the

knot vector is specified as

T = [f„ tx 12 t3 tA fs] = [-2 -10123] are given by the following formulas:

T— [fo h t2 t3 t4 t5 t6 ^7] — [ 3 2 1 0 1 2 3 4].

(6.16)

Fig. 6.2. Example of a B-spline curve (M = 4)

5'î(0 = ó(1-02^Àãî.çÌ

Y1(t)= -t2 + t + ~ = N13{t)

(6.17)

Y2(t) = ^t2 = N2,3(t)

6.1 Uniform Cubic Á-Spline Curves

237

Fig. 6.3. Example of a Â-spline curve (M = 3)

N3.4(0

Fig. 6.4. Uniform Á-spline basis functions (M = 4)

NiAt)

ËÛÎ

N2,3(t)

Fig. 6.5. Uniform Á-spline basis functions (M = 3)

The curve segment formed from these Á-spline functions becomes: ^(0 = Ó0Ø-1 + Y1(t)Ql+Y2(t)Qt + 1

-1-2 1- ã Qt-n

1—1 w 1 1 -2 2 0 Qi

1 1 o_ — Qi + 1 _

(i= 1,2, w-1). (6.18)

An example of such a curve is shown in Fig. 6.3.

Graphs of Ni 3(t) and NiA(t) in the range O^t^l are shown in Figs. 6.4 and

6.5 respectively.

In order to express the Â-spline curve segment of Eq. (6.14) by a Ferguson curve segment, we take:

238

6. The Â-Spline Approximation

0-Ã “0o

[r3 r2 f 1] MR 0, = [r3 r2 t 1] Mc 0!

0,+i Oo

_0, + 2 _ _6r_

which gives:

“Oo “0-Ã

Oi = MclMR 0,

Oo 0+i

_0i Oi + 2

1 2 l 0

6" Ó J

0 1 6* 2 J 1 ~6

l T 0 1 J 0

0 l 0 1 T-

O-i Î 0,+1

0, +2

1 2 1 g-e.-1 + ç e,+gft+1

1 2 1 6 ft+3ft + l + 6ft + 2

-Q,+

0,-1

l l òé«-òé

(6.19)

Alternatively, to express a Â-spline curve segment by a Bezier curve segment we take:

or Oo

[?3 ?2 t 1] MR 0, 0,-+i II 01 02

_0, + 2 03

Then the Bezier curve vectors 0O, 03 are expressed in terms of the Â-spline

curve vectors 0, -1, 0,, Qi+1» 0,+2 as:

6.1 Uniform Cubic Â-Spline Curves

239

Qo

Q i = MB1MR a

e2 a+i

e3 ft + 2

ft-r ft ft+1

Qi + 2

^ft+

ó @i + 1

1 ë 2 3 é+ 3

1 2 1 6e, + 3a+1 + 6a+

(6.20)

From this we obtain:

co4a-i-4ft4ft+

ei=ya+{ft+i

= eo+g-(e,+i-e,-i) e2=y a+yft+i

=e3-g-(ft+2-ei)

1 2 1 e3=-g a+yft+i+g-ei+2.

240 6. The Á-Spline Approximation

â+1

Fig. 6.6. Expression of a uniform cubic Â-spline curve segment in terms of a Bezier curve segments, fi-i, Qit Qi +1, Qi + 2: Curve defining vectors of the Á-spline curve segment. Q0,

Qz » áç: Curve defining vectors of the Bezier curve segment.

The geometric relation between the Â-spline curve defining vectors Qt-U Qb Qi + u Qi+2 and the Bezier curve defining vectors is shown in Fig. 6.6. To express a curve defined by given cubic uniform Â-spline curve defining vectors in terms of a Bezier curve formula, first trisect the 3 sides Qi-iQt, QiQi+i, Qt + iQi+2- 111 this case, on side QiQi+1 the trisecting point closer to Qt is Ql and the trisecting point closer to Qi + 1 is Q2. Next, join the trisecting point on side Qi-iQi that is closer to Qt to Q1 by a straight line. The midpoint of that line segment becomes the starting point of the curve, Q0. Similarly, join the trisecting point on side Qi + iQi + 2 closer to Qi+i to Q2 by a straight line; the midpoint of that line segment becomes the end point of the curve, Q3.

6.1.2 Properties of Curves26)

The uniform cubic Â-spline curve formula (6.14) is relatively simple and has superior properties, so it is often used in practical applications.

In formula (6.14), the Qj{j = 0,1, ..., n) are the vectors that define the curve, so they are called curve defining vectors. As in the case of a Bezier curve segment, the vectors Qj define the vertices of the polygon that determines the characteristics of the curve shape, that is, the characteristic polygon. The points Pt shown in Fig. 6.1 (i= 1,2, correspond to parameter values t = 0 or

t= 1. These are the position vectors corresponding to knots on the curve, as will be discussed below. In this book the Pt will be called knot point vectorsw

Normally, when the curve defining vectors Q} are given (/ = 0, 1, ..., n) and the knot point vectors Pt are found from them, the transformation is called a normal transformation; conversely, when the Pt are given and the Qt are found from them, it is called an inverse transformation.

6.1 Uniform Cubic Á-Spline Curves

241

As shown below, a uniform cubic Á-spline curve has superior curve design properties.

© Ñ2 Class Continuity

As is clear from the derivation of Á-spline functions in Sect. 6.1.1, at a point where uniform cubic Á-spline curve segments are connected, P(t) and P(t) are continuous regardless' of the values of the vectors Q}\ in other words, not only the slopes but also the curvature vectors are continuous. Because of this property, there are no difficult problems in connecting the ends of curve segments, and a curve consisting of connected segments can be treated as a single unit.

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