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

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ai+iQi-i + 2(al+l + ai)Q'i + aiQi+l

3

(4.66)

and also in the various end conditions discussed in Sect. 4.8.

For the unit tangent vectors Q'i(i = 0,1, ...,n) which are obtained, the curve segment Pt(t) in the i-th span can be expressed as follows:

= [Äî>î(0 H0A(t) Hl 0(t) Hul(W

ft-i

Qt

a, 6l-i . atQ\

Qi-1 Qi

O-i

Qt

(1 SiSn).

(4.67)

4.10 5-Splines

A spline function can be determined which satisfies the following conditions between the knots xJt xj+l, xj+2, xJ + 3.

Condition © is that the function S satisfies the following equations at the two end knots:

S(Xj) = S(Xj) = 0 S (xj + ç)= $ (Xj + 3) = 0.

Condition © is that at intermediate knots, the function S takes a specified value, for example:

S(xJ + 1) = h.

A function which satisfies these conditions is a C-spline of degree 2 (Fig. 4.16(a)).

4.10 5-Splines

161

(a)

(b)

Fig. 4.16. B-splines

Similarly, a spline function can be determined which satisfies the following conditions between the knots Xj, xJ+1, xJ + 2, xj+3 and x,+4.

Condition © is that the function S satisfies the following equations at the two end knots:

S(Xj) = S(Xj) = ?(*j) = 0

S(Xj+4.) = S(xJ+4) = S(xJ + 4) = 0.

Condition © is that at intermediate knots, the function S takes a specified value, for example:

S{xj+l) = h.

A function which satisfies these conditions is a C-spline of degree 3 (Fig. 4.16(b)).

Let us now derive a polynomial spline function of degree 2 which satisfies the first condition.

The following quadratic S(x) satisfies S (xj) = S (x) = 0 and S(xJ + l) = h.

From spline Eq. (4.5), calling the unknown constant bx we can write:

(4.68)

(4.69)

162 4. Spline Interpolation

Calling the unknown constant b2l a quadratic expression that satisfies the condition S(xj+3) = S(xJ + 3) = 0 can be written as follows:

S(x) = b2(x-xj+3)2. (4.70)

By adding the condition that the curves in the 2-nd and 3-rd spans must have both position and slope continuity at x = xj+2, the unknown constants bx and b2 can be determined as follows:

b -(^+2-*Æ+ç-^)

(Xj+ ! - Xj)2 {XJ + 2~XJ+ t) (XJ + 3~ Xj+ J

b2=--------------XJ+2~XJ-------------h.

(xj+l-Xj) (xJ + 3-xJ+l) (xj+3-xj+2)

Then the curves in each span can be found from (4.71), (4.72) and (4.73).

Xj^X^Xj+ 1

S(x) = ----?!_—(x-x/ (4.71)

(xJ+1-x/

õ]+õé*éõ} + 2

,M- h L Y (xJ + 2-xJ)(xi + z-xJ) "I

(xJ+1-x/ L J (xj+2-xj+l)(xJ+3-xJ+1) J + 1 J

(4.72)

(x,+ ? — x.) h -

S(x) = ?----------—^--------------------------------------------,1-- (x + 3) . (4.73)

{xJ + l-x) (xj+3-xJ+l) (xJ+3-xJ+2)

There is another method in which, as a condition for determining the function, in place of specifying the value h the area enclosed by the curve and the x-axis is specified. Since the area is:

T S(x)dx = A(*i+»-*Jfr + 3Z*<L.

3 XJ+1~XJ

Schoenberg used a method in which he specified a unit area. In this case, we have:

h 3(xJ + 1-x,)

(Xj + 2-Xj) (xj+3-X,)

4.11 Generation of Spline Surfaces

163

Cox and DeBoor specified the area as follows so that Cauchy’s relation holds (refer to Eq. (1.4)) and normalized:

h (xJt2-x,)(xJt3-xJ) = xJ + 3-Xj

3 Xj+1-Xj 3

In this case we have:

= (4.75,

Xj +2 Xj

In this case, the spline function is expressed as follows in each interval:

SW=(X ~-~x-\(X~X’)2 (476)

'O+l Xj)\XJ + 2 Xj)

õ]+1éõéõ]+2

1

S(x) =

(Xj+l-Xj) (Xj+2-Xj)

L \xj + 2 Xj+i)\xj+3 XJ + 1) j

S(x)=7x~ -x ~ux~ -7 > <4-78)

\xj+ 3 xj+l)Kxj + 3 xj + 2)

(Equations (4.76) to (4.78) agree with Eq. (6.90) to be derived in Chap. 6). The function defined by Eqs. (4.76) to (4.78), with spans continued indefinitely to the left and right and all but 3 spans having functions values of 0, as shown in Fig. 4.16, is called a 5-spline (Basis spline) or fundamental spline of degree 2. A normalized 5-spline of degree 3 can be derived by a similar method. In the case

of degree 3, in the right-hand side of Eq. (4.74) the area is specified as (xJ+4

— Xj)/4. A Ë-spline is a special case of a C-spline. For further details on B-splines refer to Chap. 6.

4.11 Generation of Spline Surfaces

The method used to generate spline curves can be used to generate spline surfaces that are continuous up to the curvature.

164

4. Spline Interpolation

Qm-l ÿ

J

Fig. 4.17. Generation of a spline surface

Assume that a lattice of points Qtj(i = 0,1,..., m; j = 0,1,..., n) is given (Fig. 4.17).

© In Fig. 4.17 there are (n + 1) rows of points in the I-direction. If conditions are given for the unit tangent vectors in the /-direction at the start and end points of each of these rows of points, then, letting ht be the chord length in the /-directi on the (n + l) spline curves in the /-direction are determined by solving the following condition equations:

(i= 1, 2, ..., m — 1; j = 0, 1, ..., n).

In this case, if normalization is done in each interval using a parameter u, then the tangent vectors with respect to the parameter è are:

(Refer to equations (4.45) and (4.46).)

© Following a similar procedure to the above, for the (m +1) rows of points in the J-direction, the following condition equations hold:

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