A note on convexity-preserving interpolation subdivision

"-~-~ N

COMPUTER AIDED GEOMETRIC DESIGN

EI£EVIRR

Computer Aided Geometric Design 15 (1998) 863--865

Short communication A note on c o n v e x i t y - p r e s e r v i n g interpolation subdivision Shongtao Liu Institute q[ Mathematics. Fudan University, Shanghai 200433, Peoph,'~ Republic o[ China Received April 1997: revised February 1998

M6haut6 and Utreras (1994) introduced a subdivision algorithm for shape preserving function interpolation in R and R 2. Starting from data points, a sequence of piecewise linear function is shown to be convergent to a C ~ function. But the procedure for Case 3 (Mdhautd and Utreras, 1994, p. 21) of the subdivision algorithm can not guarantee that the limiting function is C r. In L e m m a 3.2, Case 3b of (M6haut6 and Utreras. 1994, p. 29): IL.l~)(:r//) ) - .q~~ > IL (;~), ',,.ri(~')') - V}~:)[. We have a~2i -2

~(~'+~) 2,3{~) a'a2i- I

-

(~,k(/) (1

-

-

-

-

-

(~, ,(k) )(li+l :

12)

SO

~- 2i

~

2i 2

~

=

i

--"-~i

I)4-

A i ,)+(1

(#)

(,){ (1

(t,)

,(I,)

I

'~'

, ,)} Sign of the second term of the right side of Eq. (3) can not be determined, so (3.13) of (Mdhautd and Utreras, 1994) is not correct. In order to reach the similar L e m m a to Lemma 3.2 of (Mdhaut6 and Utreras, 1994), we modify the subdivision algorithm as following: Algorithm in Case 1, Case 2 (see (Mdhautd and Utreras, 1994, p. 20)) are not changed. Case 3 (see Fig. 3 of (M6hautd and Utreras 1994)): There exists no solution to L i(a,/ r(*) I or the solution lies outside gfm =

[.5~0., '~ a.~ (# i1. Let

+ (1

o),0,i ':)

0167 8396/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PH S 0 1 6 7 - 8 3 9 6 ( 9 8 ) 0 0 0 1 8 - 1

S. Liu / Computer Aided Geometrie Design 15 (1998) 863-865

864

with I.~ k ) - y ~ } ) l = max{IL~ k)/~x2~ (k+x)~)-~}k) I r(k) ~x(k+,),

_(k) (4)

f a,

max{a, b}

a/> b,

l b, a
Definition. Ix} k), Xi+l (k)] is called inflection interval if sign(A~ 'k, ) / sign(A~'k), and {5(k) _ is defined as Id~k) A~k)l + ~rt +(k) _ A I k) I. Let's consider Case 3 of (Mdhautd and Utreras, 1994) now. Suppose that sign(A~LkI ) < 0, sign(A~ 'k) > 0, then [xl k) ;Li+ -'(~)n1] is inflection interval. It is easy to find that no intersection of r~ i (k) ~ ~ri +(k) _.(k) ~ Assume that 1 lies in ~ [Xi(k) ~ ;Ci+l]"

L(k)~_(k), then ix2.i

, x2~+l

A(k+l)

A 2i(k+j)

r(k) (~Ik)) _ 9!k) J =

[x2i

. a(k) .

is inflection interval.

~ xi+~]

a)d!k)

2z1~ . k) . a±~ . k)

(1

~,~,(k) ct ) ~G; '

(6) _

• 2k < O, slgn(A • 2k notice that slgn(Ai'_l) i' ) > O, then sign(A~ k) •

(k+,)

sign(Al~) - ~3i +(~) ~ and s~gn(A2i . 1/'

-

(~+]) d2i

.,(/c)

d} k)) is the same with •

) is the same with slgn(A2i

(k+~)

3(~+r)

- ~2i+1 ) '

SO

5(k+l) A(~+') 2i A(k+l)

r,(~+')

-- ~2i

,q(k+l)

,~A(~+~)

a(~+~)

,~(~+~)

~ (~+~),a(~+~)A(h+~)~ ~z'a2i -- ~ 2 i - - 1 ]

~ W2i

d ( k - r 1)

rl(k)

(,~)

(5(~+1) = 12(1 - a~C .(~+~) +

- 4

~ (~+1)~, t,l

~ ~cv2i

..,,(k)

(1

+

--

a) ( j l ~ ) d ~ ) ) ,

O~) ( A { k)

-

)1

(7)

d~ k) ) (8)

.(~-+~) ~ (0, 1/2), a ff [1/2 + ¢, 1] from (5) results A! ~) - d}k) I t> IA~~) _ ,4(~) ~+~], if w2~ where ~ ~ ( 0 , 1 / 2 ) (g is a very small positive real number) then ~(~+') < I( 1 - 2e) (Al ~) - d~~)) + (zll ~) - ~i+~) I~'(~) I(A} ~)

d~~)) + (A} k) _

A ( ~ ) ) I - 12e(A} ~)

d}~))l

(9)

_ d (~)

yielding 6 (~+~) < (1 - e ) 5 (~).

(10)

865

S, Liu I Computer Aided Geometric Design 15 (1998; 863-865

Next we consider [(IvTI)

( :,i

A ,q>*-I ) - ~a2i-I q = I ( I

2(1 " 2i+1 ]

t2i+l

~"2~ ~

">--.,

(,)(I ,t~2i ia,+' ))1Alia, -

"*'i+1

,~/, ).

(11)

I = '"1AI~`/ i~l - d(a:! /~1[

'*~ ' ) ~ ,

thus we have (12) (IU,:~I / [ 2i+1

4(~:+1) aa2i,

<~ L I A I ~ ) 1 --

d}~;lI

while i~i~(~/'bri(~") ,01~':)] ~< ]r(~:)~i+,(5i (/~)) -;Yii /')~ , if ' " x(k+ _ (10) is still valid. By (10), (12) we obtain

C (l/2,1),(~

¢ [I/~3.1.._

Lemma.

Without loss o f generalio', a s s u m e that there extsts only one c h a n g e q[ sign in -/A2'01-'"-2 the s e q u e n c e t" ~ Ji=l a n d A 2'0 ~ < 0 (i = 0 . . . . . q - 1), ~ 2.0 > 0 ( i = q . . . . . i~ 2), i,Tflection interval is d e n o t e d by [r! ~:) ~'(~:) ], i]'L - a,i~, I >~ --i~ ,,.lim ...

6 (~:) = 0,

~lira

max{2(l - o ). c~} / 1, and ~,,~t,;~,:,

"~:k4 l J, else ~/,2,:~

dl~~) - A (~) ~ ~l

:0,

~ (1/2.1), then

~lirn [d(~,/~+l

~(~:) ~i~+tl=0

II.,)"

With help of the 1emma above and the lemmas in (M6haut6 and Utreras, 1994), Theorem 3.4 of (M6haut6 and Utreras, 1994) is held. References Mdhaut~, A,L. and Utreras, F.L. (1994), Convexily-preserving interpolatory subdivision. Computer Aided Geometric Design 11, 17-37.