C1 interpolation on a polygon

Cl Interpolation on a Polygon H. B. Said

School of Mathematical G Computer Science.s

Unbxsiti Sains Malaysia 11800 Penung, Malay&

Transmitted by John Casti

AEM’FbKT We construct a two-sided Boolean sum interpolant which interpolates function values and derivatives given on two adjacent sides of a polygon. There are n such interpolants for a polygon of n sides. A scheme which interpolates function values and derivatives given on all sides of the polygon is then developed by using a convex combination of all those interpolants.

1.

INTRODUCTION The purpose of this paper is to present a scheme for interpolating

function

values and derivatives

given on the boundary of a polygon. An interpolation method of this type might be useful in such areas as finite-element analysis and computer aided geometric design. Herron [4] proposed a scheme whereby the polygon is subdivided into triangles, and then introduced an interpolant which satisfies interpolation properties on each triangle. The scheme is a convex combination of all these triangular interpolants. In this paper, we construct a Boolean-sum interpolant which satisfies the interpolation properties for two sides of the polygon. The construction is analogous to the Gregory-Charrot scheme for defining a vector-valued pentagonal surface patch [2]. For the polygon of n sides, we will have n such interpolants. Hence, we use a convex combination of all these interpolants to define our scheme. In Section 2, we define the domain for the construction of the scheme, and introduce various symbols to be used in the following sections. In Section 3, we construct a two-sided Boolean sum interpolant which requires rigorous analysis in basic geometry. In Section 4 we look at the errors in the scheme. APPLIED

MATHEMATICS

AND COMPUTATION

Q Elsevier Science Publishing Co., Inc., 1988 52 Vanderbilt Ave., New York, NY 10017

27:217-229

217 ooQfWo3/88/$03.50

H. B. SAID

218

FIG. 1. Domain.

2.

DOMAIN

AND NOTATION

For the definition of the scheme, we shall choose a convex polygon with n sides to be the domain 8; see Figure 1. Let V be any point on S& and V,, i = 1,2,. . . , n, be the vertices of the polygon in anticlockise order. We define R, to be the intersection point of the lines produced by sides V,, ,V,+, and V,_ ,V,. We also define E, as the point of intersection between line VR, and side qq+l, and set rr = IIErVlL St =

IlYm*

tc= IlWr-Al.

219

C’ Znterporct&m on a Polygon

BEMARK Weinterpretiasi-lmodn+ltobringitintotherange 1~ i d n. Points E,_, and E, remain on the boundary of the polygon 6) for al,lVEQ. 3.

THE SCHEME

3.1. Local !khf3mf39 Let F be a C’ function on a neighborhood of Q, whose second partial derivatives exist on 8Q. We consider two sides of Q, namely r, = 0 and r*_,=o. For each point V E B lying on the line Ri_lEi_l, we define a Taylor interpolant

Q-#l(V) = FtE,-I)+ r,-,F,_,tEd

(34

where F,,_, = t3F/i3ri_, denotes the derivative along the radial line R, _ ,E, _ 1. Then Q1_ 1 defines a projection operator satisfying the following interpolation properties on the side r, _ 1 = 0:

e,-l[wLl)

=wLl)~

~Pl(E’-‘) f

= q_pi-1).

1

Similarly, for the side r, = 0, we can define the Taylor interpolant as

e*wtw = W,)+ r,F,@,).

(3.2)

Equation (3.2) satisfies interpolation properties on the side r, = 0.

DEFNTION 3.1 (Boolean sum interpolant). Let Q1_ 1 ,and QI be interpolation projectors defined by (3.1) and (3.2) respectively. Then Boolean+um interpolant is defined as

tef-l~ei)[Fl(w = (ei-l + ei - eI-lei)[W),

(3.3)

and it satisfies the interpolation properties of Q, _ I. The expression Qi _ &I, is called the tensor-product projector. Further details of the general theory of Boolean sum interpolants can be found in Gordon [3] and Bar&ill and Gregory [l].

H. B. SAID

220

In order to construct the Boolean-sum interpolant of (3.1) and (3.2), we require an expression for the tensor product which is given in Theorem 3.1. We need the following lemma:

LEMMA 3.1.

6)

(See Figure 2.) We have

lim

6q_,-ro

asi-

6ri_l

II~j-lvll ll’i-IEill

IIRi-lvill llviEill IlRi-,Ei-Ill



ssi

i3ri

lim lsz,_,+o Sri_1 ’

-=cosafcos/3

&!$+O 6ri_l

asi IIRiXll - IIRi%‘ll =cosp lim Sr,_,+O &q-,+0 6Q' 6ri-1 lim

(iii)

(iv)

Il’i-YII

lim Sr,_,+O

Ft,+st,(Vi’) - F,,(Y)=- F,,(Y) Sri-1

6Si

llRiVJl sl;yo

Sri_1

F,,(vi)

IIRiXll - IIRivi’ll

+ llRiVJl a..50

sr,_l

+

lim

F,*(Y) - F,,(v,) 8% SSi

h-,+0

6rj_1’

PROOF. (i): By using cross-ratio property, we have

IlRi-,E!-,II lIEi-lvll IlRi-911 IIEi-1EI-III

IIRi-lvi’ll = IIRi-IEill

lIVEill IlvitT,‘II ’

Hence,

IlYv,‘ll E;%O lIEi-J(_JI

IIRi-lvi’ll IIviEill IIRi-IVII = E;!FO IIRi_lEill llRi-1EI-III lIEi-lvll ’

which gives the required result.

C' Znterpokation on a Polygon

(ii): By projection Ilvi’El-III =

INJLIII case+ ll~~‘llcos(p + E)+ JIE,_,E;_,IJcos(a- E)

= )IqEi_lll+

II~~‘(Icos~~ +

IIE~_~E;_~IIws~+O(E~).

Hence, the result follows. (iii): By projection IIRJQ = IIR,V,‘llcos~+

SS,COS/~

= IIRi~‘ll+ 6sicosj3 + 0(e2). Hence, the result follows, (iv):

H. B. SAID

222

where Fm, is the normal derivative along the line DE:_ 1’ Also,

II%- III IIDD’II = - %(v,‘) llr>tE;_lll+ Fn‘(x’) llD’E,Ql ’

(3.5)

By eliminating F,,, in (3.4) and (3.5) we have

Sine V,‘D’E;_,

and V,V,‘R, are similar triangles, we have

6% F,,(W) = KJv’)m +-IIR*~‘ll 1 IIWII +

- IIW’II~

F,,(V).

Hence, the result follows. We have proved the lemma.

THEOREM 3.1. Q+lQi[F](V)

The tensor~uct

projector Qi- 1Qiis given by

= F(V,)+ s,mF IlWtll II%l~ll +

sftfIIh- A-dl



(v,)+ t “Ri-lv” ‘II~d4-,II

II%-ml II%-All

qw

(34

Fs,*,W

PROOF. From the definition of Q, _ 1 and Qi, we have

Q,-lQ$[Fl(V)

=Q,[Fl@+,)+r,-I

I

V-

. (3.7) E,_I

223

C’ Znteipoldon on a Polygon From (3.2) we have

eiPwL)

(3.8)

= m3+ vm

and

aF’ (Ei) F,(4) + 5%

(3.9)

Hence

+t,

F,,+a,,(V’) - F,,(K)

lim

ari_l

h,_,-rO

.

(3.10)

On substituting the results of Lemma 3.1 into (3.10), and after further simplification of the coefficient of derivatives, we have proved the theorem. n The dual expression to (3.6) is given by

Q,Q&F](V)

= F(V,)+

t,

IIWII + “llR,Eill

“Rf-1v”

IILA111

F,,(W

F (VI “

f

F,,*,w* + sit’ llR,ErllIIWi-111

(3.11)

H. B. SAID

R. 1 FIG.3

THEOREM3.2.

In Figure 3,

IIWI IIWJI IIR*-lV,ll llR~-1Ei-~II IlRi-IEill = IlRiEill IIRiEi-Ill ’ IIRi-Ill

htoo~.

By using the law of sines,

llRiVllS~4 II%-911 llR~-~E~-qll = IIRi’i- ,IIsin’, ’

Hence, the result follows. From Theorem 3.2, we have the following corollary:

COROLLARY3.1 (Commutivity).

ei-Iei[Flw = eie*-1[Fl(W

C’ Znterpohtion on a Polygon provided that F satisfh

225

the compatibilitycondition

F,,t,w = F,,,,(v,)*

(3.12)

From Equations (34, (3.2) and (3.6), we can now establish a local scheme based on a Boolean-sum interpolant which is defined as

P,[Fl(v)=(Q*-leQ,)[Fl(v) =QI-1[FI(V)+Qi[FI(V)-Qi-IQI[FI(V). (3.13) By Definition 3.1 and Corollary 3.1, Equation (3.13) will satisfy interpolation properties on sides r,_ r = 0 and r1= 0 if the compatibility condition is SidiSfid.

BEMARIL If the sides V,_,V_, and V,y+a are parallel, then II,_, is a point at infinity. In this case it is easily shown that IIRi-lvll

VLdL~ll

=’

and

II%XII IIRi-IEill = ’

in Equation (3.6). Furthermore, if the sides V,V,_ 1 and V+ lV+z are parallel, then Equation (3.6) reduces to

3.2. Final Scheme The scheme is now constructed schemes, and defined as

as a convex combination

WIW = i 49P,[FIwh

of all local

(3.15)

i-1

where q(V)

is the weight function defined by

(3.16)

H. B. SAID

228

The weight function (3.16) satisfies these properties:

(1)

w,(v)

2

0

t w,(v)=1 i-1

(2)

o,(v) = 0

(3)

for

VE51,

for

VEQ,

for

Vsuchthat

r,=O,

j#i-1,i.

The squared terms in (3.16) are introduced in order that Pi does not contribute to the derivatives of P on rj = 0, j # i - 1, i. Hence (3.15) wilI interpolate function values and derivatives on ah sides of the polygon. 4.

ERROR ANALYSIS The interpohmt (3.15) has the following remainder form:

R[Fl(V)=(Z--P)[Fl(V) = t 4v)(z - P,)[FlW)~ i-l

(44

where Pi is defined by (3.13). Thus, an error analysis of (4.1) requires as analysis of the individual remainders

R@](V) = (I- P,)[Fl(V).

(4.2)

As in Barnhih and Gregory [l], we have

The Langrange form of Taylor remainder gives

(I- Q,)[F](V) = where e,(V) have

$,,,(&(v)),

is a point on the open line segment E,V. Thus, from (4.3) we

R,[F](V)

=,$&

4 1

[ rX,,MV))]

“__$,(“)

(4.5)

C’ Zntsrpokrtion on a Po~gon

227

where &_1(V) is different from &1(V). Since r, is dependent on r,_,, we cannot simplify (4.5) into a nicer forin. Thus, if we let the polygon a2 be contained in a circle of radius h, then (4.5) shows that

and from (4.1)

R[F](V) = O(P). 5.

PRACTICAL

(4.7)

IMPLEMENTATION

For the implementation of the scheme, we shall use two test functions: 2 2 (a) F(W)=II, (b) F(r, y) = z3y2 + ye.

Fro. 4.

Test fuations: (a) F(x, v) - &/I, (b) F(r, v) - z3ye + 11’.

H. B. SAID

(4 FIG. 5.

\

(b)

Results of the scheme for the test functions in Figure 4.

These functions are used to construct boundary conditions for a hexagon. Figure 4 shows surfaces of these functions, while Figure 5 shows surfaces obtained from the scheme. These surfaces are plotted along x = constant and y = constant lines.

6.

CONCLUSION

In this paper, we have proposed a scheme which interpolates function values and derivatives on the boundary of a polygon. The method is based on the convex combination of two-sided Boolean-sum interpolants. An open question related to this scheme is whether we can improve its accuracy to a remainder of higher order than 0(h2). A change in the variables used to define Boolean sum interpolant may answer this question.

C’ Znterpolution on a Polygon

229

I would like to thank Professor S. L. Lee for going over earlier drafts of this paper and giving me valuable sugge&ms.

REFERENCES 1 2 3 4 5

R. E. Barnhill and J. A. Gregory, Compatible smooth interpolation in triangles, I. Approx. Theory 15:214-22.5 (1975). P. Charrot and J. A. Gregory, A pentagonal surface patch for computer aided geometric design, Comput. Aided Geum. Design 1:87-94 (1984). W. S. Gordon, Blending function methods of bivariate and multivariate interpolation and approximation, SIAM J. Numer. At&. 8:158-177 (1971). G. J. Herron, Triaugular and Multisided Patch Schemes, Ph.D. Dissertation, Univ. of Utah, 1979. G. M. Nielson, The side vertex method for interpolation in triangles, J. Approx. Theory. 25:318-336 (1979).