Bernstein-Bézier methods for the construction of bivariate spline approximants

Computer Aided Geometric North-Holland

29

Design 2 (1985) 29-36

Bernstein-Bkzier methods for the construction of bivariate spline approximants

Presented at Qberaolfach Revised 3 April 1985

12 November

1984

Abstract. The aim of this paper

IS to show that the BernateinBCzier method for representing polynomials over triangles may be useful for the construction of bivariate spline approximants on regular or arbitrary triangulations of a planar domam. Two different examples are given: (1) Quast-interpolants on a 3-dtrectional mesh. (2) Quadratic splines for Lagrange interpolation on a criss-cross triangulation.

Keywords. BernsteinBCzier

methods,

splines. approximation.

quasi interpolants

1. Introduction

The BernsteinBtzier (BB) representation of polynomials over triangles is particularly convenient for the construction of bivariate spline approximants (in the sense of ppf = piecewise polynomial functions) of various types on regular or arbitrary triangulations of the plane. The aim of this paper is to illustrate this assertion by showing two examples of application: (1) B-splines and quasi-interpolants on a 3-direction mesh. (2) Quadratic splines for Lagrange interpolation on a criss-cross triangulation. This technique is also useful for constructing new finite elements of high degree of smoothness [Sablonniere ‘841.

2. B-coefficients

and B-net associated with a ppf

Let T= A,A2A, be some triangle with barycentric coordinates u = (u,, u:, ui), 0 G U, G 1, 1u 1 = u, + u2 + u3 = 1. Let p,,(T) be the space of polynomial functions of degree < n on T. Every P E p,,,(T) -may be expressed in terms of the Bernstein basis:

P(u)=

c

h,B:(u),

lil=n

for i = (i,, i,. i,)~N’and 0167-X396/85/$3.30

]i] =i,+iz+ii=n.

(‘: 19X5, Elsevier Science Publishers

B.V. (North-Holland)

P. SahlonniPre

2

/ Birwrute

.rplrnr approxrmunrs

3

Fig. 1

The { 6,) are the B-coefficients of P and the points A, = (i,/n, i/n. i,/n; 6,) E R’ form the B-ner of the surface defined by P: its projection on T is the n simplicial triangulation of T. Thus, the polynomial P may be associated with the triangle T where the values b, are placed at the vertices of this triangulation. For example. P(u) = 2~: + 6u,u, + 5~: is associated with the triangle shown in Fig. 1 where b 200

--

2.

b,,,=O.

b,,,=

b 110

--

3,

b,,, = 0,

b,,, =O.

5.

Every ppf on an arbitrary triangulation is well defined once its B-coefficients are given on each triangle. There exist algorithms for the computation of P and its partial derivatives on T. General conditions on the B-coefficients of P, E P,,(T,) and P2 E p,,(T2), for two adjacent triangles T, and T,. insure that the continuity be C’ along their common edge (see [Farin ‘79, ‘X0] for details and the survey [Boehm et al. ‘841). This BB representation and the associated techniques are systematically used in the construction of the following ppf.

3. Quasi-interpolants

on a 3-direction

mesh

mesh A (giving a Let P,:(A) be the space of ppf of degree n, class Ch, on the 3-direction regular tessellation of the plane formed by equilateral triangles). It is well known that splines with bounded support exist iff 3k < 2n - 1 (see [de Boor et al. ‘831 for the basic results). The

Table 1 B-splines

with mimmal

and hexagonal

supports

(the B-coefficients

IP’:

are given in [Sablonnibe

‘84b])

1PZ.l

most interesting cases for applications are those corresponding to (n, k) = (3r. 2r - 1) and (3~ + 1. 2~). The catalog [Sablonniere ‘84b], available from the author. gives the B-coefficients of B-splines with minimal supports or with hexagonal supports (some of them are Box-splines) for r = 1, 2. 3 (see Table 1). They can be also computed using the technique given by Boehm in these proceedings. Let N,, E P,:(A) be some B-spline with hexagonal support u,, centered at the vertex A,, of A. Define the quasi-ir7terpolunt.c of a given function f by: S,&.

J)=&:,(f) ‘.I

where the coefficients A,(r‘>

.Y,(X

J)

(p-1.2.3)

are respectively: CLS,(f)=f(A,,)-h?u,lijf(A,,).

=!(A,,).

~:,(f)=~~,(f)+h4h,~3~~f(A,,). where:

a2f/ap,

dftx,

Y) = alfkz

+

d2f(x,

I‘) = a4f/a_u4

+ 2d4f/ax’a1.2

u; = l/6.

0; = l/8.

h4’= 1/12x,

h; = 83/2880.

+ Pf/ay4,

u; = 11,‘48,

0; = 3/16,

h; = 3/160

Definition

1. S,] is exact on the space E (of polynomials)

Definition 3.

2. S,] interpolutes

E on A if S,>f ( A, ,) = f ( A,,)

iff S,,f = f for all f E E. for every f E E and every vertex A,, of

Comparing the B-coefficients of f and S,f. we get the following results (P’,,, is the space of bilinear polynomials). See Table 2. Table 3 gives some error bounds (for the sup-norm) computed for the following functions on [O. l]* with the mesh A* (there is a simple change of variables to do in this case; Fig. 2 shows A* when h = l/4): f,(.x.

y) = ln(3 +x

f>( x. J’) = l/(3 fi(x.

+.v).

+ .Y +_v).

.v)=sin(~(x+~)).

exacton

P,,,

exact on PL interpolates

P,

exact on P,,,

exact on P,

exact on P,

exact on P,,,

exact on P,

exact on P,

exact on P,,,

exact on P,

exact on P5

interpolates interpolates

P, ff,

I

fl

f2

s3

l/16

-~___~ 112 ----114 -___ l/8

I

I 3. -03 4.5-03 __~______~ l/2 1.5-04 2. -04 _~___~~~ s2 l/4 1.2-05 1.3-05 -pp---118 1.1-06 1.3-06

--

sI

h

I 2.6-02 1.9-02 -~ ~___ l/2 6.3-03 4.2-03 _~______~~ l/4 1.6-03 I. -03 __----118 3.9-04 2.6-04 ~~___~~~ -___ l/16 6.4-05

__I_l-------

I

I-

Table 3

fl

2.9-04

3.4-02

1.9-02 ~___ 4.7-03 1.2-03

7. -05 4.1-06 2.8-07

1.4-01 1.1-02 7.2-04

5.5-08

4.4-08 3.7-09

7.5-07

7.2-06

5.5-04

2.4-08

3.6-07

5.5-06

9.6-05

4.8-05 ~~~ 2.2-03

1.9-04

7.8-04

3.2-03

1.4-02

6.1-07

6.9-06

2.1-04

1.4-03

I

8.5-03

f2

f3

2.1-05

5.1-03

9.3-07

1.5-05

2.6-04

6.8-03

2.1-03 ~___ 5.3-04

8.7-03

3.8-02

fl

I

4.6-05

1.9-02

7.8-08

1.2-06

2. -05

3.8-04

1.3-02

0.9-05

3.6-04

1.4-03

5.9-03

2.9-02

f2

3

Ip6

3.9-06

1.1-03 2.8-07 5.6-07 ______ 6.5-05 I 5. -09 I 8.2-09

2.4-05 ~___ I ___~ 2.1-02

---

3.8-04

5.5-03

_______ 6.4-02

6.4-03

2.5-02

I ~___ 3.3-01 ~9.8-02

I

/

1.3-01

4.4-01

I

f3

/

I

f3

1.7-02 ___~ 1.2-03 ~~ 7.6-05 ______ I ~~ 5.9-02 ~~ 1.3-03

1.7-01 -~ 4.6-02 ~~ 1.2-02 ~~ I ~___ 2. -01

1 ~-5.4-01

-‘-

1.3-07

1. -05

1

7. -03 1.9-03

6.6-04 1.1-05

2.7-07

..

4.4-09

/ 1.8-07

2.9-02

2.2-05

. -___

.

_~___

4.9-05

5.1-08

8.1-07

_~___

7.7-04

1.2-02

1.3-05

_______

1.4-01

I

9.6-03

3.8-02

1.4-01

4.7-01

1

f3

7.1-03 ~___ 2.4-04

_

4.8-03 ___~ 1.2-03 ______ 2.9-04 ______ 7.2-05

f2 ___~ kr

3.8-08

6. -07

9.9-06

1.7-04

4. -03

1.1-04

4.4-04

1.8-03

7. -03

3. -02

fl

7

M

33

Fig. 2

Remarks. (1) Partial derivatives can be replaced by finite differences. (2) Exactness of S,, on P, implies If- S,Jj r = O(h’+‘) when f~ Cr+‘, thus we have smooth and good approximants. (3) For a detailed study of error bounds when n = 3, 4, 6. see chapter 5 of [Sablonniere ‘82a]. For the general theory of the multidimensional B-splines and associated quasi-interpolants, we refer to the complete works of Dahmen and Micchelli. e.g. their survey papers in the books [Barnhill et al. ‘831 and [Chui et al. ‘841. Other interesting partial results are given in [Frederickson ‘70/71], [Sabin ‘771 and [Boehm et al. ‘841.

4. Quadratic splines for Lagrange interpolation

on a criss-cross triangulation

For 11= 2p + 1, let pi( R,,) be the space of Cl-quadratic splines on the triangulation of R = [u. b] x [c, d] obtained by drawing the diagonals in each of the rectangles determined by the partitions u = { CI= xc1< X, < . . < ,Y,>= h} and 7 = {c =J‘(, _~. x,~+,+,]x[~;,-,, R li+l and CIA+, =R?A+3\RZi+l. Let h,=x,+l

-xx,.

k,=4;+1

-_“;,

o,=h,/(h,_,+h,)

and

T,=k,/(k

,.., +k,).

We want to interpolate a given function u on the set z, = L, U L, U . U L,, where L,, + , := {the four vertices of RzA+, and the middle points on the 4(2k + 1) intervals subdividing r 2A+, }. Then 1q, 1= 4( p + l)( p + 2) = dim pi( R,,). and we have the following: Theorem.

There exists a unique v E Pd( R,, ) interpolating

Proof. The computation Step (a). On each Lagrange interpolation of order 2k + 1 with 0 ,< k
u on q,.

of the B-coefficients of 11can be described as follows: side of R,, + ,, the problem is reduced to solving a one-dimensional problem by quadratic splines [Kammerer et al. ‘741. i.e. a linear system a tridiagonal matrix. Therefore the B-coefficients of c’ on rZh+,, for compute: it remains to compute those in Cl, + ,. For this, we need the

Lemma 1 (Fig. 3). Let { a,, b,, w } be the B-coefficients a,‘s are known, the other B-coefficients me given by 2b, = ux + uz,

2b, = u2 + ad,

2b, = ah + a,,

2w = b, + 6, = b, + b.,.

of

L’ on

some

2b, = a4 + ah.

subrectangle

of R,,. Once the

34

a,

a2

a3

Ezz3 b,

b,

b,

b,

a,

a4

Lemma 2 (Fig 4). For two adjcrcent rectungles, _x=P+(P-a)h,/h,.

a5

a6

a7

Fig. 3

we haue:

.r’=6+(by)h,/h,

Step (b). By Step (a) and Lemma 1. we get the B-coefficients of (1 in R,. Now suppose that we have already computed the B-coefficients of c in Rzk + 1 and that we want to compute those Lemma 2 gives those on the boundaries of in Czi+,. Step (a) gives the B-coefficients on I;,,,, subrectangles in Cl, + 1 and finally Lemma 1 gives the interior B-coefficients of these rectangles. The simplicity of the algorithm has a counterpart: the rather quick growth of the sup-norm of the interpolation operator U,, defined by c’= n,,u. Some numerical computations suggest that for a regular partition. the norm of II,, is proportional to n, i.e. to the square root of ] 5$ 1. (See [Zedek ‘X5].) For other Lagrange interpolation problems, one can use either directly the BB representation of c (as is done above) or its representation in terms of B-splines: c(x,

.,I)=

I?+ I c* u,,fl,,,(x, ,.,=(I

_v)

Fig. 5 shows the B-coefficients of N,,, whose octagonal support is centered at ((x0 + x,)/2, (I;,+J,)/~) (we must add grid lines ,Y_, x,,, y-l 4;,). There are (17 + 2)* linearly dependent B-splines and we suppress one of them, say N,],], to get a basis of tPi( R,,) (see e.g. the paper by Chui & Wang in [Chui et al. ‘841). whence the (*) in the above sum. In our Lagrange problem, the linear system obtained from the interpolation data is completely decomposed into small linear subsystems. Finally, let us give an SB ulgorithm computing the B-coefficients of u from the S-coefficients u,,:Let R,,=[x,,x,+,]X[4;.):,+,1 and {br+r,,+,; r. s=O, i. 1: (r,s)+(:,t)} be the

P. SuhlonniPre

/ B~rwiuie

35

.splrne uppm.uimanf.t

Y2-

k,

k.

Y-l-

h-1

;0

ho Fig. 5. Quadratic

B-coefficients

B-spllne

of c’ on dR,,. (SBl)compute:

(SB2) compute:

c,,=~,u,,+(l-~,)a,,,,,

(OGiGfl+l,OGjGfl),

d,,=a,a,,+(l-a,)u,+,,,

(Odi
b,,=v,,+(l

-+,+l., (Ogi.

=7,d,,-t(l

The interior The author

B-coefficients thanks

j
-+c,+,

h ,+1/c./ = L’,+1,,

(0
b r.,+1/2 -d -

(0

I./+1

are given by Lemma


O
1.

the referees for their helpful

suggestions.

References R.E. Barnhill and W. Boehm. eds. (1983) Surfores in CAGD, North-Holland, Amsterdam. W. Boehm, G. Farin and J. Kahmann (1984), A survey of curve and surface methods in CAGD. Computer Aided Geometric Design 1 (1). l-60. C. de Boor and K. Hoellig (1983). Bivariate Box-splines and smooth pp functions on a three-direction mesh. J. Computational Appl. Math. 9, 13-28. C.K. Chui, L.L. Schumaker and J. Ward, eds. (1984). Approxmation Theor?; IV Academic Press. New York. G. Farin (1979). Subsplines fiber Dreiecken, Dissertation. Braunschweig. G. Farin (1980), B&ier polynomials over triangles and the construction of piecewise C’-polynomials. TR/91, Dept. of Math.. Brunel University.

P.O. FrederIckson (1970/71). Triangular spline interpolation. Generalized triangular splines. Math. Reports #6/70 and *t/71. Lakehead University. W.J. Kammerer. G.W. Reddien and R.S. Varga (1974), Quadratic interpolatory aplines. Numer. Math. 22. 241-259. M.J.D. Powell and M.A. Sabin (1977). PiecewIse quadratic approximation on triangles ACM Trans. Math. Software 3. 316-325. M.A. Sabin (1977). The use of pieceuise forms for the numerical representation of shapes, Dissertation. Budapest. P. Sahlonni&re (1980). Interpolation d’Hermite par des surfaces de classe C’ quadrattques par morceaux. Proc. 2” Congres International sur les MCthodes NumCriques de I’In&nieur. Dunod, Parls, 175-185. P. Sahlonm&re (1982a). Bases de Bernstein et approximanta splines. These de Doctorat. UniversitC de Lille I. P. Sahlonn&e (1982h). Interpolation by Quadratic Splines on Triangles and Squares. Computers in Industry. 3. pp. 45-52. P. Sahlonniere (1984a), Composite finite elements of class Ch. InternatIonal Congress on Computational and Applied Mathematics. Leuven. Report AN0 133. Lille. to appear in J. Computational Appl. Math. P. Sahlonmbe (1984h). A catalog of B-splines of degree < 10 on a three-direction mesh. Report AN0 132, Lille. F. Zedek (1985). Interpolants splines quadratiques. Th&e de 3eme Cycle, Lille I (to appear).