Interpolation in triangles

B U L L . AUSTRAL. MATH. SOC. VOL. 20 ( 1 9 7 9 ) ,

65D05,

4IA05

I 15-130.

Interpolation in triangles G.M. Nielson, D.H. Thomas, and J.A. Wixom Several new methods of approximation which assume arbitrary values on the boundary of a triangular domain are presented.

All

of the methods are affine invariant and have optimal algebraic precision.

Nine parameter discrete interpolants which result

from these methods are also given.

1.

Introduction

This report is concerned with bivariate approximations which assume arbitrary values on the entire boundary of a triangular domain.

This type

of approximation has application in such areas as finite element analysis and computer aided geometric design.

Examples of these two applications

are given in [/] and [/5] respectively. The first investigation into this subject was by Barnhill, Birkhoff, and Gordon [2]. Several other papers, [3], [4], [5], [6], [7], [«], [9], [//], [J3], have dealt with this same general topic. In this paper, several new interpolation schemes are defined.

Rather

than give an exhaustive list of interpolants, we have selected methods on the basis of the techniques used in their development.

It is hoped that

this approach will lead to the extension of these interpolants to the caseof interpolation to both position and slope on the boundary. For the most part, the standard triangle (0, 1) , and

(1, 0)

is used.

T

with vertices

(0, 0 ) ,

The three boundary functions are assumed to

be evaluation of a given bivariate function

/

and the interpolation

process is viewed as mapping of the data function

/

to the interpolant

Received 12 December 1978. The research of Professor Ilielson was supported by the United States Office of Naval Research.

II5

116

G.M. N i e l s o n ,

P[f]

D.H. Thomas,

a n d J . A . . Wixom

with the property that P[/](0, y) = / ( 0 , y) ,

(1.1)

PlfUx,

0) = f(x, 0) ,

P[f](x, 1-x) = /(x, 1-x) . All of the methods are of the form N

P[f](x, y) = Y. ) •

(1.2)

The functions

l i=l l * £ = 1, . . . , ff are called the weight functions and

w. , Is

fa.(x, iy), 3-(^, J/)) € 3T , i = 1, ..., # , are collectively referred to as the stencil.

It is often useful to describe the stencil graphically.

Figure 1 depicts the stencil of the trilinear interpolant of [ 2 ] ,

d.3) e*[f] = £ p ^ f ( o , »)+igH-/(*, o) 1%" f ( l " y ' ^

+

x%

f{x+h

'

0)

- xf(l, 0) - (l-x-y)f(O, The affine transformations which relate A with vertices

v. = [x., y.) ,

x^

0) - yf(O, 1)J .

T and an arbitrary triangle

£ = 1, 2, 3 , can be used to obtain what

is called an affine equivalent interpolant.

For

F defined on

A , this

process is described by the formula (l.*0

PL[PUs,_

t) = P[f][bj(s,

t), bk(s,

t))

,

( s , t) i A ,

where f(x,

y) = F[xxi+yxMl-x-y)xk, (i,

and

b. = b.{s, 1r

t) ,

xyi+yy;.+(l-x-y)yk) j , k)

represents a permutation of

( l , 2, 3) ,

i = 1, 2, 3 , represent t h e barycentric coordinates

If

defined by

(1.5)

,

t=

Interpolation

in

17

triangles

CO, 1 )

: , 1-*)

(0, x+y)

(0, y)

_

-y> y)

(0, 0)

( 1 , 0)

(x+y", 0) Figure 1

If each of the six affine transformations lead to the same interpolant, then the method is said to be affine invariant. As an example, we apply (1.1*) to Q* and obtain Q*IF] - i

3

Tb.F[b .v.+(l-b.)v,)+b .F(b.v.+ [l-b.)v .} -*—*-* * \_b-l " ^ _ _

I

If an operator P satisfies

hF{Vi)

P[