- Email: [email protected]
A bivariate rational interpolation and the properties
Applied Mathematics and Computation 179 (2006) 190–199 www.elsevier.com/locate/amc
A bivariate rational interpolation and the properties Qi Duan
a,*
, Yunfeng Zhang a, E.H. Twizell
b
a b
School of Mathematics and System Science, Shandong University, Jinan 250100, China School of Information System, Computing and Mathematics, Brunel University, Uxbridge, Middlesex, England UB8 3PH, United Kingdom
Abstract In this paper a bivariate rational interpolation is constructed using both function values and partial derivatives of the function being interpolated as the interpolation data. The interpolation function has a simple and explicit rational mathematical representation with parameters, and it can be expressed by the symmetric bases. It is proved that the interpolation is stable. The concept of integral weights coefficients of the interpolation is given, which describes the ‘‘weight’’ of the interpolation points and the quantity as the interpolation data in the local interpolating region. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Bivariate interpolation; Rational spline; Computer-aided geometric design
1. Introduction The construction method of the curve and surface and the mathematical description of them is a key issue in computer-aided geometric design. There are many ways to tackle this problem [1–10], for example, the polynomial spline method, the NURBS method and the Be´zier method. These methods are effective and applied widely in shape design of industrial products. Generally speaking, most of the polynomial spline methods are the interpolating methods, which means that the curves or surfaces constructed by the methods pass through the interpolating points. The NURBS and Be´zier methods are the so-called ‘‘no-interpolating type’’ methods; this means that the constructed curve and surface do not pass through the given data, and the given points play the role of the control points. In general, therefore, the interpolation method gives a better approximation to the function being approximated than that constructed by the other methods. In recent years, the univariate rational spline interpolation with parameters has been constructed [11–19]. Those kinds of interpolation spline not only have simple mathematical representation, they can be used for the modification of local curves by selecting suitable parameters under the condition that the interpolating data are not changed. The bivariate spline interpolation method does not appear very often in the literature. Indeed, so far there are few such bivariate interpolating splines which have simple and explicit mathematical *
Corresponding author. E-mail address: [email protected] (Q. Duan).
0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.11.094
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
191
representation and can be modified by the parameters. Motivated by the univariate rational spline interpolation, the bivariate rational interpolation with parameters, based only on the values of the function being interpolated, has been studied in [20]. This paper will deal with the bivariate interpolation based on both function values and partial derivative values of the function being interpolated. The paper is arranged as follows. In Section 2, the new bivariate rational spline based on function values and partial derivatives with parameters is constructed. More important is that the symmetric bases of this bivariate interpolation can be derived from the process of its construction, they are discussed in Section 3. Section 4 is about some properties of the interpolation, the concept of integral weights coefficients of the interpolation is given, which describes the ‘‘weight’’ of the interpolation points and interpolating quantities in the local interpolating region. Section 5 is about the stability of the interpolation, it have been proved that the values of the interpolation function must be in an interval related to the interpolating data no matter what the positive parameters might be. An example is given in Section 6, which shows that this interpolation is with good approximation to the function being interpolated. 2. Interpolation n
Let X : [a, b; c, d] be the plane region, and
xi ; y j ; fi;j ;
ofi;j ox
;
ofi;j oy
o ; i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; m be a given set
of data points, where a = x1 < x2 < < xn = b and c = y1 < y2 < < ym = d are the knot spacings, of of ðx;yÞ of ðx;yÞ ; oy at the point (xi, yj) respectively. Let hi = xi+1 xi, lj = yj+1 yj, fi;j ; oxi;j ; oyi;j represent f ðxi ; y j Þ; of ox yy i and g ¼ lj j . First, for each and for any point (x, y) 2 [xi, xi+1; yj, yj+1] in the (x, y)-plane, and let h ¼ xx hi y = yj, j = 1, 2, . . . , m, construct the x-direct interpolating curve P i;j ðxÞ in [xi, xi+1] [18]; this is given by P i;j ðxÞ ¼
pi;j ðxÞ ; qi;j ðxÞ
i ¼ 1; 2; . . . ; n 1;
ð1Þ
where 3
2
pi;j ðxÞ ¼ ð1 hÞ ai;j fi;j þ hð1 hÞ V i;j þ h2 ð1 hÞW i;j þ h3 bi;j fiþ1;j ; qi;j ðxÞ ¼ ð1 hÞai;j þ hbi;j ; and ofi;j ; ox ofiþ1;j ; W i;j ¼ ðai;j þ 2bi;j Þfiþ1;j hi bi;j ox with ai;j > 0, bi;j > 0. This interpolation is called the rational cubic interpolation based on function values and derivatives which satisfies V i;j ¼ ð2ai;j þ bi;j Þfi;j þ hi ai;j
P i;j ðxi Þ ¼ fi;j ;
P i;j ðxiþ1 Þ ¼ fiþ1;j ;
0
P i;j ðxi Þ ¼
ofi;j ; ox
0
P i;j ðxiþ1 Þ ¼
ofiþ1;j . ox
n o of Obviously, the interpolating function P i;j ðxÞ on [xi, xi+1] is unique for the given data xr ; fr;j ; oxr;j ; r ¼ i; i þ 1 and positive parameters ai;j ; bi;j . Using the x-direction interpolation function, P i;j ðxÞ, i = 1, 2, . . . , n 1; j = 1, 2, . . . , m defines the bivariate rational interpolating function in [x1, xn; y1, ym]. For each pair (i, j), i = 1, 2, . . . , n 1 and j = 1, 2, . . . , m 1, let ai,j > 0, bi,j > 0, define the bivariate interpolating function Pi,j(x, y) on [xi, xi+1; yj, yj+1] as follows: P i;j ðx; yÞ ¼
pi;j ðx; yÞ ; qi;j ðyÞ
i ¼ 1; 2; . . . ; n 1; j ¼ 1; 2; . . . ; m 1;
where 3
2
pi;j ðx; yÞ ¼ ð1 gÞ ai;j P i;j ðxÞ þ gð1 gÞ V i;j þ g2 ð1 gÞW i;j þ g3 bi;j P i;jþ1 ðxÞ; qi;j ðyÞ ¼ ð1 gÞai;j þ gbi;j ;
ð2Þ
192
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
and V i;j ¼ ð2ai;j þ bi;j ÞP i;j ðxÞ þ lj ai;j fi;j ðx; y j Þ; W i;j ¼ ðai;j þ 2bi;j ÞP i;jþ1 ðxÞ lj bi;j fi;jþ1 ðx; y jþ1 Þ;
with fi;s ðx; y s Þ ¼ ð1 hÞ
ofi;s ofiþ1;s þh ; oy oy
h 2 ½0; 1; s ¼ j; j þ 1.
ð3Þ
It is obvious that fi;s ðx; y s Þ satisfy fi;s ðxr ; y s Þ ¼
ofr;s ; oy
r ¼ i; i þ 1; s ¼ j; j þ 1.
The term Pi,j(x, y) is called the bivariate rational interpolation based on function values and partial derivative values which satisfies oP i;j ðxr ; y s Þ ofr;s ¼ ; ox ox
P i;j ðxr ; y s Þ ¼ f ðxr ; y s Þ;
oP i;j ðxr ; y s Þ ofr;s ¼ ; oy oy
r ¼ i; i þ 1; s ¼ j; j þ 1.
It is easy to understand that this form of the interpolating function Pi,j(x, y) on [xi, xi+1; yj, yj+1] is unique for ofr;s ofr;s the given data xr ; y s ; fr;s ; ox ; oy ; r ¼ i; i þ 1; s ¼ j; j þ 1 and parameters ai;j ; bi;j , ai;jþ1 ; bi;jþ1 and ai,j, bi,j. 3. The bases of the interpolation From Eqs. (1)–(3), the interpolating function Pi,j(x, y) defined by (2) can be written as follows: jþ1 iþ1 X X ofr;s ofr;s þ cr;s ðh; gÞlj P i;j ðx; yÞ ¼ ar;s ðh; gÞfr;s þ br;s ðh; gÞhi ; ox oy r¼i s¼j
ð4Þ
where 2
ai;j ðh; gÞ ¼
2
ð1 gÞ ð1 hÞ ðð1 þ gÞai;j þ gbi;j Þðð1 þ hÞai;j þ hbi;j Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;j þ hbi;j Þ
ð5Þ
2
aiþ1;j ðh; gÞ ¼
h2 ð1 gÞ ðð1 þ gÞai;j þ gbi;j Þðð1 hÞai;j þ ð2 hÞbi;j Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;j þ hbi;j Þ
ð6Þ
ai;jþ1 ðh; gÞ ¼
g2 ð1 hÞ2 ðð1 gÞai;j þ ð2 gÞbi;j Þðð1 þ hÞai;jþ1 þ hbi;jþ1 Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;jþ1 þ hbi;jþ1 Þ
ð7Þ
aiþ1;jþ1 ðh; gÞ ¼
g2 h2 ðð1 gÞai;j þ ð2 gÞbi;j Þðð1 hÞai;jþ1 þ ð2 hÞbi;jþ1 Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;jþ1 þ hbi;jþ1 Þ 2
bi;j ðh; gÞ ¼
ð8Þ
2
hð1 hÞ ð1 gÞ ai;j ðð1 þ gÞai;j þ gbi;j Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;j þ hbi;j Þ
ð9Þ
h2 ð1 hÞð1 gÞ2 bi;j ðð1 þ gÞai;j þ gbi;j Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;j þ hbi;j Þ
ð10Þ
hð1 hÞ2 g2 ai;jþ1 ðð1 gÞai;j þ ð2 gÞbi;j Þ bi;jþ1 ðh; gÞ ¼ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;jþ1 þ hbi;jþ1 Þ
ð11Þ
h2 ð1 hÞg2 bi;jþ1 ðð1 gÞai;j þ ð2 gÞbi;j Þ biþ1;jþ1 ðh; gÞ ¼ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;jþ1 þ hbi;jþ1 Þ
ð12Þ
biþ1;j ðh; gÞ ¼
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
193
2
ð1 hÞð1 gÞ gai;j ; ð1 gÞai;j þ gbi;j
ð13Þ
ciþ1;j ðh; gÞ ¼
hgð1 gÞ2 ai;j ; ð1 gÞai;j þ gbi;j
ð14Þ
ci;jþ1 ðh; gÞ ¼
ð1 hÞð1 gÞg2 bi;j ; ð1 gÞai;j þ gbi;j
ð15Þ
hg2 ð1 gÞbi;j . ð1 gÞai;j þ gbi;j
ð16Þ
ci;j ðh; gÞ ¼
ciþ1;jþ1 ðh; gÞ ¼
(5)–(16) are called the bases of the interpolation defined by (2) and (4) is the more explicit form of the interpolation. When ai,j = bi,j, ai;j ¼ bi;j and ai;jþ1 ¼ bi;jþ1 , (5)–(16) become ai;j ðh; gÞ ¼ ð1 gÞ2 ð1 þ 2gÞð1 hÞ2 ð1 þ 2hÞ; 2
ð18Þ
2
ð19Þ
aiþ1;j ðh; gÞ ¼ ð1 gÞ ð1 þ 2gÞh ð3 2hÞ; 2
ai;jþ1 ðh; gÞ ¼ g ð3 2gÞð1 hÞ ð1 þ 2hÞ; 2
2
aiþ1;jþ1 ðh; gÞ ¼ g ð3 2gÞh ð3 2hÞ; 2
ð20Þ
2
bi;j ðh; gÞ ¼ ð1 gÞ ð1 þ 2gÞð1 hÞ h; 2
ð21Þ 2
biþ1;j ðh; gÞ ¼ ð1 gÞ ð1 þ 2gÞð1 hÞh ; 2
2
ð17Þ
2
bi;jþ1 ðh; gÞ ¼ g ð3 2gÞð1 hÞ h;
ð23Þ 2
2
ð22Þ
biþ1;jþ1 ðh; gÞ ¼ g ð3 2gÞð1 hÞh ; 2
ci;j ðh; gÞ ¼ ð1 hÞgð1 gÞ ; 2
ciþ1;j ðh; gÞ ¼ hgð1 gÞ ; 2
ci;jþ1 ðh; gÞ ¼ g ð1 gÞð1 hÞ; 2
ciþ1;jþ1 ðh; gÞ ¼ hg ð1 gÞ.
ð24Þ ð25Þ ð26Þ ð27Þ ð28Þ
Eqs. (17)–(28) are called the bases of bivariate Herimite interpolation on a rectangle net. 4. Some properties of the interpolation For the bivariate rational interpolating function Pi,j(x, y) defined by (2), since the base functions ar,s(h, g), r = i, i + 1, s = j, j + 1 satisfy jþ1 iþ1 X X r¼i
ar;s ðh; gÞ ¼ 1;
ð29Þ
s¼j
there are the following unity property. Property 1. If f(x, y) = 1, (x, y) 2 X, Pi,j(x, y) is its interpolating function in [xi, xi+1; yj, yj+1] defined by (2), no matter what positive number the parameters ai,j, bi,j take, the unity property holds, namely Z Z P i;j ðx; yÞ dx dy ¼ hi lj ; D
where D denotes the subregion [xi, xi+1; yj, yj+1]. From (4), define Z Z jþ1 iþ1 X X ofr;s ofr;s þ cr;s lj P i;j ðx; yÞ dx dy ¼ hi lj ar;s fr;s þ br;s hi ; ox oy D r¼i s¼j
ð30Þ
194
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
where ar;s
¼
Z Z
ar;s ðh; gÞ dh dg;
r ¼ i; i þ 1; s ¼ j; j þ 1;
br;s ðh; gÞ dh dg;
r ¼ i; i þ 1; s ¼ j; j þ 1;
cr;s ðh; gÞ dh dg;
r ¼ i; i þ 1; s ¼ j; j þ 1;
½0;1;0;1
br;s ¼ cr;s ¼
Z Z Z Z
½0;1;0;1
½0;1;0;1
calling ar;s , br;s and cr;s the integral weights coefficients of the interpolation defined by (2). It is easy to see that ar;s > 0;
r ¼ i; i þ 1; s ¼ j; j þ 1;
bi;s cr;j
> 0;
biþ1;s < 0;
s ¼ j; j þ 1;
> 0;
cr;jþ1 < 0;
r ¼ i; i þ 1.
For estimating the values of the integral weight coefficient, it is easy to see that P i;j ðxÞ can be rewritten as P i;j ðxÞ ¼ x0;0 ðh; ai;j ; bi;j Þfi;j þ x1;0 ðh; ai;j ; bi;j Þfiþ1;j þ x0;1 ðh; ai;j ; bi;j Þhi
ofi;j ofiþ1;j þ x1;1 ðh; ai;j ; bi;j Þhi ; ox ox
where 2
x0;0 ðh; ai;j ; bi;j Þ ¼
ð1 hÞ ðð1 þ hÞai;j þ hbi;j Þ ; ð1 hÞai;j þ hbi;j
x1;0 ðh; ai;j ; bi;j Þ ¼
h2 ðð1 hÞai;j þ ð2 hÞbi;j Þ ; ð1 hÞai;j þ hbi;j
x0;1 ðh; ai;j ; bi;j Þ ¼
hð1 hÞ ai;j ; ð1 hÞai;j þ hbi;j
2
x1;1 ðh; ai;j ; bi;j Þ
h2 ð1 hÞbi;j ¼ ð1 hÞai;j þ hbi;j
and, from (5)–(16), ar,s(h, g), br,s(h, g), cr,s(h, g), r = i, i + 1, s = j, j + 1 can be rewritten as ai;j ðh; gÞ ¼ x0;0 ðg; ai;j ; bi;j Þx0;0 ðh; ai;j ; bi;j Þ; aiþ1;j ðh; gÞ ¼ x0;0 ðg; ai;j ; bi;j Þx1;0 ðh; ai;j ; bi;j Þ; ai;jþ1 ðh; gÞ ¼ x1;0 ðg; ai;j ; bi;j Þx0;0 ðh; ai;jþ1 ; bi;jþ1 Þ; aiþ1;jþ1 ðh; gÞ ¼ x1;0 ðg; ai;j ; bi;j Þx1;0 ðh; ai;jþ1 ; bi;jþ1 Þ; bi;j ðh; gÞ ¼ x0;0 ðg; ai;j ; bi;j Þx0;1 ðh; ai;j ; bi;j Þ; biþ1;j ðh; gÞ ¼ x0;0 ðg; ai;j ; bi;j Þx1;1 ðh; ai;j ; bi;j Þ; bi;jþ1 ðh; gÞ ¼ x1;0 ðg; ai;j ; bi;j Þx0;1 ðh; ai;jþ1 ; bi;jþ1 Þ; biþ1;jþ1 ðh; gÞ ¼ x1;0 ðg; ai;j ; bi;j Þx1;1 ðh; ai;jþ1 ; bi;jþ1 Þ; ci;j ðh; gÞ ¼ ð1 hÞx0;1 ðg; ai;j ; bi;j Þ; ciþ1;j ðh; gÞ ¼ hx0;1 ðg; ai;j ; bi;j Þ; ci;jþ1 ðh; gÞ ¼ ð1 hÞx1;1 ðg; ai;j ; bi;j Þ; ciþ1;jþ1 ðh; gÞ ¼ hx1;1 ðg; ai;j ; bi;j Þ.
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
195
Define xr;s ¼
Z
1
xr;s ðh; ai;j ; bi;j Þ dh;
r ¼ 0; 1; s ¼ 0; 1;
0
it is easy to show that for any positive parameters ai;j ; bi;j 1 x0;0 2x0;1 ¼ ; 3
x0;0 þ x1;0 ¼ 1;
1 x0;1 x1;1 ¼ ; 6
then 1=3 < x0;0 < 2=3;
1=3 < x1;0 < 2=3;
0 < x0;1 < 1=6;
1=6 < x1;1 < 0.
Thus, it is easy to get the following property: Property 2. For the bivariate interpolation Pi,j(x, y) defined by (2) on the rectangle net, the integral weight coefficients of interpolation satisfy 1=9 < ar;s < 4=9; 0<
bi;s
< 1=9;
r ¼ i; i þ 1; 1=9 <
0 < cr;j < 1=12;
biþ1;s
s ¼ j; j þ 1; < 0;
1=12 < cr;jþ1 < 0;
s ¼ j; j þ 1; r ¼ i; i þ 1.
Furthermore, since ai;j ðh; gÞ þ aiþ1;j ðh; gÞ þ ai;jþ1 ðh; gÞ þ aiþ1;jþ1 ðh; gÞ ¼ 1;
ð31Þ
bi;j ðh; gÞ biþ1;j ðh; gÞ þ bi;jþ1 ðh; gÞ biþ1;jþ1 ðh; gÞ ¼ hð1 hÞ;
ð32Þ
ci;j ðh; gÞ þ ciþ1;j ðh; gÞ ci;jþ1 ðh; gÞ ciþ1;jþ1 ðh; gÞ ¼ gð1 gÞ;
ð33Þ
there is the following property. Property 3. If Pi,j(x, y) is the interpolating function in [xi, xi+1; yj, yj+1] defined by (2), no matter what positive number the parameters ai;j ; bi;j ; ai;jþ1 ; bi;jþ1 ; ai;j ; bi;j take, the integral weight coefficients of interpolation Pi,j(x, y) satisfy the following equations: jþ1 iþ1 X X r¼i
jþ1 iþ1 X X r¼i
s¼j
jþ1 iþ1 X X r¼i
ar;s ¼ 1;
s¼j
s¼j
1 jbr;s j ¼ ; 6 1 jcr;s j ¼ . 6
As shown in Section 3, (17)–(28) are the bases of the bivariate Hermite interpolation on the rectangle net. In this case, it is easy to derive that jþ1 iþ1 X X r¼i
jþ1 iþ1 X X r¼i
br;s ðh; gÞ ¼ hð1 hÞð1 2hÞ;
s¼j
s¼j
cr;s ðh; gÞ ¼ gð1 gÞð1 2gÞ;
196
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
and by the definition of br;s and cr;s it can be shown that jþ1 iþ1 X X br;s ¼ 0; r¼i
s¼j
jþ1 iþ1 X X r¼i
cr;s ¼ 0.
s¼j
Thus, for the bases of the bivariate Hermite interpolation on the rectangle net, there is the following property. Property 4. For the bivariate Hermite interpolation Pi,j(x, y) on the rectangle net, the integral weight coefficients of interpolation satisfy the following equations: jþ1 iþ1 X X ½ar;s þ br;s þ cr;s ¼ 1. r¼i
s¼j
5. Stability of the interpolation This bivariate interpolation is a local interpolation in each sub-rectangle, which depends on the function values and the partial derivative values of the function being interpolated at four vertexes of the sub-rectangle. There are six parameters ai;j , bi;j , ai;jþ1 , bi;jþ1 and ai,j, bi,j in each interpolating region, and when the parameters vary, the interpolating surface changes accordingly. Generally speaking, one do not wish the interpolating surface undulate greatly when the parameters vary so as to keep the surface’s original shape. The theorems in this section show that the values of the interpolating function in each interpolating region must be in an interval related to the interpolating data, it is called the stability of the interpolation. n o of of Theorem 1. For the given interpolation data fr;s ; oxr;s ; oyr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1 , let Pi,j(x, y) be the bivariate interpolation defined by (2) in the rectangle region [xi, xi+1; yj, yj+1], define Q1 ¼ maxffr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1g; ofr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1 ; Q2 ¼ max hi ox ofr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1 Q3 ¼ max lj oy and Q2 þ Q3 4 then, Pi,j(x, y) 6 M for all (x, y) 2 [xi, xi+1; yj, yj+1]. M ¼ Q1 þ
Proof. Since P i;j ðx; yÞ ¼
jþ1 iþ1 X X r¼i
ar;s ðh; gÞfr;s þ br;s ðh; gÞhi
s¼j
and, from (31)–(33) jþ1 iþ1 X X ar;s ðh; gÞ ¼ 1; r¼i
s¼j
jþ1 iþ1 X X r¼i
jþ1 iþ1 X X r¼i
jbr;s ðh; gÞj ¼ hð1 hÞ;
s¼j
s¼j
jcr;s ðh; gÞj ¼ gð1 gÞ;
ofr;s ofr;s þ cr;s ðh; gÞlj ; ox oy
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
197
thus, P i;j ðx; yÞ 6 Q1 þ hð1 hÞQ2 þ gð1 gÞQ3 6 Q1 þ The proof completes.
Q2 þ Q3 . 4
h
Similarly, there is the following Theorem 2. n o of of Theorem 2. For the given interpolation data fr;s ; oxr;s ; oyr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1 , let Pi,j(x, y) be the bivariate interpolation defined by (2) in the rectangle region [xi, xi+1; yj, yj+1], define Q1 ¼ minffr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1g; ofr;s Q2 ¼ min hi ; r ¼ i; i þ 1; s ¼ j; j þ 1 ; ox ofr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1 Q3 ¼ min lj oy and Q2 þ Q3 4 then, Pi,j(x, y) P N for all (x, y) 2 [xi, xi+1; yj, yj+1]. N ¼ Q1 þ
6. Numerical example qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Assume f ðx; yÞ ¼ 1 ð1 xÞ2 ð1 yÞ2 , (x, y) 2 [0.5, 1.5; 0.5, 1.5], let hi = 1/5, li = 1/5, namely, xi = 0.5 + i/5, yi = 0.5 + i/5, and let ai;j ¼ ai;j ¼ 1:1, bi;j ¼ bi;j ¼ 0:99 for all i = 1, 2, . . . , 5, j = 1, 2, . . . , 5, P(x, y) is the interpolating function defined by (2) of f(x, y) in above region. Fig. 1 shows the surface P(x, y). To see how the interpolating function P(x, y) approximate f(x, y), let us compute the values of P(x, y) and f(x, y) at some points. For example, let x = 0.9 and x = 0.98 respectively, y = 0.5 + i/50, i = 0, 1, 2 . . . , 25, the following Table 1 shows that P(x, y) approximates f(x, y) very well in the interpolating region.
Fig. 1. The graph of the interpolating surface.
198
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
Table 1 The values of the interpolating function P(x, y) x
y
P(x, y)
f(x, y)
x
y
P(x, y)
f(x, y)
.9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000
.5000 .5200 .5400 .5600 .5800 .6000 .6200 .6400 .6600 .6800 .7000 .7200 .7400 .7600 .7800 .8000 .8200 .8400 .8600 .8800 .9000 .9200 .9400 .9600 .9800 1.0000
.8602 .8716 .8823 .8925 .9021 .9111 .9196 .9276 .9351 .9421 .9487 .9548 .9604 .9656 .9704 .9747 .9786 .9821 .9851 .9877 .9899 .9918 .9932 .9942 .9948 .9950
.8602 .8716 .8823 .8924 .9020 .9110 .9196 .9276 .9351 .9421 .9487 .9548 .9604 .9656 .9704 .9747 .9786 .9820 .9851 .9877 .9899 .9918 .9932 .9942 .9948 .9950
.9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800
.5000 .5200 .5400 .5600 .5800 .6000 .6200 .6400 .6600 .6800 .7000 .7200 .7400 .7600 .7800 .8000 .8200 .8400 .8600 .8800 .9000 .9200 .9400 .9600 .9800 1.0000
.8658 .8771 .8878 .8979 .9075 .9164 .9249 .9328 .9402 .9472 .9537 .9598 .9655 .9706 .9754 .9797 .9835 .9870 .9900 .9926 .9948 .9966 .9980 .9990 .9996 .9998
.8658 .8770 .8877 .8978 .9073 .9163 .9248 .9327 .9402 .9472 .9537 .9598 .9654 .9706 .9753 .9796 .9835 .9869 .9899 .9926 .9948 .9966 .9980 .9990 .9996 .9998
7. Conclusions This paper gives the simple and explicit expression of the bivariate spline; the spline depends on the function values and partial derivatives of the function being interpolated. There are some parameters in the interpolating functions: ai;j , bi;j and ai,j, bi,j; ai;j , bi;j are for univariate interpolation on the x-direction as described by (1); ai,j, bi,j are for bivariate interpolating surface as described by (2). This bivariate spline has two expressions: form (2) and form (4). Form (2) is convenient for use in practical applications, and form (4) is convenient in theoretical analysis. For each pitch of the interpolating surface, the value of the interpolating function depends on the interpolant data, the function values and partial derivatives. Property 2 shows that the function values and the partial derivative values play not the same roles in the interpolation. That is, the magnitude of the integral weight coefficient values describe the magnitude of ‘‘function’’ of the interpolating knots and data. Acknowledgements The support of the National Nature Science Foundation of China, the Nature Science Foundation of Shandong Province of China and the Specialized Research Fund for the Doctoral Program of Higher Education of China are gratefully acknowledged. References [1] [2] [3] [4]
R.H. Wang, Numerical Rational Approximation, Shanghai Science-Technology Press, China, 1980. G. Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, Academic press, 1988. C.K. Chui, Multivariate Spline, SIAM, 1988. R.H. Wang, Multivariate Spline Functions and Their Applications, Kluwer Academic Publishers, Beijing/New York/Dordrecht/ Boston/London, 2001.
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
199
[5] P.E. Be´zier, The Mathematical Basis of The UNISURF CAD System, Butterworth, London, 1986. [6] P. Dierck, B. Tytgat, Generating the Be´zier points of b-spline curve, Comput. Aided Geom. Des. 6 (1989) 279–291. [7] B.A. Narsky, The beta-spline: a local representation based on shape parameters and fundamental geometric measure, Ph.D. thesis, University of Utah, 1981. [8] L. Piegl, On NURBS: A survey, IEEE Comput. Graph. Appl. 11 (1) (1991) 55–71. [9] G.M. Nielson, CAGD’s top ten: What to watch, IEEE Comput. Graph. Autom. January (1993) 35–37. [10] K. Konno, H. Chiyokura, An approach of designing and controlling free-form surfaces by using NURBS boundary Gregory patches, Comput. Aided Geom. Des. 13 (1996) 825–849. [11] P. Comninos, An interpolating piecewise bicubic surface with shape parameters, Comput. Graph. 25 (2001) 463–481. [12] R. Mu¨ller, Universal parametrization and interpolation on cubic surfaces, Comput. Aided Geom. Des. 19 (2002) 479–502. [13] J.W. Schmidt, W. Hess, Positive interpolation with rational quadratic spline, Computing 38 (1987) 261–267. [14] M. Sarfraz, Convexity preserving piecewise rational interpolation for planar curves, Bull. Kor. Math. Soc. 29 (1992) 193–200. [15] J.A. Gregory, M. Sarfraz, P.K. Yuen, Interactive curve design using C2 rational splines, Comput. Graph. 18 (1994) 153–159. [16] Q. Duan, K. Djidjeli, W.G. Price, E.H. Twizell, Rational cubic spline based on function values, Comput. Graph. 22 (1998) 479–486. [17] Q. Duan, K. Djidjeli, W.G. Price, E.H. Twizell, Weighted rational cubic spline interpolation and its approximation, J. Comput. Appl. Math. 117 (2000) 121–135. [18] Q. Duan, A. Liu, F. Cheng, Constrained rational cubic spline and its application, J. Comput. Math. 19 (2001) 143–150. [19] Q. Duan, L. Wang, E.H. Twizell, A novel approach to the convexity control of interpolant curves, Commun. Numer. Meth. Eng. 19 (2003) 833–845. [20] Q. Duan, L. Wang, E.H. Twizell, A new bivariate rational interpolation based on function values, Inform. Sci. 166 (2004) 181–191.
A bivariate rational interpolation and the properties Qi Duan
a,*
, Yunfeng Zhang a, E.H. Twizell
b
a b
School of Mathematics and System Science, Shandong University, Jinan 250100, China School of Information System, Computing and Mathematics, Brunel University, Uxbridge, Middlesex, England UB8 3PH, United Kingdom
Abstract In this paper a bivariate rational interpolation is constructed using both function values and partial derivatives of the function being interpolated as the interpolation data. The interpolation function has a simple and explicit rational mathematical representation with parameters, and it can be expressed by the symmetric bases. It is proved that the interpolation is stable. The concept of integral weights coefficients of the interpolation is given, which describes the ‘‘weight’’ of the interpolation points and the quantity as the interpolation data in the local interpolating region. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Bivariate interpolation; Rational spline; Computer-aided geometric design
1. Introduction The construction method of the curve and surface and the mathematical description of them is a key issue in computer-aided geometric design. There are many ways to tackle this problem [1–10], for example, the polynomial spline method, the NURBS method and the Be´zier method. These methods are effective and applied widely in shape design of industrial products. Generally speaking, most of the polynomial spline methods are the interpolating methods, which means that the curves or surfaces constructed by the methods pass through the interpolating points. The NURBS and Be´zier methods are the so-called ‘‘no-interpolating type’’ methods; this means that the constructed curve and surface do not pass through the given data, and the given points play the role of the control points. In general, therefore, the interpolation method gives a better approximation to the function being approximated than that constructed by the other methods. In recent years, the univariate rational spline interpolation with parameters has been constructed [11–19]. Those kinds of interpolation spline not only have simple mathematical representation, they can be used for the modification of local curves by selecting suitable parameters under the condition that the interpolating data are not changed. The bivariate spline interpolation method does not appear very often in the literature. Indeed, so far there are few such bivariate interpolating splines which have simple and explicit mathematical *
Corresponding author. E-mail address: [email protected] (Q. Duan).
0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.11.094
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
191
representation and can be modified by the parameters. Motivated by the univariate rational spline interpolation, the bivariate rational interpolation with parameters, based only on the values of the function being interpolated, has been studied in [20]. This paper will deal with the bivariate interpolation based on both function values and partial derivative values of the function being interpolated. The paper is arranged as follows. In Section 2, the new bivariate rational spline based on function values and partial derivatives with parameters is constructed. More important is that the symmetric bases of this bivariate interpolation can be derived from the process of its construction, they are discussed in Section 3. Section 4 is about some properties of the interpolation, the concept of integral weights coefficients of the interpolation is given, which describes the ‘‘weight’’ of the interpolation points and interpolating quantities in the local interpolating region. Section 5 is about the stability of the interpolation, it have been proved that the values of the interpolation function must be in an interval related to the interpolating data no matter what the positive parameters might be. An example is given in Section 6, which shows that this interpolation is with good approximation to the function being interpolated. 2. Interpolation n
Let X : [a, b; c, d] be the plane region, and
xi ; y j ; fi;j ;
ofi;j ox
;
ofi;j oy
o ; i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; m be a given set
of data points, where a = x1 < x2 < < xn = b and c = y1 < y2 < < ym = d are the knot spacings, of of ðx;yÞ of ðx;yÞ ; oy at the point (xi, yj) respectively. Let hi = xi+1 xi, lj = yj+1 yj, fi;j ; oxi;j ; oyi;j represent f ðxi ; y j Þ; of ox yy i and g ¼ lj j . First, for each and for any point (x, y) 2 [xi, xi+1; yj, yj+1] in the (x, y)-plane, and let h ¼ xx hi y = yj, j = 1, 2, . . . , m, construct the x-direct interpolating curve P i;j ðxÞ in [xi, xi+1] [18]; this is given by P i;j ðxÞ ¼
pi;j ðxÞ ; qi;j ðxÞ
i ¼ 1; 2; . . . ; n 1;
ð1Þ
where 3
2
pi;j ðxÞ ¼ ð1 hÞ ai;j fi;j þ hð1 hÞ V i;j þ h2 ð1 hÞW i;j þ h3 bi;j fiþ1;j ; qi;j ðxÞ ¼ ð1 hÞai;j þ hbi;j ; and ofi;j ; ox ofiþ1;j ; W i;j ¼ ðai;j þ 2bi;j Þfiþ1;j hi bi;j ox with ai;j > 0, bi;j > 0. This interpolation is called the rational cubic interpolation based on function values and derivatives which satisfies V i;j ¼ ð2ai;j þ bi;j Þfi;j þ hi ai;j
P i;j ðxi Þ ¼ fi;j ;
P i;j ðxiþ1 Þ ¼ fiþ1;j ;
0
P i;j ðxi Þ ¼
ofi;j ; ox
0
P i;j ðxiþ1 Þ ¼
ofiþ1;j . ox
n o of Obviously, the interpolating function P i;j ðxÞ on [xi, xi+1] is unique for the given data xr ; fr;j ; oxr;j ; r ¼ i; i þ 1 and positive parameters ai;j ; bi;j . Using the x-direction interpolation function, P i;j ðxÞ, i = 1, 2, . . . , n 1; j = 1, 2, . . . , m defines the bivariate rational interpolating function in [x1, xn; y1, ym]. For each pair (i, j), i = 1, 2, . . . , n 1 and j = 1, 2, . . . , m 1, let ai,j > 0, bi,j > 0, define the bivariate interpolating function Pi,j(x, y) on [xi, xi+1; yj, yj+1] as follows: P i;j ðx; yÞ ¼
pi;j ðx; yÞ ; qi;j ðyÞ
i ¼ 1; 2; . . . ; n 1; j ¼ 1; 2; . . . ; m 1;
where 3
2
pi;j ðx; yÞ ¼ ð1 gÞ ai;j P i;j ðxÞ þ gð1 gÞ V i;j þ g2 ð1 gÞW i;j þ g3 bi;j P i;jþ1 ðxÞ; qi;j ðyÞ ¼ ð1 gÞai;j þ gbi;j ;
ð2Þ
192
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
and V i;j ¼ ð2ai;j þ bi;j ÞP i;j ðxÞ þ lj ai;j fi;j ðx; y j Þ; W i;j ¼ ðai;j þ 2bi;j ÞP i;jþ1 ðxÞ lj bi;j fi;jþ1 ðx; y jþ1 Þ;
with fi;s ðx; y s Þ ¼ ð1 hÞ
ofi;s ofiþ1;s þh ; oy oy
h 2 ½0; 1; s ¼ j; j þ 1.
ð3Þ
It is obvious that fi;s ðx; y s Þ satisfy fi;s ðxr ; y s Þ ¼
ofr;s ; oy
r ¼ i; i þ 1; s ¼ j; j þ 1.
The term Pi,j(x, y) is called the bivariate rational interpolation based on function values and partial derivative values which satisfies oP i;j ðxr ; y s Þ ofr;s ¼ ; ox ox
P i;j ðxr ; y s Þ ¼ f ðxr ; y s Þ;
oP i;j ðxr ; y s Þ ofr;s ¼ ; oy oy
r ¼ i; i þ 1; s ¼ j; j þ 1.
It is easy to understand that this form of the interpolating function Pi,j(x, y) on [xi, xi+1; yj, yj+1] is unique for ofr;s ofr;s the given data xr ; y s ; fr;s ; ox ; oy ; r ¼ i; i þ 1; s ¼ j; j þ 1 and parameters ai;j ; bi;j , ai;jþ1 ; bi;jþ1 and ai,j, bi,j. 3. The bases of the interpolation From Eqs. (1)–(3), the interpolating function Pi,j(x, y) defined by (2) can be written as follows: jþ1 iþ1 X X ofr;s ofr;s þ cr;s ðh; gÞlj P i;j ðx; yÞ ¼ ar;s ðh; gÞfr;s þ br;s ðh; gÞhi ; ox oy r¼i s¼j
ð4Þ
where 2
ai;j ðh; gÞ ¼
2
ð1 gÞ ð1 hÞ ðð1 þ gÞai;j þ gbi;j Þðð1 þ hÞai;j þ hbi;j Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;j þ hbi;j Þ
ð5Þ
2
aiþ1;j ðh; gÞ ¼
h2 ð1 gÞ ðð1 þ gÞai;j þ gbi;j Þðð1 hÞai;j þ ð2 hÞbi;j Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;j þ hbi;j Þ
ð6Þ
ai;jþ1 ðh; gÞ ¼
g2 ð1 hÞ2 ðð1 gÞai;j þ ð2 gÞbi;j Þðð1 þ hÞai;jþ1 þ hbi;jþ1 Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;jþ1 þ hbi;jþ1 Þ
ð7Þ
aiþ1;jþ1 ðh; gÞ ¼
g2 h2 ðð1 gÞai;j þ ð2 gÞbi;j Þðð1 hÞai;jþ1 þ ð2 hÞbi;jþ1 Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;jþ1 þ hbi;jþ1 Þ 2
bi;j ðh; gÞ ¼
ð8Þ
2
hð1 hÞ ð1 gÞ ai;j ðð1 þ gÞai;j þ gbi;j Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;j þ hbi;j Þ
ð9Þ
h2 ð1 hÞð1 gÞ2 bi;j ðð1 þ gÞai;j þ gbi;j Þ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;j þ hbi;j Þ
ð10Þ
hð1 hÞ2 g2 ai;jþ1 ðð1 gÞai;j þ ð2 gÞbi;j Þ bi;jþ1 ðh; gÞ ¼ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;jþ1 þ hbi;jþ1 Þ
ð11Þ
h2 ð1 hÞg2 bi;jþ1 ðð1 gÞai;j þ ð2 gÞbi;j Þ biþ1;jþ1 ðh; gÞ ¼ ; ðð1 gÞai;j þ gbi;j Þðð1 hÞai;jþ1 þ hbi;jþ1 Þ
ð12Þ
biþ1;j ðh; gÞ ¼
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
193
2
ð1 hÞð1 gÞ gai;j ; ð1 gÞai;j þ gbi;j
ð13Þ
ciþ1;j ðh; gÞ ¼
hgð1 gÞ2 ai;j ; ð1 gÞai;j þ gbi;j
ð14Þ
ci;jþ1 ðh; gÞ ¼
ð1 hÞð1 gÞg2 bi;j ; ð1 gÞai;j þ gbi;j
ð15Þ
hg2 ð1 gÞbi;j . ð1 gÞai;j þ gbi;j
ð16Þ
ci;j ðh; gÞ ¼
ciþ1;jþ1 ðh; gÞ ¼
(5)–(16) are called the bases of the interpolation defined by (2) and (4) is the more explicit form of the interpolation. When ai,j = bi,j, ai;j ¼ bi;j and ai;jþ1 ¼ bi;jþ1 , (5)–(16) become ai;j ðh; gÞ ¼ ð1 gÞ2 ð1 þ 2gÞð1 hÞ2 ð1 þ 2hÞ; 2
ð18Þ
2
ð19Þ
aiþ1;j ðh; gÞ ¼ ð1 gÞ ð1 þ 2gÞh ð3 2hÞ; 2
ai;jþ1 ðh; gÞ ¼ g ð3 2gÞð1 hÞ ð1 þ 2hÞ; 2
2
aiþ1;jþ1 ðh; gÞ ¼ g ð3 2gÞh ð3 2hÞ; 2
ð20Þ
2
bi;j ðh; gÞ ¼ ð1 gÞ ð1 þ 2gÞð1 hÞ h; 2
ð21Þ 2
biþ1;j ðh; gÞ ¼ ð1 gÞ ð1 þ 2gÞð1 hÞh ; 2
2
ð17Þ
2
bi;jþ1 ðh; gÞ ¼ g ð3 2gÞð1 hÞ h;
ð23Þ 2
2
ð22Þ
biþ1;jþ1 ðh; gÞ ¼ g ð3 2gÞð1 hÞh ; 2
ci;j ðh; gÞ ¼ ð1 hÞgð1 gÞ ; 2
ciþ1;j ðh; gÞ ¼ hgð1 gÞ ; 2
ci;jþ1 ðh; gÞ ¼ g ð1 gÞð1 hÞ; 2
ciþ1;jþ1 ðh; gÞ ¼ hg ð1 gÞ.
ð24Þ ð25Þ ð26Þ ð27Þ ð28Þ
Eqs. (17)–(28) are called the bases of bivariate Herimite interpolation on a rectangle net. 4. Some properties of the interpolation For the bivariate rational interpolating function Pi,j(x, y) defined by (2), since the base functions ar,s(h, g), r = i, i + 1, s = j, j + 1 satisfy jþ1 iþ1 X X r¼i
ar;s ðh; gÞ ¼ 1;
ð29Þ
s¼j
there are the following unity property. Property 1. If f(x, y) = 1, (x, y) 2 X, Pi,j(x, y) is its interpolating function in [xi, xi+1; yj, yj+1] defined by (2), no matter what positive number the parameters ai,j, bi,j take, the unity property holds, namely Z Z P i;j ðx; yÞ dx dy ¼ hi lj ; D
where D denotes the subregion [xi, xi+1; yj, yj+1]. From (4), define Z Z jþ1 iþ1 X X ofr;s ofr;s þ cr;s lj P i;j ðx; yÞ dx dy ¼ hi lj ar;s fr;s þ br;s hi ; ox oy D r¼i s¼j
ð30Þ
194
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
where ar;s
¼
Z Z
ar;s ðh; gÞ dh dg;
r ¼ i; i þ 1; s ¼ j; j þ 1;
br;s ðh; gÞ dh dg;
r ¼ i; i þ 1; s ¼ j; j þ 1;
cr;s ðh; gÞ dh dg;
r ¼ i; i þ 1; s ¼ j; j þ 1;
½0;1;0;1
br;s ¼ cr;s ¼
Z Z Z Z
½0;1;0;1
½0;1;0;1
calling ar;s , br;s and cr;s the integral weights coefficients of the interpolation defined by (2). It is easy to see that ar;s > 0;
r ¼ i; i þ 1; s ¼ j; j þ 1;
bi;s cr;j
> 0;
biþ1;s < 0;
s ¼ j; j þ 1;
> 0;
cr;jþ1 < 0;
r ¼ i; i þ 1.
For estimating the values of the integral weight coefficient, it is easy to see that P i;j ðxÞ can be rewritten as P i;j ðxÞ ¼ x0;0 ðh; ai;j ; bi;j Þfi;j þ x1;0 ðh; ai;j ; bi;j Þfiþ1;j þ x0;1 ðh; ai;j ; bi;j Þhi
ofi;j ofiþ1;j þ x1;1 ðh; ai;j ; bi;j Þhi ; ox ox
where 2
x0;0 ðh; ai;j ; bi;j Þ ¼
ð1 hÞ ðð1 þ hÞai;j þ hbi;j Þ ; ð1 hÞai;j þ hbi;j
x1;0 ðh; ai;j ; bi;j Þ ¼
h2 ðð1 hÞai;j þ ð2 hÞbi;j Þ ; ð1 hÞai;j þ hbi;j
x0;1 ðh; ai;j ; bi;j Þ ¼
hð1 hÞ ai;j ; ð1 hÞai;j þ hbi;j
2
x1;1 ðh; ai;j ; bi;j Þ
h2 ð1 hÞbi;j ¼ ð1 hÞai;j þ hbi;j
and, from (5)–(16), ar,s(h, g), br,s(h, g), cr,s(h, g), r = i, i + 1, s = j, j + 1 can be rewritten as ai;j ðh; gÞ ¼ x0;0 ðg; ai;j ; bi;j Þx0;0 ðh; ai;j ; bi;j Þ; aiþ1;j ðh; gÞ ¼ x0;0 ðg; ai;j ; bi;j Þx1;0 ðh; ai;j ; bi;j Þ; ai;jþ1 ðh; gÞ ¼ x1;0 ðg; ai;j ; bi;j Þx0;0 ðh; ai;jþ1 ; bi;jþ1 Þ; aiþ1;jþ1 ðh; gÞ ¼ x1;0 ðg; ai;j ; bi;j Þx1;0 ðh; ai;jþ1 ; bi;jþ1 Þ; bi;j ðh; gÞ ¼ x0;0 ðg; ai;j ; bi;j Þx0;1 ðh; ai;j ; bi;j Þ; biþ1;j ðh; gÞ ¼ x0;0 ðg; ai;j ; bi;j Þx1;1 ðh; ai;j ; bi;j Þ; bi;jþ1 ðh; gÞ ¼ x1;0 ðg; ai;j ; bi;j Þx0;1 ðh; ai;jþ1 ; bi;jþ1 Þ; biþ1;jþ1 ðh; gÞ ¼ x1;0 ðg; ai;j ; bi;j Þx1;1 ðh; ai;jþ1 ; bi;jþ1 Þ; ci;j ðh; gÞ ¼ ð1 hÞx0;1 ðg; ai;j ; bi;j Þ; ciþ1;j ðh; gÞ ¼ hx0;1 ðg; ai;j ; bi;j Þ; ci;jþ1 ðh; gÞ ¼ ð1 hÞx1;1 ðg; ai;j ; bi;j Þ; ciþ1;jþ1 ðh; gÞ ¼ hx1;1 ðg; ai;j ; bi;j Þ.
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
195
Define xr;s ¼
Z
1
xr;s ðh; ai;j ; bi;j Þ dh;
r ¼ 0; 1; s ¼ 0; 1;
0
it is easy to show that for any positive parameters ai;j ; bi;j 1 x0;0 2x0;1 ¼ ; 3
x0;0 þ x1;0 ¼ 1;
1 x0;1 x1;1 ¼ ; 6
then 1=3 < x0;0 < 2=3;
1=3 < x1;0 < 2=3;
0 < x0;1 < 1=6;
1=6 < x1;1 < 0.
Thus, it is easy to get the following property: Property 2. For the bivariate interpolation Pi,j(x, y) defined by (2) on the rectangle net, the integral weight coefficients of interpolation satisfy 1=9 < ar;s < 4=9; 0<
bi;s
< 1=9;
r ¼ i; i þ 1; 1=9 <
0 < cr;j < 1=12;
biþ1;s
s ¼ j; j þ 1; < 0;
1=12 < cr;jþ1 < 0;
s ¼ j; j þ 1; r ¼ i; i þ 1.
Furthermore, since ai;j ðh; gÞ þ aiþ1;j ðh; gÞ þ ai;jþ1 ðh; gÞ þ aiþ1;jþ1 ðh; gÞ ¼ 1;
ð31Þ
bi;j ðh; gÞ biþ1;j ðh; gÞ þ bi;jþ1 ðh; gÞ biþ1;jþ1 ðh; gÞ ¼ hð1 hÞ;
ð32Þ
ci;j ðh; gÞ þ ciþ1;j ðh; gÞ ci;jþ1 ðh; gÞ ciþ1;jþ1 ðh; gÞ ¼ gð1 gÞ;
ð33Þ
there is the following property. Property 3. If Pi,j(x, y) is the interpolating function in [xi, xi+1; yj, yj+1] defined by (2), no matter what positive number the parameters ai;j ; bi;j ; ai;jþ1 ; bi;jþ1 ; ai;j ; bi;j take, the integral weight coefficients of interpolation Pi,j(x, y) satisfy the following equations: jþ1 iþ1 X X r¼i
jþ1 iþ1 X X r¼i
s¼j
jþ1 iþ1 X X r¼i
ar;s ¼ 1;
s¼j
s¼j
1 jbr;s j ¼ ; 6 1 jcr;s j ¼ . 6
As shown in Section 3, (17)–(28) are the bases of the bivariate Hermite interpolation on the rectangle net. In this case, it is easy to derive that jþ1 iþ1 X X r¼i
jþ1 iþ1 X X r¼i
br;s ðh; gÞ ¼ hð1 hÞð1 2hÞ;
s¼j
s¼j
cr;s ðh; gÞ ¼ gð1 gÞð1 2gÞ;
196
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
and by the definition of br;s and cr;s it can be shown that jþ1 iþ1 X X br;s ¼ 0; r¼i
s¼j
jþ1 iþ1 X X r¼i
cr;s ¼ 0.
s¼j
Thus, for the bases of the bivariate Hermite interpolation on the rectangle net, there is the following property. Property 4. For the bivariate Hermite interpolation Pi,j(x, y) on the rectangle net, the integral weight coefficients of interpolation satisfy the following equations: jþ1 iþ1 X X ½ar;s þ br;s þ cr;s ¼ 1. r¼i
s¼j
5. Stability of the interpolation This bivariate interpolation is a local interpolation in each sub-rectangle, which depends on the function values and the partial derivative values of the function being interpolated at four vertexes of the sub-rectangle. There are six parameters ai;j , bi;j , ai;jþ1 , bi;jþ1 and ai,j, bi,j in each interpolating region, and when the parameters vary, the interpolating surface changes accordingly. Generally speaking, one do not wish the interpolating surface undulate greatly when the parameters vary so as to keep the surface’s original shape. The theorems in this section show that the values of the interpolating function in each interpolating region must be in an interval related to the interpolating data, it is called the stability of the interpolation. n o of of Theorem 1. For the given interpolation data fr;s ; oxr;s ; oyr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1 , let Pi,j(x, y) be the bivariate interpolation defined by (2) in the rectangle region [xi, xi+1; yj, yj+1], define Q1 ¼ maxffr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1g; ofr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1 ; Q2 ¼ max hi ox ofr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1 Q3 ¼ max lj oy and Q2 þ Q3 4 then, Pi,j(x, y) 6 M for all (x, y) 2 [xi, xi+1; yj, yj+1]. M ¼ Q1 þ
Proof. Since P i;j ðx; yÞ ¼
jþ1 iþ1 X X r¼i
ar;s ðh; gÞfr;s þ br;s ðh; gÞhi
s¼j
and, from (31)–(33) jþ1 iþ1 X X ar;s ðh; gÞ ¼ 1; r¼i
s¼j
jþ1 iþ1 X X r¼i
jþ1 iþ1 X X r¼i
jbr;s ðh; gÞj ¼ hð1 hÞ;
s¼j
s¼j
jcr;s ðh; gÞj ¼ gð1 gÞ;
ofr;s ofr;s þ cr;s ðh; gÞlj ; ox oy
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
197
thus, P i;j ðx; yÞ 6 Q1 þ hð1 hÞQ2 þ gð1 gÞQ3 6 Q1 þ The proof completes.
Q2 þ Q3 . 4
h
Similarly, there is the following Theorem 2. n o of of Theorem 2. For the given interpolation data fr;s ; oxr;s ; oyr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1 , let Pi,j(x, y) be the bivariate interpolation defined by (2) in the rectangle region [xi, xi+1; yj, yj+1], define Q1 ¼ minffr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1g; ofr;s Q2 ¼ min hi ; r ¼ i; i þ 1; s ¼ j; j þ 1 ; ox ofr;s ; r ¼ i; i þ 1; s ¼ j; j þ 1 Q3 ¼ min lj oy and Q2 þ Q3 4 then, Pi,j(x, y) P N for all (x, y) 2 [xi, xi+1; yj, yj+1]. N ¼ Q1 þ
6. Numerical example qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Assume f ðx; yÞ ¼ 1 ð1 xÞ2 ð1 yÞ2 , (x, y) 2 [0.5, 1.5; 0.5, 1.5], let hi = 1/5, li = 1/5, namely, xi = 0.5 + i/5, yi = 0.5 + i/5, and let ai;j ¼ ai;j ¼ 1:1, bi;j ¼ bi;j ¼ 0:99 for all i = 1, 2, . . . , 5, j = 1, 2, . . . , 5, P(x, y) is the interpolating function defined by (2) of f(x, y) in above region. Fig. 1 shows the surface P(x, y). To see how the interpolating function P(x, y) approximate f(x, y), let us compute the values of P(x, y) and f(x, y) at some points. For example, let x = 0.9 and x = 0.98 respectively, y = 0.5 + i/50, i = 0, 1, 2 . . . , 25, the following Table 1 shows that P(x, y) approximates f(x, y) very well in the interpolating region.
Fig. 1. The graph of the interpolating surface.
198
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
Table 1 The values of the interpolating function P(x, y) x
y
P(x, y)
f(x, y)
x
y
P(x, y)
f(x, y)
.9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000 .9000
.5000 .5200 .5400 .5600 .5800 .6000 .6200 .6400 .6600 .6800 .7000 .7200 .7400 .7600 .7800 .8000 .8200 .8400 .8600 .8800 .9000 .9200 .9400 .9600 .9800 1.0000
.8602 .8716 .8823 .8925 .9021 .9111 .9196 .9276 .9351 .9421 .9487 .9548 .9604 .9656 .9704 .9747 .9786 .9821 .9851 .9877 .9899 .9918 .9932 .9942 .9948 .9950
.8602 .8716 .8823 .8924 .9020 .9110 .9196 .9276 .9351 .9421 .9487 .9548 .9604 .9656 .9704 .9747 .9786 .9820 .9851 .9877 .9899 .9918 .9932 .9942 .9948 .9950
.9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800 .9800
.5000 .5200 .5400 .5600 .5800 .6000 .6200 .6400 .6600 .6800 .7000 .7200 .7400 .7600 .7800 .8000 .8200 .8400 .8600 .8800 .9000 .9200 .9400 .9600 .9800 1.0000
.8658 .8771 .8878 .8979 .9075 .9164 .9249 .9328 .9402 .9472 .9537 .9598 .9655 .9706 .9754 .9797 .9835 .9870 .9900 .9926 .9948 .9966 .9980 .9990 .9996 .9998
.8658 .8770 .8877 .8978 .9073 .9163 .9248 .9327 .9402 .9472 .9537 .9598 .9654 .9706 .9753 .9796 .9835 .9869 .9899 .9926 .9948 .9966 .9980 .9990 .9996 .9998
7. Conclusions This paper gives the simple and explicit expression of the bivariate spline; the spline depends on the function values and partial derivatives of the function being interpolated. There are some parameters in the interpolating functions: ai;j , bi;j and ai,j, bi,j; ai;j , bi;j are for univariate interpolation on the x-direction as described by (1); ai,j, bi,j are for bivariate interpolating surface as described by (2). This bivariate spline has two expressions: form (2) and form (4). Form (2) is convenient for use in practical applications, and form (4) is convenient in theoretical analysis. For each pitch of the interpolating surface, the value of the interpolating function depends on the interpolant data, the function values and partial derivatives. Property 2 shows that the function values and the partial derivative values play not the same roles in the interpolation. That is, the magnitude of the integral weight coefficient values describe the magnitude of ‘‘function’’ of the interpolating knots and data. Acknowledgements The support of the National Nature Science Foundation of China, the Nature Science Foundation of Shandong Province of China and the Specialized Research Fund for the Doctoral Program of Higher Education of China are gratefully acknowledged. References [1] [2] [3] [4]
R.H. Wang, Numerical Rational Approximation, Shanghai Science-Technology Press, China, 1980. G. Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, Academic press, 1988. C.K. Chui, Multivariate Spline, SIAM, 1988. R.H. Wang, Multivariate Spline Functions and Their Applications, Kluwer Academic Publishers, Beijing/New York/Dordrecht/ Boston/London, 2001.
Q. Duan et al. / Applied Mathematics and Computation 179 (2006) 190–199
199
[5] P.E. Be´zier, The Mathematical Basis of The UNISURF CAD System, Butterworth, London, 1986. [6] P. Dierck, B. Tytgat, Generating the Be´zier points of b-spline curve, Comput. Aided Geom. Des. 6 (1989) 279–291. [7] B.A. Narsky, The beta-spline: a local representation based on shape parameters and fundamental geometric measure, Ph.D. thesis, University of Utah, 1981. [8] L. Piegl, On NURBS: A survey, IEEE Comput. Graph. Appl. 11 (1) (1991) 55–71. [9] G.M. Nielson, CAGD’s top ten: What to watch, IEEE Comput. Graph. Autom. January (1993) 35–37. [10] K. Konno, H. Chiyokura, An approach of designing and controlling free-form surfaces by using NURBS boundary Gregory patches, Comput. Aided Geom. Des. 13 (1996) 825–849. [11] P. Comninos, An interpolating piecewise bicubic surface with shape parameters, Comput. Graph. 25 (2001) 463–481. [12] R. Mu¨ller, Universal parametrization and interpolation on cubic surfaces, Comput. Aided Geom. Des. 19 (2002) 479–502. [13] J.W. Schmidt, W. Hess, Positive interpolation with rational quadratic spline, Computing 38 (1987) 261–267. [14] M. Sarfraz, Convexity preserving piecewise rational interpolation for planar curves, Bull. Kor. Math. Soc. 29 (1992) 193–200. [15] J.A. Gregory, M. Sarfraz, P.K. Yuen, Interactive curve design using C2 rational splines, Comput. Graph. 18 (1994) 153–159. [16] Q. Duan, K. Djidjeli, W.G. Price, E.H. Twizell, Rational cubic spline based on function values, Comput. Graph. 22 (1998) 479–486. [17] Q. Duan, K. Djidjeli, W.G. Price, E.H. Twizell, Weighted rational cubic spline interpolation and its approximation, J. Comput. Appl. Math. 117 (2000) 121–135. [18] Q. Duan, A. Liu, F. Cheng, Constrained rational cubic spline and its application, J. Comput. Math. 19 (2001) 143–150. [19] Q. Duan, L. Wang, E.H. Twizell, A novel approach to the convexity control of interpolant curves, Commun. Numer. Meth. Eng. 19 (2003) 833–845. [20] Q. Duan, L. Wang, E.H. Twizell, A new bivariate rational interpolation based on function values, Inform. Sci. 166 (2004) 181–191.