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On interpolating multivariate rational splines
Applied Numerical Mathematics North-Holland
12 (1993) 357-372
357
APNUM 406
On interpolating splines * Ren-Hong
multivariate
rational
Wang
Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
Jie-Qing
Tan
Department of Mathematics and Mechanics, Hefei University of Technology, Hefei 230009, People’s Republic of China
Abstract
Wang, R.H. and J.Q. Tan, On interpolating (1993) 357-372.
multivariate
rational splines, Applied Numerical Mathematics
12
Over the past years little has been achieved for the well-known theoretical and practical difficulties in treating multivariate rational splines. In this paper, by means of Wachspress’ rational basis functions we obtain the necessary and sufficient conditions for the existence of interpolating bivariate rational splines consisting of piecewise rational functions with linear numerators as well as denominators in triangulation and quadrilateral partition respectively and give a geometric interpretation of these conditions. Moreover by constructing basis functions that are accurate up to order two, we also get the necessary and sufficient conditions for the existence of interpolating bivariate rational splines with quadratic numerators as well as denominators in triangulation. All the above-mentioned rational splines possess explicit representations in the cells associated with them. Keywords. Interpolation;
multivariate
spline; rational function.
1. Introduction
During the past decades, multivariate splines, which are composed of piecewise polynomials over collections of convex polygons, usually triangles or rectangles, have been developed rapidly and found wide applications in various practical problems such as computer-aided geometric design (CAGD) and curved surface fitting. A great many results have been achieved in this area (see [4,5]). Multivariate rational splines, however, lead to theoretical as well as practical difficulties. Very few papers touch this field. As we know, the first derivative of a rational function of (n, mktype is of (n + m - 1, 2m)-type, which brings us the difficulties in patching together piecewise rational functions as smoothly as possible. In most cases we require a rational function to be nonsingular in the corresponding domain. This frequently gets us into big trouble in treating a rational function whose denominator is a polynomial of degree & 3. Correspondence to: R.H. Wang, Institute
of Mathematical People’s Republic of China. * Supported by the National Natural Science Fundation
016%9274/93/$06.00
Sciences, Dalian University of Technology, Dalian 116024, of China.
0 1993 - Elsevier Science Publishers B.V. All rights reserved
358
R. H. Wang, J. Q. Tan / Multiuaria te rational splines
Last but not least, it is easy to understand that dealing with rational splines, whatever means used, will result in solving nonlinear equations. There is no doubt that all these difficulties become serious obstacles to stimulating the development of multivariate rational splines. In this paper, by means of Wachspress’ rational wedge functions, we find an effective approach to the construction of bivariate rational splines of lower order by seeking suitable interpolation data. In the next section we obtain, for triangulation, the necessary and sufficient conditions for the existence of Co and C1 interpolating bivariate rational splines which consist of piecewise rational functions with linear numerators as well as denominators and provide a geometric interpretation. In Section 3 concerning triangulation and Section 4 regarding quadrilateral partition, we give, without detailed proofs, the necessary and sufficient conditions for the existence of Co interpolating bivariate rational splines with quadratic numerators as well as denominators and with linear numerators as well as denominators respectively. In the last section we give two numerical examples.
2. Rational splines of (l,l)-type
in triangulation
Let a triangular partition A of a polygonal domain D be given. Suppose that there are N interior vertices in D and denote them by A,, A,, , . . , A,. The union of all triangular cells with Ai as their common vertex is called the domain incident with Ai or briefly the Ai-domain, denoted by B(A,, A >. Suppose that B(A,, A) contains Mj cells and denote them by Al,, A;,..., A’&, i=1,2 , . . . , N, in the counterclockwise direction. We say a rational function is of (m, n)-type if its numerator and denominator are polynomials of degree m and n respectively, and denote by R,,, the set of all the rational functions of (m, n)-type. For a triangular cell ai we label its vertices and some point distinct from the vertices on each edge as lk,i, 5k,i, 2k,i, 6k,i, 3k*i, and qk,’ in counterclockwise direction, where lk,j, 2k,i, and 3k,’ serve as the three vertices with the convention that lkj’ =Ai. Definition
2.1. If the rational function
Rk,‘(x, y) in R,,,(x,
y) which has no singular points in
ai satisfies Rk,‘(
pk,i)
p=l,2 Rk+,
= +i
,..., 6, Y) -Rk+‘+,
= u;.i/u;.‘,
k=l,2,...,Mi,
i-1,2
y) = o((l(x,
where u:,~/v~“‘~is an irreducible fraction = 1 if ~$3~= 0, and that u,“( > 0 and ~pk,~ &’ A;,,, then we call the union of rational spline associated with A’ of SR:j’(x, Y; f&4,, A>).
y))‘),
,..., N,
(2.1)
t = I, 2,
(2.2)
given at point pky’ = (xf’, y,“,“) with the convention I(x, y) denotes the linear form whose locus contains the Mi rational functions an incident interpolating (l,l)-type with C’- ‘-smoothness and denote it by
Definition 2.2. If B(A,, A) fIB(A,, A) 3 Af,,= A', implies R”,‘(x, y) =R""(x, y>, then We call U ~lSR~;l(x, y; B(A,, A>> the interpolating rational spline defined in D of (L&type with C’-’ smoothness and denote it by SR$,‘(x, y; D, A>.
R.H. Wang, J.Q. Tan / Multiuariate rational splines
359
For simplicity, the sub- or superscripts k and i in what follows may be dropped-and so will be done with later introduced symbols-wherever the context makes them clear. We shall use the following terminology throughout this paper: l l l l
S denotes the line A,x + B,y + C, = 0. _sdenotes the linear form of the line mentioned above. s is the value of _sat point p. pq denotes the distance from point p to point q.
For the triangular cell
A,
I& = 2/&T
let
w* = 3/3, >
which can be written in the following determinantal
1 w, =
x
1
Y
1
x2
Yz
1
x3
Y3
/S?
w, =
(2.3)
w3 = 1/L,
x
forms: Y
1
x
Y
w, = 1
x1
Yl
x2
Y2
1
x3
Y3
1
x1
Yl
/S7
1
/SY
(24
where s=
1
Xl
Yl
1
x2
Y,
1
x3
Y3
W)
.
Obviously we have (refer to Fig. 1): I%(q) = $pq,
P,4
where S,, is the Kronecker represented as
(2.6)
= 1, 293,
symbol. It is easy to see that R(x, y) satisfying (2.1) can be
(2.7) where dP, p = 1, 2, 3, are the constants to be determined. following simultaneous equations
3
Fig. 1.
By (2.1) and (2.71, we can yield the
360
R.H. Wang, J.Q. Tan / Mdtivariate rational splines
Denote by M the matrix composed of the coefficients with respect to d,, d,, and d,, then the necessary and sufficient condition for the unique R(x, y> satisfying (2.1) to exist is rank(M) = 2, which leads to --((Ye- CX~)(CQ -(Y&Q - a,)43.51*62 -= ((Ye- aq)(cyg - (Y~)((Y~ - a$i.52.63. (2.9) Therefore
we obtain three groups of equivalent solutions as follows d, = C,(%Q - %%)(%%
- %%)IG(4)&(5)~
d, = C&A
- V&V1
- %%)K(4)K(5),
d, = C,(V,
- V&V%
- &%)K(4)K(5);
(2.10)
where C,, C,, and C, are arbitrary non-zero constants. In the particular case when 4, 5, and 6 are mid-points, formulae (2.9)-(2.12) will be somewhat simplified because W,(4) = W,(5) = -W,(5) = W,(6) = W,(6) = W,(4) = $ as well as a= 41,51= 3, and a= a. By the way we point out that the equivalent forms (2.10)-(2.12) are necessary in practical applications, which is illustrated in our numerical examples in the last section. We know that the necessary and sufficient condition for A to contain no singular points of R(x, y) is that d,, d,, and d, possess the same sign, which implies that the following inequalities hold: min(cr,, (Y*)<(Ye < max(a,,
(Y*),
(2.13)
min(a,,
a,),
(2.14)
CX~)< aa < max(a,,
(2.15)
min(cy,, (Ye)
It is worth mentioning that one and only one of the inequalities (2.13H2.15) can be changed into the identity relation without losing the validity of our assertion. For two adjacent triangular cells ak and hk+ 1 the following conditions k = &r+1 a1
1
k_ a3--2
k+l
k_
k+
a4 -a15
7
(2.16)
7
)
are not only necessary but also sufficient to guarantee Rk(x, y) with Rk+‘(x, y> over the common boundary of proved as follows.
the Co-smoothness connection of and L&+1. This statement can be
ak
R.H. Wang, J. Q. Tan / Multivariate
361
rational splines
Let Y) =
Rk(X, y) -P+l(X,
%A-? Y) Q&G
(2.17)
Y) ’
then N,(x, y)/D,(x, y) is a rational function of (2,2)-type. If conditions then it follows from (2.1) that
(2.16) are satisfied,
R&(P) =P+‘(lk+‘), (2.18)
Rk(3k) = Rk+‘(2k+‘), Rk(4k) =P+l(Sk+r), which, according to our convention N,(P)
= &(3k)
that lk = lk+‘, 3“ = 2k+1, and 4k = 5k+1, are equivalent to
= N,(4k) = 0.
(2.19)
Since lk, 3k, and 4k are located on the common boundary r = {(x, y> E Ak fl A~+r ( 1(x, y) = 0) of Ak and Ak+r, it follows by Bezout’s theorem (see, e.g., [3]) that N,(x, y> contains Z(x, y> as its factor. Moreover the fact that R,(x, y) and Rk+i(X, y) are nonsingular over their respective triangular cells ensures that Dk(x, y> vanishes nowhere on r. Therefore Rk(&
Y> -Rkfl(X,
Y) = qqx,
(2.20)
Y)),
and the sufficiency is proved. The necessity of (2.16) is evident. Thus we have: Theorem 2.3. Conditions (2.9) and (2.13H2.16) s&(x, y; D, A).
are necessary and sufficient for the existence of
We can, to a certain extent, interpret the meaning of the conditions (2.9) and (2.13)-(2.16) in a geometric way. As shown in Fig. 2, if we think of (p, a,) as a point in three-dimensional space, then A can be regarded as a projection of the triangular plane a composed of points (1, a,), (2, (Y,), and (3, a,> into the xoy-olane. If at most one among three boundary lines of n is parallel to the xoy-plane and three groups of points (1, (~~1,(5, CYJ,(2, a,); (2, a,>, (6, cu,>, (3, C-Q);and (3, aJ, (4, (~~1,(1, a,) are collinear respectively, then CX~,p = 1, 2,. . . ,6, obviously satisfy (2.9) and (2.13)-(2.151, while conditions (2.16) hold if nk and ak+r share a common boundary line.
3
k+l
lk+l=lk
Fig. 2.
362
R.H. Wang, J.Q. Tan / Multivariate rational splines
Of course, the geometric interpretation we give above only partially reflects the algebraic character of conditions (2.9) and (2.13)-(2.16). Suppose that 1(x, y> = Ax + By + C, and define LM;+*;.
(2.21)
With (2.17) in mind, if we require Rk(& Y) -Rk+‘(X,
Y) = ((qxY))z),
then by Bezout’s theorem the following conditions are also necessary and sufficient in addition to conditions (2.9) and (2.13)-(2.16): D&(P)
= DN,(3k) = 0,
(2.22)
which leads to the following equalities ~(A,)P~,,(A(y,k+'-y:+l)+B(X,k+'-X:+l)) +S(Ak)F:,S(~(y:+l-y,k+')+B(x:+l-X,k+'))
+s
(2.23)
9’3”) +B(X2k -x3k))
++k+&(A(Y$
-Y;)
++3k-X:))
+S( A,,~~l(A(y~+l--y~+l)+~(X~+l-JT~+l)) +S(A,)F,k,,(~(Y~+l-y;+l)+~(X;+l-~;+l))=~,
where S(A) denotes the area of Fkm,n = u,$;+‘d;d;+‘(ai
A
(2.24)
and -a,+‘).
Theorem 2.4. The necessary and sufficient conditions for the existence of SR&(x, consist of (2.9), (2.13)-(2.16), (2.201, and (2.21).
(2.25) y; D, A)
3. Rational splines of (2,2)-type in triangulation
For a triangular cell A we label all its vertices and three different points distinct from the vertices on each edge as 1, 10, 5, 11, 2, 12, 6, 7, 3, 8, 4, and 9 in counterclockwise direction, where 1, 2, and 3 stand for the three vertices respectively. For the sake of convenience we might assume that these points are uniformly distributed. In general the vertex with label 1 will be always assumed to be the common vertex of the corresponding incident domain in which the
R.H. Wang, J. Q. Tan / Multivariate
rational splines
triangular cell A lies. Suppose that a rational function satisfies the interpolating conditions R( P> = Qp = up/up 7
p =
R(x, y) in &,,(x,
1, 2,. . . ) 12,
363
y) defined in
A
(3.1)
where up/up is an irreducible fraction given at point p = (f,, yJ with the convention that up > 0 and up = 1 if ayp= 0. In what follows we try to derive the conditions to which the interpolating data must subject, such that (a) R(x, y) uniquely exists; (b) R(x, y) is nonsingular in a; cc> Rk(X, y> and R“+~(x, y) are of Co-smoothness connection over the common boundary of ak and A~+ ,. Let _s=A,x + B,y + c, 1 = 1
x XS
1
x,+1
Y Ys
,
s=l,
(3.2)
2 ,..., 6,
Ys+l
where we make the convention that s + 1 = 1 if s = 3 and s + 1 = 4 if s = 6, and introduce the following basis functions (refer to Fig. 3):
(3.3) It is not difficult to show wP(q)=&, Therefore
p,q=l,2,...,6.
(3 *4)
the rational function R(x, y) satisfying (3.1) can be expressed as
where the fP’s are the coefficients to be’ determined. We observe that R(p) = up/up holds for 1, 2,. . . ) 6 due to (3.4). It remains to extend this relation for p = 7, 8,. , . ,12. Making use of (3.51, R(p) =u&,, p = 7, 8 ,..., 12,
p =
1
i
5
Fig. 3.
2
364
R.H. Wang, J.Q. Tan / Multiuariate rational splines
are equivalent to ( u2”7
-
u2”7)w2(7)f2
(UP8
-
v%3)~l(~)fl+
(U3%
-
~3&3)~3W3
+
(U4%
-
~4%PxW4
=
09
(UP9
-
w,m9fl+
(u39
-
~3u9)~3Pv3
+
(U4%
-
u4%Pww4
=
09
=
09
=
09
=
0.
( WJlO
-
wlowlwvl+
-
( WJll
( u2u12
+
-
u3”7)w3(7)f3
(U2UlO
wll)Jwvl+
-
(“3u7
u2u12)~2Wf2+
-
~2ulo)~2w)f2
+ (WI0
-
wlow5w)f5
(u2+
-
U2Ull)~2Wf2
+ (Wll
-
%u*lPxwf5
-
~3u12w3w)f3
-
w12w6w.&
b3u12 + bwl2
+
(‘6’7
-
u6”7)w6(7)f6
=
‘7
P-6)
Since w,(7) : w,(7) : w,(7) = w,(8) : W,(8) : W,(8) = WI(ll):W,(ll):W,(ll) We
: W,(9) : W,(9)
= -1:3:6,
= W,(lO) : W,(lO) : W,(lO) = W2(12) : k&(12) : W&2) = 3 : - 1: 6,
we have from above that aIf2
+ w3
+
Clf6
=
a2f2
+
b2f3
+ c2.h
=
0,
a3.L
+
b3f3
+ c3f4
=
0,
a4.h
+
b4f3
+
c4f4
=
0,
%fl+
w2
+
c5f5
=
0,
%L+
kf2
+
Gif5= 0.
The above’equations
0,
(3.9)
can be written in matrix form as follows:
0
a,
b,
0
0
Cl
\ /,fl
0
a2
b,
0
0
c2
f2
a3
0
b,
c3
0
0
f3
a4
0
b,
c4
0
0
a5
b,
0
0
cg
0
a6
b,
0
0
c6
0
I
\
f4 f5 f-5
(3 .lO)
RH. Wang, J. Q. Tan / Multivariate rational splines
which is equivalent
365
to '0
(
‘0
21
0
0
0
z*
0
0
=1
0
0
z3
24
0
0
25
0
0
0
0
24
z(j
0
0
27
0
0
0
Z8
0
I
&O
=7
0
0
zg
0
j-l ~f* I
’
f3
0 0
=
f4
0
f5
0
’ \f6
0
(3.11)
,
where z1 = a,b, - a2bl,
z2 = qb,
.z4 = a,b, - a4b3,
.z5 = c3b4 - c4b3,
z6 = a3c4 - a4c3,
.z7 = a,b, - a6bs,
z8 = csb, - c6b5,
zg = a5c6 - a6c5,
- c2bl,
a, = u2v7 - u7v2,
b, = 3(~3
a2 = 3(%2U2 - U2%2),
z3 =
(3.12)
cl=
6&u,
b2 = u$12 - u12u37
c2 =
6(%2%
a3 = ups - u8v1,
b, = 3(~,~3 - u3+3),
~3 = 6(~4
- u4+J,
a4 = 3(u9v, - ~9),
b4 = u3vg - ugv3,
~4 = 6(u,v,
- u4u9),
a5 = 3(%U1
- WIO),
b, =
~2~10
ullvl~
b,
3(%p2
a6 =
ulvll
-
=
- u3+),
alcz - a2c1,
-
~10~2 -
9
U2ull),
c5 =
+,ou,
c6 = 6(u,,+
- ug+), -
-
%~12)~
(3.13)
WIO)~
- u5ull).
It follows from (3.11) that f2=
-;f6.
f3= -;f69 f4 = - tf,
= =,, ZlZ6
f5 = - zf2
fl
= - ;f4
= zf6,
(3.14)
= - Zf6,
fl = - Zf5 = - Y!f,. =I=9 The last two equalities
in (3.14) imply
z2zgzs =23zgz9.
(3.15)
f6 = cz&,z9,
(3.16)
Letting
R.H. Wang, J.Q. Tan / Multivariate rational splines
366
we have
and at this time rank(M) = 5, where M denotes the 6 X 6 matrix occurring in (3.11). Hence we have: Theorem
3.1. Condition (3.15) is necessary and sufficient for the unique existence of R(x, y)
satisfying
(3.1), and R(x, y) has the explicit representation (3.5), where the fp’s are given in (3.16)
and (3.17).
A.
Next we consider the conditions for which R(x, y) satisfying (3.1) is nonsingular From (3.2) and (3.3) it follows that x +&Y + C&4,x WI = CA2
+&Y
+ G>
-w02 w
_
in the cell
’
(A+ + J&Y + C,)(&
+ BSY+ cs)
2-(SW2 w
_
(&+B,Y
=
(A,x
’
+C,)(A,x+&y
+C,)
3-w4)2 w
+
B,Y
’
+ C&42x
+ B2y
+
,
4 N
w
(3.18)
c2)
_
AN’
(A*x+B~Y+C*)(A~X+B~Y+C~) 9
5(S(fd)* w
=
(A,~~-B,Y+C,)(A,~+B,Y+C,) 7
6 w412
where S(A) denotes the area of the triangular cell Let G(X, Y) =
-(f,+&y,
+f&x&
A.
+f3v3x1y6
-f4hxly2
-f5v5x2y3
-f6v6x3yd
,
(w))2
(3.19) ‘(G(A, 2
B) + G(B,
A)),
a31 = $(G(A,
C) + G(C,
A)),
a23 = a3* = +(G(B,
C) + G(C,
B)).
all = G(A,
A),
a12 = a21 --
a22 = G(B,
B),
a13 =
a33 = G(C,
C),
(3.20)
R.H. Wang, J. Q. Tan / Multioariate rational splines
367
Then we have 6 c
fpupWp
=
u&
2a,,xy
+
+ u,,y* + 2a,,x
+ 2a,,y
(3.21)
+ u33.
p=l
Let a11
a12
a13
a21
a22
‘23
a31
‘32
a33
(3.22)
.
(3.23)
then we obtain by a certain affine transformation
5fpvpwp= I*(
11 -
+?=q
I,(11
+
x1*+
21
3
p=l
(see, e.g., [l])
where (x ‘, y ‘1 is the affine coordinate
1/1:-41,)
y,*
+
1 >
o f
2*
(3.24)
3
corresponding
to (x, y ).
Theorem 3.2. Condition (3.23) is sufficient to guarantee the nonsingularity of R(x, y) satisfying (3.1) and (3.15) in the cell A.
Furthermore
we have
Theorem 3.3. Suppose that Rk(x, y) and Rk+‘(x, y) satisfy (3.1), (3.15), and (3.23) on hk and ak + 1 respectively, then the necessary and sufficient conditions for R k(x, y ) and Rk + ‘(x, y ) to be of Co-smoothness connection over the common boundary r of L& and Ak + 1 are *I
k_
-aI
k+l
3
k_ a9 a8
k_
k+l “10
-%l
k+l
,
k_ ff4-a5
ktl
7
k_ (Y3-cI*
k+l
9 .
Proof. The necessity is obvious. Next we verify the sufficiency.
Let Rk(x,
y) - Rk+‘(x,
Then Nk(x7 y)/D,(x,
&(x3 Y>
y) = _ D,(x,
y) *
y> iS a rational function of (4,4)-type. Conditions (3.25) imply
Rk(lk) = Rk+‘(lk+‘), Rk(9k) = Rk+‘(lOktl), Rk(4k) = Rk+‘(Sk+‘), Rk(Sk) = Rk+‘(llk+‘),
Rk(3k) = Rk+‘(2k+1),
(3.25)
368
R.H. Wang, J.Q. Tan / Multivariate rational sphes
which are equivalent to N,(P)
= &(9k)
= Nk(4k) = A$(@) = Rk(3k) = 0,
due to our convention on the label. Since lk, 9k, 4k, 8&, and 3k lie on the common boundary r Of ak and A~+~, it follows by Bezout’s theorem that Nk(x, y) vanishes on r. On the other hand Rk(x, y) and Rk+l (x, y) satisfy (3.23). Therefore D,(x, y) vanishes nowhere on r. 17 Theorem 3.3 is proved.
4. Rational splines of (l,l)-type
in quadrilateral
partition
Let a polygon D on the plane be partitioned in such a way that each cell of D is composed of a quadrilateral. (It is clear that we require that the vertices of every quadrilateral lie neither in the interior nor on the boundary of any other quadrilateral.) Suppose that A is an interior vertex in D under the partition 0 and B(A, 0) indicates the incident domain associated with point A which consists of the union of all quadrilaterals sharing the common vertex A. For each 0 in B(A, 01, we label all its vertices and a certain point distinct from the vertices on every edge as 1, 6,2,7, 3, 8,4, and 5 in counterclockwise direction, where 1,2, 3, and 4 are the four vertices with the convention that the label 1 corresponds to the vertex A. Definition
4.1. SRy,Jx, y; HA,
0)) is called the incident interpolating rational spline associA of (&&type belonging to the smoothness class Co under the quadrilateral if its restriction R(x, y) to the cell A has the following properties: Y) E R,,,k Y). = cxp= u,/u,, p = 1, 2,. . . , 8, where up/up is an irreducible fraction given at point p = (xP, yP) with th e convention that up > 0 and up = 1 if (Ye= 0. (c) R(x, y) is nonsingular in A. (4 R(x, y> a 11ows a Co-smoothness connection with its neighbouring interpolating rational functions.
ated with partition 0 (a> Nx, (b) R(p)
Definition 4.2. If two incident interpolating rational splines SR&(x, y; B(A,, 0)) and SRy,,(x, y; B(A,, 0)) associated with arbitrary interior vertices Aj and Aj respectively are coincident in B(A,, 0) n B(A,, O), then we call U f?,SR$(x, y; HA,, 0)) the interpolating rational spline of (l,l)-type with Co-smoothness defined in D under the quadrilateral partition 0 and de-note it by bSR$x, y; D, 0) where M is the number of all interior vertices contained in D.
In order to obtain the explicit representation functions as follows (see [2] and refer to Fig. 4):
of R(x, y), we introduce
rational
basis
369
RH. Wang, J. Q. Tan / Multioariate rational splines 3
5
2
4 b
2 \
,I 4
_ i ,'l'.
II'
'1, '.
I/’
\
‘.
\
\ ‘.’
5 Fig. 4.
Let
(4.2) Then R(x, y> can be expressed as
(4.3) Theorem 4.3. The necessary and sufficient conditions for the existence of SRy,,(x, y; D, 0) composed of the following items: el”1s234
+ e3”3s124
elv1s234
+ e3v3s124
= e2”2s134 = e2v2s134
+ +
are
(4.4)
e4”4s123,
(4.5)
e4”4s123,
---
61.72.83
min(cu,, LYE)< cyg< max(a,,
(Ye),
(4.7)
min(a,,
(Ye)< cy7< max(a2, a3),
(4.8)
min(%,
a4)
< a8
< max(a37
a4)p
(4.9)
min(a,,
al)
<
q),
ma(a4,
(4.10)
where sImn denotes the area of the triangle formed by points I, m, and n while C is an arbitrary non-zero constant.
Conditions (4.4)-(4.10) are obtained in a way similar to those in Section 2 except that here additional equalities (4.4) and (4.5) are required to ensure the linearity of C~=leP~PWP as well as C;=,e,u,W,. Moreover, what needs to be mentioned is that one and only one of the
370
R.H. Wang, J.Q. Tan / Multivariate rational splines
ij
A
Fig.5.
inequalities (4.7)-(4.10) may be replaced by an identity relation without losing the validity of our assertion.
5. Numerical examples In this section, we give two examples explaining how to construct rational (l&type over a triangulated domain by the proper choice of interpolating data.
splines of
Example 5.1. Shown in Fig. 5 is a partition consisting of four triangular cells denoted by A,, A27 A37 and A~. By using (2.3) we construct basis functions and choose (or, cz2, and a3 as follows: w, =x, (Y2= 1,
on A3:
W,=l-x-y, (Y1= 1, w,=1+x-y, (Yr= 1, W,=l+x+y,
w,=y, ff2 = 2, w,= -x,
(Y3= 2, w, = -x; LX3 = 3, w, = -y;
a2=3,
a3
on Ad:
(Y1= 1, W,=l-x+y,
on A,: on A2:
a1
=
w, = -y,
1,
a2=
3 7,
w,=y;
=
3 2,
w, =x; (Y3=
1.
Step 1. By (2.9) and (2.13)-(2.15), we choose (Ye= z, (Ye= 1, and CQ= + on A,. By (2.12) we have d, = C, d, = 2C, and d, = C, and by (2.7) we obtain
MXY y>=
1+x+y l+n
*
Step 2. By (2.161, we should choose (Ye= i on A~. According to (2.141, we take czq= +. Then
by (2.9) we solve (Ye= y. Hence we obtain by (2.7) l-sx+y R2G7
y>=
1 _zx
*
R.H. Wang, J.Q. Tan / Multiuariate rational splines
371
Step 3. BY (2.16), we should choose (Ye= i on A~. According to (2.14), we take IY~= +. Then by (2.9) we solve (Ye= $. Hence we obtain by (2.7) R,(%
y>=
2-16x-y 2_4X
.
Step 4. By (2.16), we should choose (Ye= 1 and (Ye= : on A~. By (2.15) we may take LYE = $. Thus by (2.7) we yield R,(x,
Y> = #-Y).
It is easy to verify that for i = 1,2,3,4, Co-smoothly connected over the boundaries. Example
5.2. Consider
the triangular
cells
Ri(x, y) is nonsingular
A~
in ai and they are
and a4 in Fig. 5 and suppose that we have
obtained R,(x,
Y) =
l+x+y 1 +X
as in Example 5.1. Our purpose now is to construct a ratiqnal function R&x, y) defined on a4 which allows to be C’-smoothly patched with R,(x, y) over A, n A~. This question, in fact, reduces to the choice of interpolating data (Ye, p = 1, 2,. . . ,6. On a4 we choose (pi = (Ye= 1 and, by (2.16), cyq= 1. Therefore it remains to determine cy2, cy5, and (Ye. Through the application of (2.20)-(2.22), we solve: 1 IX5
2-(Y,
=y-Iq-
a6
=
3 - 2cu, *
If we take cz2= n/(1 + n), where y1 is a positive integer, then it follows from (2.11) and (2.7) that 1+x+(1-n)y R,(x,
Y> =
&(x7
Y> -R,(-?
1+x--y
’
Thus, 2 Y> =
(1+x
-:Y)(l+x)'
as required. Remark. In Example 5.2, one can only apply (2.11), instead of (2.10) or (2.12), to determine d,, d,, and d, in (2.7), since (Ye= (Ye= ayq= 1, which explains the reason why the equivalent forms
(2.10)-(2.12) are necessary.
Acknowledgement
The authors would like to thank the referees for their helpful suggestions on simplification of the symbols and on rewriting of Sections 3 and 5.
372
R.H. Wang, J.Q. Tan / Multiuatiate rational splines
References [l] G. Kowalewski, Einfihrung in die Analytische Geometrie (Walter de Gruyter, Berlin, 1953). [2] E.L. Wachspress, A rational basis for function approximation, J. Inst. Math. Appl. 8 (1971) 57-68. [3] R.J. Walker, Algebraic Curues (Springer, New York, 1978). [4] R.H. Wang, The structural characterization and interpolation for multivariate splines, Acta Math. Sinica 18 (1975) 91-106. [S] R.H. Wang, The dimension and basis of spaces of multivariate spline, J. Comput. Appl. Math. 12-13 (1985) 163-177.
12 (1993) 357-372
357
APNUM 406
On interpolating splines * Ren-Hong
multivariate
rational
Wang
Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
Jie-Qing
Tan
Department of Mathematics and Mechanics, Hefei University of Technology, Hefei 230009, People’s Republic of China
Abstract
Wang, R.H. and J.Q. Tan, On interpolating (1993) 357-372.
multivariate
rational splines, Applied Numerical Mathematics
12
Over the past years little has been achieved for the well-known theoretical and practical difficulties in treating multivariate rational splines. In this paper, by means of Wachspress’ rational basis functions we obtain the necessary and sufficient conditions for the existence of interpolating bivariate rational splines consisting of piecewise rational functions with linear numerators as well as denominators in triangulation and quadrilateral partition respectively and give a geometric interpretation of these conditions. Moreover by constructing basis functions that are accurate up to order two, we also get the necessary and sufficient conditions for the existence of interpolating bivariate rational splines with quadratic numerators as well as denominators in triangulation. All the above-mentioned rational splines possess explicit representations in the cells associated with them. Keywords. Interpolation;
multivariate
spline; rational function.
1. Introduction
During the past decades, multivariate splines, which are composed of piecewise polynomials over collections of convex polygons, usually triangles or rectangles, have been developed rapidly and found wide applications in various practical problems such as computer-aided geometric design (CAGD) and curved surface fitting. A great many results have been achieved in this area (see [4,5]). Multivariate rational splines, however, lead to theoretical as well as practical difficulties. Very few papers touch this field. As we know, the first derivative of a rational function of (n, mktype is of (n + m - 1, 2m)-type, which brings us the difficulties in patching together piecewise rational functions as smoothly as possible. In most cases we require a rational function to be nonsingular in the corresponding domain. This frequently gets us into big trouble in treating a rational function whose denominator is a polynomial of degree & 3. Correspondence to: R.H. Wang, Institute
of Mathematical People’s Republic of China. * Supported by the National Natural Science Fundation
016%9274/93/$06.00
Sciences, Dalian University of Technology, Dalian 116024, of China.
0 1993 - Elsevier Science Publishers B.V. All rights reserved
358
R. H. Wang, J. Q. Tan / Multiuaria te rational splines
Last but not least, it is easy to understand that dealing with rational splines, whatever means used, will result in solving nonlinear equations. There is no doubt that all these difficulties become serious obstacles to stimulating the development of multivariate rational splines. In this paper, by means of Wachspress’ rational wedge functions, we find an effective approach to the construction of bivariate rational splines of lower order by seeking suitable interpolation data. In the next section we obtain, for triangulation, the necessary and sufficient conditions for the existence of Co and C1 interpolating bivariate rational splines which consist of piecewise rational functions with linear numerators as well as denominators and provide a geometric interpretation. In Section 3 concerning triangulation and Section 4 regarding quadrilateral partition, we give, without detailed proofs, the necessary and sufficient conditions for the existence of Co interpolating bivariate rational splines with quadratic numerators as well as denominators and with linear numerators as well as denominators respectively. In the last section we give two numerical examples.
2. Rational splines of (l,l)-type
in triangulation
Let a triangular partition A of a polygonal domain D be given. Suppose that there are N interior vertices in D and denote them by A,, A,, , . . , A,. The union of all triangular cells with Ai as their common vertex is called the domain incident with Ai or briefly the Ai-domain, denoted by B(A,, A >. Suppose that B(A,, A) contains Mj cells and denote them by Al,, A;,..., A’&, i=1,2 , . . . , N, in the counterclockwise direction. We say a rational function is of (m, n)-type if its numerator and denominator are polynomials of degree m and n respectively, and denote by R,,, the set of all the rational functions of (m, n)-type. For a triangular cell ai we label its vertices and some point distinct from the vertices on each edge as lk,i, 5k,i, 2k,i, 6k,i, 3k*i, and qk,’ in counterclockwise direction, where lk,j, 2k,i, and 3k,’ serve as the three vertices with the convention that lkj’ =Ai. Definition
2.1. If the rational function
Rk,‘(x, y) in R,,,(x,
y) which has no singular points in
ai satisfies Rk,‘(
pk,i)
p=l,2 Rk+,
= +i
,..., 6, Y) -Rk+‘+,
= u;.i/u;.‘,
k=l,2,...,Mi,
i-1,2
y) = o((l(x,
where u:,~/v~“‘~is an irreducible fraction = 1 if ~$3~= 0, and that u,“( > 0 and ~pk,~ &’ A;,,, then we call the union of rational spline associated with A’ of SR:j’(x, Y; f&4,, A>).
y))‘),
,..., N,
(2.1)
t = I, 2,
(2.2)
given at point pky’ = (xf’, y,“,“) with the convention I(x, y) denotes the linear form whose locus contains the Mi rational functions an incident interpolating (l,l)-type with C’- ‘-smoothness and denote it by
Definition 2.2. If B(A,, A) fIB(A,, A) 3 Af,,= A', implies R”,‘(x, y) =R""(x, y>, then We call U ~lSR~;l(x, y; B(A,, A>> the interpolating rational spline defined in D of (L&type with C’-’ smoothness and denote it by SR$,‘(x, y; D, A>.
R.H. Wang, J.Q. Tan / Multiuariate rational splines
359
For simplicity, the sub- or superscripts k and i in what follows may be dropped-and so will be done with later introduced symbols-wherever the context makes them clear. We shall use the following terminology throughout this paper: l l l l
S denotes the line A,x + B,y + C, = 0. _sdenotes the linear form of the line mentioned above. s is the value of _sat point p. pq denotes the distance from point p to point q.
For the triangular cell
A,
I& = 2/&T
let
w* = 3/3, >
which can be written in the following determinantal
1 w, =
x
1
Y
1
x2
Yz
1
x3
Y3
/S?
w, =
(2.3)
w3 = 1/L,
x
forms: Y
1
x
Y
w, = 1
x1
Yl
x2
Y2
1
x3
Y3
1
x1
Yl
/S7
1
/SY
(24
where s=
1
Xl
Yl
1
x2
Y,
1
x3
Y3
W)
.
Obviously we have (refer to Fig. 1): I%(q) = $pq,
P,4
where S,, is the Kronecker represented as
(2.6)
= 1, 293,
symbol. It is easy to see that R(x, y) satisfying (2.1) can be
(2.7) where dP, p = 1, 2, 3, are the constants to be determined. following simultaneous equations
3
Fig. 1.
By (2.1) and (2.71, we can yield the
360
R.H. Wang, J.Q. Tan / Mdtivariate rational splines
Denote by M the matrix composed of the coefficients with respect to d,, d,, and d,, then the necessary and sufficient condition for the unique R(x, y> satisfying (2.1) to exist is rank(M) = 2, which leads to --((Ye- CX~)(CQ -(Y&Q - a,)43.51*62 -= ((Ye- aq)(cyg - (Y~)((Y~ - a$i.52.63. (2.9) Therefore
we obtain three groups of equivalent solutions as follows d, = C,(%Q - %%)(%%
- %%)IG(4)&(5)~
d, = C&A
- V&V1
- %%)K(4)K(5),
d, = C,(V,
- V&V%
- &%)K(4)K(5);
(2.10)
where C,, C,, and C, are arbitrary non-zero constants. In the particular case when 4, 5, and 6 are mid-points, formulae (2.9)-(2.12) will be somewhat simplified because W,(4) = W,(5) = -W,(5) = W,(6) = W,(6) = W,(4) = $ as well as a= 41,51= 3, and a= a. By the way we point out that the equivalent forms (2.10)-(2.12) are necessary in practical applications, which is illustrated in our numerical examples in the last section. We know that the necessary and sufficient condition for A to contain no singular points of R(x, y) is that d,, d,, and d, possess the same sign, which implies that the following inequalities hold: min(cr,, (Y*)<(Ye < max(a,,
(Y*),
(2.13)
min(a,,
a,),
(2.14)
CX~)< aa < max(a,,
(2.15)
min(cy,, (Ye)
It is worth mentioning that one and only one of the inequalities (2.13H2.15) can be changed into the identity relation without losing the validity of our assertion. For two adjacent triangular cells ak and hk+ 1 the following conditions k = &r+1 a1
1
k_ a3--2
k+l
k_
k+
a4 -a15
7
(2.16)
7
)
are not only necessary but also sufficient to guarantee Rk(x, y) with Rk+‘(x, y> over the common boundary of proved as follows.
the Co-smoothness connection of and L&+1. This statement can be
ak
R.H. Wang, J. Q. Tan / Multivariate
361
rational splines
Let Y) =
Rk(X, y) -P+l(X,
%A-? Y) Q&G
(2.17)
Y) ’
then N,(x, y)/D,(x, y) is a rational function of (2,2)-type. If conditions then it follows from (2.1) that
(2.16) are satisfied,
R&(P) =P+‘(lk+‘), (2.18)
Rk(3k) = Rk+‘(2k+‘), Rk(4k) =P+l(Sk+r), which, according to our convention N,(P)
= &(3k)
that lk = lk+‘, 3“ = 2k+1, and 4k = 5k+1, are equivalent to
= N,(4k) = 0.
(2.19)
Since lk, 3k, and 4k are located on the common boundary r = {(x, y> E Ak fl A~+r ( 1(x, y) = 0) of Ak and Ak+r, it follows by Bezout’s theorem (see, e.g., [3]) that N,(x, y> contains Z(x, y> as its factor. Moreover the fact that R,(x, y) and Rk+i(X, y) are nonsingular over their respective triangular cells ensures that Dk(x, y> vanishes nowhere on r. Therefore Rk(&
Y> -Rkfl(X,
Y) = qqx,
(2.20)
Y)),
and the sufficiency is proved. The necessity of (2.16) is evident. Thus we have: Theorem 2.3. Conditions (2.9) and (2.13H2.16) s&(x, y; D, A).
are necessary and sufficient for the existence of
We can, to a certain extent, interpret the meaning of the conditions (2.9) and (2.13)-(2.16) in a geometric way. As shown in Fig. 2, if we think of (p, a,) as a point in three-dimensional space, then A can be regarded as a projection of the triangular plane a composed of points (1, a,), (2, (Y,), and (3, a,> into the xoy-olane. If at most one among three boundary lines of n is parallel to the xoy-plane and three groups of points (1, (~~1,(5, CYJ,(2, a,); (2, a,>, (6, cu,>, (3, C-Q);and (3, aJ, (4, (~~1,(1, a,) are collinear respectively, then CX~,p = 1, 2,. . . ,6, obviously satisfy (2.9) and (2.13)-(2.151, while conditions (2.16) hold if nk and ak+r share a common boundary line.
3
k+l
lk+l=lk
Fig. 2.
362
R.H. Wang, J.Q. Tan / Multivariate rational splines
Of course, the geometric interpretation we give above only partially reflects the algebraic character of conditions (2.9) and (2.13)-(2.16). Suppose that 1(x, y> = Ax + By + C, and define LM;+*;.
(2.21)
With (2.17) in mind, if we require Rk(& Y) -Rk+‘(X,
Y) = ((qxY))z),
then by Bezout’s theorem the following conditions are also necessary and sufficient in addition to conditions (2.9) and (2.13)-(2.16): D&(P)
= DN,(3k) = 0,
(2.22)
which leads to the following equalities ~(A,)P~,,(A(y,k+'-y:+l)+B(X,k+'-X:+l)) +S(Ak)F:,S(~(y:+l-y,k+')+B(x:+l-X,k+'))
+s
(2.23)
9’3”) +B(X2k -x3k))
++k+&(A(Y$
-Y;)
++3k-X:))
+S( A,,~~l(A(y~+l--y~+l)+~(X~+l-JT~+l)) +S(A,)F,k,,(~(Y~+l-y;+l)+~(X;+l-~;+l))=~,
where S(A) denotes the area of Fkm,n = u,$;+‘d;d;+‘(ai
A
(2.24)
and -a,+‘).
Theorem 2.4. The necessary and sufficient conditions for the existence of SR&(x, consist of (2.9), (2.13)-(2.16), (2.201, and (2.21).
(2.25) y; D, A)
3. Rational splines of (2,2)-type in triangulation
For a triangular cell A we label all its vertices and three different points distinct from the vertices on each edge as 1, 10, 5, 11, 2, 12, 6, 7, 3, 8, 4, and 9 in counterclockwise direction, where 1, 2, and 3 stand for the three vertices respectively. For the sake of convenience we might assume that these points are uniformly distributed. In general the vertex with label 1 will be always assumed to be the common vertex of the corresponding incident domain in which the
R.H. Wang, J. Q. Tan / Multivariate
rational splines
triangular cell A lies. Suppose that a rational function satisfies the interpolating conditions R( P> = Qp = up/up 7
p =
R(x, y) in &,,(x,
1, 2,. . . ) 12,
363
y) defined in
A
(3.1)
where up/up is an irreducible fraction given at point p = (f,, yJ with the convention that up > 0 and up = 1 if ayp= 0. In what follows we try to derive the conditions to which the interpolating data must subject, such that (a) R(x, y) uniquely exists; (b) R(x, y) is nonsingular in a; cc> Rk(X, y> and R“+~(x, y) are of Co-smoothness connection over the common boundary of ak and A~+ ,. Let _s=A,x + B,y + c, 1 = 1
x XS
1
x,+1
Y Ys
,
s=l,
(3.2)
2 ,..., 6,
Ys+l
where we make the convention that s + 1 = 1 if s = 3 and s + 1 = 4 if s = 6, and introduce the following basis functions (refer to Fig. 3):
(3.3) It is not difficult to show wP(q)=&, Therefore
p,q=l,2,...,6.
(3 *4)
the rational function R(x, y) satisfying (3.1) can be expressed as
where the fP’s are the coefficients to be’ determined. We observe that R(p) = up/up holds for 1, 2,. . . ) 6 due to (3.4). It remains to extend this relation for p = 7, 8,. , . ,12. Making use of (3.51, R(p) =u&,, p = 7, 8 ,..., 12,
p =
1
i
5
Fig. 3.
2
364
R.H. Wang, J.Q. Tan / Multiuariate rational splines
are equivalent to ( u2”7
-
u2”7)w2(7)f2
(UP8
-
v%3)~l(~)fl+
(U3%
-
~3&3)~3W3
+
(U4%
-
~4%PxW4
=
09
(UP9
-
w,m9fl+
(u39
-
~3u9)~3Pv3
+
(U4%
-
u4%Pww4
=
09
=
09
=
09
=
0.
( WJlO
-
wlowlwvl+
-
( WJll
( u2u12
+
-
u3”7)w3(7)f3
(U2UlO
wll)Jwvl+
-
(“3u7
u2u12)~2Wf2+
-
~2ulo)~2w)f2
+ (WI0
-
wlow5w)f5
(u2+
-
U2Ull)~2Wf2
+ (Wll
-
%u*lPxwf5
-
~3u12w3w)f3
-
w12w6w.&
b3u12 + bwl2
+
(‘6’7
-
u6”7)w6(7)f6
=
‘7
P-6)
Since w,(7) : w,(7) : w,(7) = w,(8) : W,(8) : W,(8) = WI(ll):W,(ll):W,(ll) We
: W,(9) : W,(9)
= -1:3:6,
= W,(lO) : W,(lO) : W,(lO) = W2(12) : k&(12) : W&2) = 3 : - 1: 6,
we have from above that aIf2
+ w3
+
Clf6
=
a2f2
+
b2f3
+ c2.h
=
0,
a3.L
+
b3f3
+ c3f4
=
0,
a4.h
+
b4f3
+
c4f4
=
0,
%fl+
w2
+
c5f5
=
0,
%L+
kf2
+
Gif5= 0.
The above’equations
0,
(3.9)
can be written in matrix form as follows:
0
a,
b,
0
0
Cl
\ /,fl
0
a2
b,
0
0
c2
f2
a3
0
b,
c3
0
0
f3
a4
0
b,
c4
0
0
a5
b,
0
0
cg
0
a6
b,
0
0
c6
0
I
\
f4 f5 f-5
(3 .lO)
RH. Wang, J. Q. Tan / Multivariate rational splines
which is equivalent
365
to '0
(
‘0
21
0
0
0
z*
0
0
=1
0
0
z3
24
0
0
25
0
0
0
0
24
z(j
0
0
27
0
0
0
Z8
0
I
&O
=7
0
0
zg
0
j-l ~f* I
’
f3
0 0
=
f4
0
f5
0
’ \f6
0
(3.11)
,
where z1 = a,b, - a2bl,
z2 = qb,
.z4 = a,b, - a4b3,
.z5 = c3b4 - c4b3,
z6 = a3c4 - a4c3,
.z7 = a,b, - a6bs,
z8 = csb, - c6b5,
zg = a5c6 - a6c5,
- c2bl,
a, = u2v7 - u7v2,
b, = 3(~3
a2 = 3(%2U2 - U2%2),
z3 =
(3.12)
cl=
6&u,
b2 = u$12 - u12u37
c2 =
6(%2%
a3 = ups - u8v1,
b, = 3(~,~3 - u3+3),
~3 = 6(~4
- u4+J,
a4 = 3(u9v, - ~9),
b4 = u3vg - ugv3,
~4 = 6(u,v,
- u4u9),
a5 = 3(%U1
- WIO),
b, =
~2~10
ullvl~
b,
3(%p2
a6 =
ulvll
-
=
- u3+),
alcz - a2c1,
-
~10~2 -
9
U2ull),
c5 =
+,ou,
c6 = 6(u,,+
- ug+), -
-
%~12)~
(3.13)
WIO)~
- u5ull).
It follows from (3.11) that f2=
-;f6.
f3= -;f69 f4 = - tf,
= =,, ZlZ6
f5 = - zf2
fl
= - ;f4
= zf6,
(3.14)
= - Zf6,
fl = - Zf5 = - Y!f,. =I=9 The last two equalities
in (3.14) imply
z2zgzs =23zgz9.
(3.15)
f6 = cz&,z9,
(3.16)
Letting
R.H. Wang, J.Q. Tan / Multivariate rational splines
366
we have
and at this time rank(M) = 5, where M denotes the 6 X 6 matrix occurring in (3.11). Hence we have: Theorem
3.1. Condition (3.15) is necessary and sufficient for the unique existence of R(x, y)
satisfying
(3.1), and R(x, y) has the explicit representation (3.5), where the fp’s are given in (3.16)
and (3.17).
A.
Next we consider the conditions for which R(x, y) satisfying (3.1) is nonsingular From (3.2) and (3.3) it follows that x +&Y + C&4,x WI = CA2
+&Y
+ G>
-w02 w
_
in the cell
’
(A+ + J&Y + C,)(&
+ BSY+ cs)
2-(SW2 w
_
(&+B,Y
=
(A,x
’
+C,)(A,x+&y
+C,)
3-w4)2 w
+
B,Y
’
+ C&42x
+ B2y
+
,
4 N
w
(3.18)
c2)
_
AN’
(A*x+B~Y+C*)(A~X+B~Y+C~) 9
5(S(fd)* w
=
(A,~~-B,Y+C,)(A,~+B,Y+C,) 7
6 w412
where S(A) denotes the area of the triangular cell Let G(X, Y) =
-(f,+&y,
+f&x&
A.
+f3v3x1y6
-f4hxly2
-f5v5x2y3
-f6v6x3yd
,
(w))2
(3.19) ‘(G(A, 2
B) + G(B,
A)),
a31 = $(G(A,
C) + G(C,
A)),
a23 = a3* = +(G(B,
C) + G(C,
B)).
all = G(A,
A),
a12 = a21 --
a22 = G(B,
B),
a13 =
a33 = G(C,
C),
(3.20)
R.H. Wang, J. Q. Tan / Multioariate rational splines
367
Then we have 6 c
fpupWp
=
u&
2a,,xy
+
+ u,,y* + 2a,,x
+ 2a,,y
(3.21)
+ u33.
p=l
Let a11
a12
a13
a21
a22
‘23
a31
‘32
a33
(3.22)
.
(3.23)
then we obtain by a certain affine transformation
5fpvpwp= I*(
11 -
+?=q
I,(11
+
x1*+
21
3
p=l
(see, e.g., [l])
where (x ‘, y ‘1 is the affine coordinate
1/1:-41,)
y,*
+
1 >
o f
2*
(3.24)
3
corresponding
to (x, y ).
Theorem 3.2. Condition (3.23) is sufficient to guarantee the nonsingularity of R(x, y) satisfying (3.1) and (3.15) in the cell A.
Furthermore
we have
Theorem 3.3. Suppose that Rk(x, y) and Rk+‘(x, y) satisfy (3.1), (3.15), and (3.23) on hk and ak + 1 respectively, then the necessary and sufficient conditions for R k(x, y ) and Rk + ‘(x, y ) to be of Co-smoothness connection over the common boundary r of L& and Ak + 1 are *I
k_
-aI
k+l
3
k_ a9 a8
k_
k+l “10
-%l
k+l
,
k_ ff4-a5
ktl
7
k_ (Y3-cI*
k+l
9 .
Proof. The necessity is obvious. Next we verify the sufficiency.
Let Rk(x,
y) - Rk+‘(x,
Then Nk(x7 y)/D,(x,
&(x3 Y>
y) = _ D,(x,
y) *
y> iS a rational function of (4,4)-type. Conditions (3.25) imply
Rk(lk) = Rk+‘(lk+‘), Rk(9k) = Rk+‘(lOktl), Rk(4k) = Rk+‘(Sk+‘), Rk(Sk) = Rk+‘(llk+‘),
Rk(3k) = Rk+‘(2k+1),
(3.25)
368
R.H. Wang, J.Q. Tan / Multivariate rational sphes
which are equivalent to N,(P)
= &(9k)
= Nk(4k) = A$(@) = Rk(3k) = 0,
due to our convention on the label. Since lk, 9k, 4k, 8&, and 3k lie on the common boundary r Of ak and A~+~, it follows by Bezout’s theorem that Nk(x, y) vanishes on r. On the other hand Rk(x, y) and Rk+l (x, y) satisfy (3.23). Therefore D,(x, y) vanishes nowhere on r. 17 Theorem 3.3 is proved.
4. Rational splines of (l,l)-type
in quadrilateral
partition
Let a polygon D on the plane be partitioned in such a way that each cell of D is composed of a quadrilateral. (It is clear that we require that the vertices of every quadrilateral lie neither in the interior nor on the boundary of any other quadrilateral.) Suppose that A is an interior vertex in D under the partition 0 and B(A, 0) indicates the incident domain associated with point A which consists of the union of all quadrilaterals sharing the common vertex A. For each 0 in B(A, 01, we label all its vertices and a certain point distinct from the vertices on every edge as 1, 6,2,7, 3, 8,4, and 5 in counterclockwise direction, where 1,2, 3, and 4 are the four vertices with the convention that the label 1 corresponds to the vertex A. Definition
4.1. SRy,Jx, y; HA,
0)) is called the incident interpolating rational spline associA of (&&type belonging to the smoothness class Co under the quadrilateral if its restriction R(x, y) to the cell A has the following properties: Y) E R,,,k Y). = cxp= u,/u,, p = 1, 2,. . . , 8, where up/up is an irreducible fraction given at point p = (xP, yP) with th e convention that up > 0 and up = 1 if (Ye= 0. (c) R(x, y) is nonsingular in A. (4 R(x, y> a 11ows a Co-smoothness connection with its neighbouring interpolating rational functions.
ated with partition 0 (a> Nx, (b) R(p)
Definition 4.2. If two incident interpolating rational splines SR&(x, y; B(A,, 0)) and SRy,,(x, y; B(A,, 0)) associated with arbitrary interior vertices Aj and Aj respectively are coincident in B(A,, 0) n B(A,, O), then we call U f?,SR$(x, y; HA,, 0)) the interpolating rational spline of (l,l)-type with Co-smoothness defined in D under the quadrilateral partition 0 and de-note it by bSR$x, y; D, 0) where M is the number of all interior vertices contained in D.
In order to obtain the explicit representation functions as follows (see [2] and refer to Fig. 4):
of R(x, y), we introduce
rational
basis
369
RH. Wang, J. Q. Tan / Multioariate rational splines 3
5
2
4 b
2 \
,I 4
_ i ,'l'.
II'
'1, '.
I/’
\
‘.
\
\ ‘.’
5 Fig. 4.
Let
(4.2) Then R(x, y> can be expressed as
(4.3) Theorem 4.3. The necessary and sufficient conditions for the existence of SRy,,(x, y; D, 0) composed of the following items: el”1s234
+ e3”3s124
elv1s234
+ e3v3s124
= e2”2s134 = e2v2s134
+ +
are
(4.4)
e4”4s123,
(4.5)
e4”4s123,
---
61.72.83
min(cu,, LYE)< cyg< max(a,,
(Ye),
(4.7)
min(a,,
(Ye)< cy7< max(a2, a3),
(4.8)
min(%,
a4)
< a8
< max(a37
a4)p
(4.9)
min(a,,
al)
<
q),
ma(a4,
(4.10)
where sImn denotes the area of the triangle formed by points I, m, and n while C is an arbitrary non-zero constant.
Conditions (4.4)-(4.10) are obtained in a way similar to those in Section 2 except that here additional equalities (4.4) and (4.5) are required to ensure the linearity of C~=leP~PWP as well as C;=,e,u,W,. Moreover, what needs to be mentioned is that one and only one of the
370
R.H. Wang, J.Q. Tan / Multivariate rational splines
ij
A
Fig.5.
inequalities (4.7)-(4.10) may be replaced by an identity relation without losing the validity of our assertion.
5. Numerical examples In this section, we give two examples explaining how to construct rational (l&type over a triangulated domain by the proper choice of interpolating data.
splines of
Example 5.1. Shown in Fig. 5 is a partition consisting of four triangular cells denoted by A,, A27 A37 and A~. By using (2.3) we construct basis functions and choose (or, cz2, and a3 as follows: w, =x, (Y2= 1,
on A3:
W,=l-x-y, (Y1= 1, w,=1+x-y, (Yr= 1, W,=l+x+y,
w,=y, ff2 = 2, w,= -x,
(Y3= 2, w, = -x; LX3 = 3, w, = -y;
a2=3,
a3
on Ad:
(Y1= 1, W,=l-x+y,
on A,: on A2:
a1
=
w, = -y,
1,
a2=
3 7,
w,=y;
=
3 2,
w, =x; (Y3=
1.
Step 1. By (2.9) and (2.13)-(2.15), we choose (Ye= z, (Ye= 1, and CQ= + on A,. By (2.12) we have d, = C, d, = 2C, and d, = C, and by (2.7) we obtain
MXY y>=
1+x+y l+n
*
Step 2. By (2.161, we should choose (Ye= i on A~. According to (2.141, we take czq= +. Then
by (2.9) we solve (Ye= y. Hence we obtain by (2.7) l-sx+y R2G7
y>=
1 _zx
*
R.H. Wang, J.Q. Tan / Multiuariate rational splines
371
Step 3. BY (2.16), we should choose (Ye= i on A~. According to (2.14), we take IY~= +. Then by (2.9) we solve (Ye= $. Hence we obtain by (2.7) R,(%
y>=
2-16x-y 2_4X
.
Step 4. By (2.16), we should choose (Ye= 1 and (Ye= : on A~. By (2.15) we may take LYE = $. Thus by (2.7) we yield R,(x,
Y> = #-Y).
It is easy to verify that for i = 1,2,3,4, Co-smoothly connected over the boundaries. Example
5.2. Consider
the triangular
cells
Ri(x, y) is nonsingular
A~
in ai and they are
and a4 in Fig. 5 and suppose that we have
obtained R,(x,
Y) =
l+x+y 1 +X
as in Example 5.1. Our purpose now is to construct a ratiqnal function R&x, y) defined on a4 which allows to be C’-smoothly patched with R,(x, y) over A, n A~. This question, in fact, reduces to the choice of interpolating data (Ye, p = 1, 2,. . . ,6. On a4 we choose (pi = (Ye= 1 and, by (2.16), cyq= 1. Therefore it remains to determine cy2, cy5, and (Ye. Through the application of (2.20)-(2.22), we solve: 1 IX5
2-(Y,
=y-Iq-
a6
=
3 - 2cu, *
If we take cz2= n/(1 + n), where y1 is a positive integer, then it follows from (2.11) and (2.7) that 1+x+(1-n)y R,(x,
Y> =
&(x7
Y> -R,(-?
1+x--y
’
Thus, 2 Y> =
(1+x
-:Y)(l+x)'
as required. Remark. In Example 5.2, one can only apply (2.11), instead of (2.10) or (2.12), to determine d,, d,, and d, in (2.7), since (Ye= (Ye= ayq= 1, which explains the reason why the equivalent forms
(2.10)-(2.12) are necessary.
Acknowledgement
The authors would like to thank the referees for their helpful suggestions on simplification of the symbols and on rewriting of Sections 3 and 5.
372
R.H. Wang, J.Q. Tan / Multiuatiate rational splines
References [l] G. Kowalewski, Einfihrung in die Analytische Geometrie (Walter de Gruyter, Berlin, 1953). [2] E.L. Wachspress, A rational basis for function approximation, J. Inst. Math. Appl. 8 (1971) 57-68. [3] R.J. Walker, Algebraic Curues (Springer, New York, 1978). [4] R.H. Wang, The structural characterization and interpolation for multivariate splines, Acta Math. Sinica 18 (1975) 91-106. [S] R.H. Wang, The dimension and basis of spaces of multivariate spline, J. Comput. Appl. Math. 12-13 (1985) 163-177.