- Email: [email protected]
Dual bases of a Bernstein polynomial basis on simplices
Computer
483
Aided Geometric Design 10 (1993) 383-489 North-Holland
COMAID 306
Dual bases of a Bernstein polynomial basis on simplices Wu Dong-Bing Department
ofMathematics, Beijing
University, Beijing, People’s Republic of China
Received August 199 I Revised May 1992
Abstract Wu Dong-Bing, Dual bases of a Bernstein Design 10 (1993) 483-489.
polynomial
basis on simplices,
Computer
Aided Geometric
This paper extends the theory of dual bases of a one-variate B-spline basis to the case of Bernstein polynomials and presents methods of constructing dual bases of a Bernstein pol nomial basis on a r simplex from that of a one-variate polynomial basis, {~‘}y=s and {Blfk_!(x) = (,)x’(l - ~)~-‘}f=n on the interval [0, 11. Keywords. Dual basis; Bernstein basis; simptices.
1. Introduction It is well known that the theory of dual bases of a one-variate B-spline basis, which is due to de Boor [de Boor ‘76a,b], plays an important roIe in the research of onevariate splines. Also for multi-variate splines, work has been done on dual bases. Hollig [Hollig ‘821 has worked out a differential-form dual basis of simplex B-splines, expanding at a11points. Zhao and Sun [Zhao & Sun ‘871 have constructed a dual basis of Bernstein polynomials on simpiices, expanding at one vertex. But in the one-variate case, there are not only differential-form dual bases of a B-spline basis, but also integral-form ones. Therefore, we hope that we can obtain such dual bases also in the multi-variate case. This paper solves the above problems on simplex B-polynomials. Although B-polynomials are a special case of general simplex splines, it is clear from the examples given below that the differential-form dual basis constructed using the method of this paper is not a special case of the one constructed in [Hollig ‘821. Example 2.2 shows that the dual basis given in [Zhao & Sun ‘871 is a special case of the one given in this paper. We will now list some notations that are needed in this paper.
Correspondence to: Wu Dong-Bing, Department of China. Elsevier Science Publishers
B.V.
of Mathematics,
Beijing University,
Beijing, People’s Republic
484
Wu Dong-Bing / Dual bases of a Bernstein pol_vnomial basis on simplices
R and Z+ are the set of reals and the set of positive integers, respectively. x~R~wedenotebyxitheithcomponentofx(i= l,...,k).Letcu,p~Z+.Wedefine: XQ := ,q
. .,ykak,
For any
(1.1)
Ial := a1 + ... + cxk,
(1.2)
a! := cu,!...cq!,
(1.3) (1.4)
X\i:=
x :=
(X*9. . . . Xi_*,Xi+l,...,
a :=
(x0,x),
(i = I,...,
Xk)
k),
(1.5)
(ao,cy),
(1.6)
where x0 E R and cue E Z, are to be determined. a>j?meansthatforalli= l,..., k we have ai 2 pi; if this is not the case we write Q: & j3. Similarly we define < and A. Without loss of generality we can restrict ourselves to the simplex (1.7) The Bernstein
basis of n-degree polynomials
on S, is denoted
B, := {B: 1a E 2’7, 161 = n}, B;(c)
:= $2,
u E s,,
a EZT,
by (1.8)
jtil = n
(1.9)
(cf. [de Boor ‘861). A dual basis of B, is a set of linear functions {J.: 1a E Zy, Ia/ ,< n} defined on the space n,,, (S,) of n-degree polynomials on S, such that (1.10)
l;;Bz = aaB : = fi Solsi i=l
is Kronecker’s symbol). Suppose 1 is a linear functional defined on some one-variate function space S. Then we introduce the sign (I.)! as follows. For any k-variate function f and fixed ,...,xk) E S, we prescribe that (n),f(x \ I) := x \ 1 E P-‘, if f(xi,. ..,x~-i,.,x[+i xk). It is easy to see that (A)/ is a linear operator from the &f (XI ,...,~i-l,‘,~l+I,.~., k-variable function space to the (k - 1 )-variable space. (Sj,
2. Dual bases of B, from a dual base of {xi}~=,, For any a E (0,l /m ] we define the operator on Sm, &f(Y)
:= J(1
-acYi3aY)9 i=l
It is clear that & is a linear operator.
A, as follows. For any function f defined
Y E lo,
llrn.
(2.1)
Wu Dong-Bing / Dud bases of a Bernstein polynomial basis on simplices
48.5
Theorem 2.1. Suppose that {$}f=, is a duai basis of the basis {x~)~=~offl,, (0,i] {the space of n-degree ~olyno~~afs on IO,11 1 and that 0 < a < 1jm. ~e~~e~u~ctiona~s (2.2) Then
Proof. We have
and
12.3) Clearly
(I.;, )I o -. . o (Q,
o
A,Bj
(_a)h-Bi
= $aifli (rl-
P)!(Po-
c;,l(yIi--
Pi)!)
We get by (2.21, (2.4) and (2.5) that A:Bj = 0 for cy 3 p. If Q: 2 /? then
j=l
It is easy to see that this also holds for ty & j3. So AzBj = S,b. 0
(2.5)
486
W’u Dong-Bing / Dual bases of a Bernstein polynomial basu on simpiices
Example 2.2. Consider
;-Yf =
the set {A:}:=‘=,of functionals
f
AJ"'CO),
E Cn[O,
I],
i =
(on C”[O, 11) defined by
0 ,...,
n.
It is easy to check that it is a dual basis of {xi}:=e. By Theorem in [Zhao & Sun ‘871 is obtained:
(2.6) 2.1 the dual basis of B,
(2.7)
Example 2.3. Suppose Lj(X) (Legendre
=
&$&x2._ e
polynomials
Gn =
i = 0,1,2,...
l)‘,
on [ - 1, 1 ] ). Let
((-!)/((i+j+l)(i~j)))~j=o.
(2.8)
Here we use the notation
The matrix G, is an upper triangular
matrix. Denote X” := (1,x )...) X”).
L” := (Lo(2,Y - 1) ,...) L,(2.u - l)), Since {Li}y,o that
is an orthogonal
polynomial
((-lpi(;)
=
Define functionals
system on [ - 1, 11, we obtain by calculation
/ i = O,...,n),
Ln = X”F,idiag(&
Gil
(2.9)
(i :‘)t2j + l))z,Eo’
(2.10)
li by 1
lif
Li(2X-
I=
l)f(X)dX.
J 0
Let
(I,“,..., f;JT:= (T means “transpose”).
G,'(fo,...,f,JT,
Pi(X)
Then Aif = Jd Pi(X)f(.X)
=
$(-l)q;)(i:j)
d,u, and
(2.11)
Wu Dong-Bing / Dual bases of a Bernstein polynomial basis on simpiices
{Al;,.. .,~~)TX” = G,‘C&,. ..,I,)T(L”)Tdiag(2i = G;‘G,diag(& = (n +- I)th-order
I
i = 0 ,...,
487
t 1 j i = 0,. ..,n)G, n
>
diag(2i =+1 1 i = 0, . . . . n)
unit matrix.
We get by Theorem 2.1 an integral-fog It follows that {A~}~,Ois a dual basis of {,Y~)~=~. dual basis of B,, {L,” 1a E X7, /al 6 a), where
3. Dual bases of B,, from a dual basis of (Btk_i)tSl=,
(3.2)
Praot We have
so
If a: = p, it is clear that J.zB;i’= I. Otherwise, there exists some io such that cyiO# pi, 17 and cri = fli for Cl< i < i+ Hence %iB$ = 0. This proves the theorem.
Wu Dong-Bing I Dual bases of a Bernstein poiynomial
488
basis on simplices
Example 3.2.Let
LCk’”= u-E Lx 00
SUppOSeGk E L$+”
E
L,(R),
i -_ 0,. . . ,k + I}.
satisfies 0
Gk(X) =
(RI 1f”’
1
ifx
(3.3)
Define functions ~7, 0 < j G k ,< n, by
IVjkf-4= Gk(XM$(X),x
E
[O, 11.
(3.4)
We know from [de Boor ‘76a, Section 4j and [Schumaker ‘81, Remark 4.71 that the set of functionals {,?$}~=adefined by
Akf
J
=
J 0
f(X)Dk+'tf!fk(X)~ j
j
=
0
II_ ,-..,
(3.5)
,
is a dual basis of {B[k_i}fzO fk = 0,...,n).Thus we get another integral-form dual basis (2: 1a: E it?!& Icr/ G n} of B, with
nzj-=
J
f((l
-xl)**q(l
-Xm),X,,(l
-x1)xz,...,(l
--xl)**.
(1 -x,_*)xm)
Ktll”
x ~~+l~~~(~~~}
mrjD”-c:=i.“l~ai,,
‘-c;=,
cLJ
(Xi+ I 1 dX
i=l =
J XV f(l
-
Ixl,x)D”+‘w~~
(x1)
Example 3.3. If we take in ExampIe 3.2 as duai basis of {&I,k_i}fCOthe one defined by
Iff := ~(-l)‘ok-‘O”(7i)o’(/ri),
i = O,...,k;
7jE
[0, I]
(3.7)
r=O
(cf. [de Boor ‘76b, Section ,511, we obtain a di~erential-fog any point of B,.
dual basis, expanding at
Wu Dong-Bing / Dual bases of a Bernstein polynomial basis on slmplices
489
References de Boor, C. (1976a), On local linear functionais which vanish at all B-splines but one. in: A. Law and A. Sahney, eds., Theory of ,-lppro.rimafion wirh Applications, hcademic Press, New York, 120- 145. de Boor, C. ( 1976b). Splines as linear combinations of El-splines, a survey, Report MRC TSR $1667. de Boor, C. (1986). B-form basics, in: G. Farin, ed., Geometric Modeling, SIAM, Philadelphia, PA, 131-148. Hiillig, K. (1982), Multivariate splines, SIAM J. Numer. Anal. 19, 1013-1031. Schumaker, L.L. ( 1981), Spline Funcfions. Basic Theory, Wiley, New York. Zhao, K. and Sun, J. (1987). Dual bases of multi-variate Bernstein-Bezier polynomials, Computer Aided Geometric Design 5, 119-125.
483
Aided Geometric Design 10 (1993) 383-489 North-Holland
COMAID 306
Dual bases of a Bernstein polynomial basis on simplices Wu Dong-Bing Department
ofMathematics, Beijing
University, Beijing, People’s Republic of China
Received August 199 I Revised May 1992
Abstract Wu Dong-Bing, Dual bases of a Bernstein Design 10 (1993) 483-489.
polynomial
basis on simplices,
Computer
Aided Geometric
This paper extends the theory of dual bases of a one-variate B-spline basis to the case of Bernstein polynomials and presents methods of constructing dual bases of a Bernstein pol nomial basis on a r simplex from that of a one-variate polynomial basis, {~‘}y=s and {Blfk_!(x) = (,)x’(l - ~)~-‘}f=n on the interval [0, 11. Keywords. Dual basis; Bernstein basis; simptices.
1. Introduction It is well known that the theory of dual bases of a one-variate B-spline basis, which is due to de Boor [de Boor ‘76a,b], plays an important roIe in the research of onevariate splines. Also for multi-variate splines, work has been done on dual bases. Hollig [Hollig ‘821 has worked out a differential-form dual basis of simplex B-splines, expanding at a11points. Zhao and Sun [Zhao & Sun ‘871 have constructed a dual basis of Bernstein polynomials on simpiices, expanding at one vertex. But in the one-variate case, there are not only differential-form dual bases of a B-spline basis, but also integral-form ones. Therefore, we hope that we can obtain such dual bases also in the multi-variate case. This paper solves the above problems on simplex B-polynomials. Although B-polynomials are a special case of general simplex splines, it is clear from the examples given below that the differential-form dual basis constructed using the method of this paper is not a special case of the one constructed in [Hollig ‘821. Example 2.2 shows that the dual basis given in [Zhao & Sun ‘871 is a special case of the one given in this paper. We will now list some notations that are needed in this paper.
Correspondence to: Wu Dong-Bing, Department of China. Elsevier Science Publishers
B.V.
of Mathematics,
Beijing University,
Beijing, People’s Republic
484
Wu Dong-Bing / Dual bases of a Bernstein pol_vnomial basis on simplices
R and Z+ are the set of reals and the set of positive integers, respectively. x~R~wedenotebyxitheithcomponentofx(i= l,...,k).Letcu,p~Z+.Wedefine: XQ := ,q
. .,ykak,
For any
(1.1)
Ial := a1 + ... + cxk,
(1.2)
a! := cu,!...cq!,
(1.3) (1.4)
X\i:=
x :=
(X*9. . . . Xi_*,Xi+l,...,
a :=
(x0,x),
(i = I,...,
Xk)
k),
(1.5)
(ao,cy),
(1.6)
where x0 E R and cue E Z, are to be determined. a>j?meansthatforalli= l,..., k we have ai 2 pi; if this is not the case we write Q: & j3. Similarly we define < and A. Without loss of generality we can restrict ourselves to the simplex (1.7) The Bernstein
basis of n-degree polynomials
on S, is denoted
B, := {B: 1a E 2’7, 161 = n}, B;(c)
:= $2,
u E s,,
a EZT,
by (1.8)
jtil = n
(1.9)
(cf. [de Boor ‘861). A dual basis of B, is a set of linear functions {J.: 1a E Zy, Ia/ ,< n} defined on the space n,,, (S,) of n-degree polynomials on S, such that (1.10)
l;;Bz = aaB : = fi Solsi i=l
is Kronecker’s symbol). Suppose 1 is a linear functional defined on some one-variate function space S. Then we introduce the sign (I.)! as follows. For any k-variate function f and fixed ,...,xk) E S, we prescribe that (n),f(x \ I) := x \ 1 E P-‘, if f(xi,. ..,x~-i,.,x[+i xk). It is easy to see that (A)/ is a linear operator from the &f (XI ,...,~i-l,‘,~l+I,.~., k-variable function space to the (k - 1 )-variable space. (Sj,
2. Dual bases of B, from a dual base of {xi}~=,, For any a E (0,l /m ] we define the operator on Sm, &f(Y)
:= J(1
-acYi3aY)9 i=l
It is clear that & is a linear operator.
A, as follows. For any function f defined
Y E lo,
llrn.
(2.1)
Wu Dong-Bing / Dud bases of a Bernstein polynomial basis on simplices
48.5
Theorem 2.1. Suppose that {$}f=, is a duai basis of the basis {x~)~=~offl,, (0,i] {the space of n-degree ~olyno~~afs on IO,11 1 and that 0 < a < 1jm. ~e~~e~u~ctiona~s (2.2) Then
Proof. We have
and
12.3) Clearly
(I.;, )I o -. . o (Q,
o
A,Bj
(_a)h-Bi
= $aifli (rl-
P)!(Po-
c;,l(yIi--
Pi)!)
We get by (2.21, (2.4) and (2.5) that A:Bj = 0 for cy 3 p. If Q: 2 /? then
j=l
It is easy to see that this also holds for ty & j3. So AzBj = S,b. 0
(2.5)
486
W’u Dong-Bing / Dual bases of a Bernstein polynomial basu on simpiices
Example 2.2. Consider
;-Yf =
the set {A:}:=‘=,of functionals
f
AJ"'CO),
E Cn[O,
I],
i =
(on C”[O, 11) defined by
0 ,...,
n.
It is easy to check that it is a dual basis of {xi}:=e. By Theorem in [Zhao & Sun ‘871 is obtained:
(2.6) 2.1 the dual basis of B,
(2.7)
Example 2.3. Suppose Lj(X) (Legendre
=
&$&x2._ e
polynomials
Gn =
i = 0,1,2,...
l)‘,
on [ - 1, 1 ] ). Let
((-!)/((i+j+l)(i~j)))~j=o.
(2.8)
Here we use the notation
The matrix G, is an upper triangular
matrix. Denote X” := (1,x )...) X”).
L” := (Lo(2,Y - 1) ,...) L,(2.u - l)), Since {Li}y,o that
is an orthogonal
polynomial
((-lpi(;)
=
Define functionals
system on [ - 1, 11, we obtain by calculation
/ i = O,...,n),
Ln = X”F,idiag(&
Gil
(2.9)
(i :‘)t2j + l))z,Eo’
(2.10)
li by 1
lif
Li(2X-
I=
l)f(X)dX.
J 0
Let
(I,“,..., f;JT:= (T means “transpose”).
G,'(fo,...,f,JT,
Pi(X)
Then Aif = Jd Pi(X)f(.X)
=
$(-l)q;)(i:j)
d,u, and
(2.11)
Wu Dong-Bing / Dual bases of a Bernstein polynomial basis on simpiices
{Al;,.. .,~~)TX” = G,‘C&,. ..,I,)T(L”)Tdiag(2i = G;‘G,diag(& = (n +- I)th-order
I
i = 0 ,...,
487
t 1 j i = 0,. ..,n)G, n
>
diag(2i =+1 1 i = 0, . . . . n)
unit matrix.
We get by Theorem 2.1 an integral-fog It follows that {A~}~,Ois a dual basis of {,Y~)~=~. dual basis of B,, {L,” 1a E X7, /al 6 a), where
3. Dual bases of B,, from a dual basis of (Btk_i)tSl=,
(3.2)
Praot We have
so
If a: = p, it is clear that J.zB;i’= I. Otherwise, there exists some io such that cyiO# pi, 17 and cri = fli for Cl< i < i+ Hence %iB$ = 0. This proves the theorem.
Wu Dong-Bing I Dual bases of a Bernstein poiynomial
488
basis on simplices
Example 3.2.Let
LCk’”= u-E Lx 00
SUppOSeGk E L$+”
E
L,(R),
i -_ 0,. . . ,k + I}.
satisfies 0
Gk(X) =
(RI 1f”’
1
ifx
(3.3)
Define functions ~7, 0 < j G k ,< n, by
IVjkf-4= Gk(XM$(X),x
E
[O, 11.
(3.4)
We know from [de Boor ‘76a, Section 4j and [Schumaker ‘81, Remark 4.71 that the set of functionals {,?$}~=adefined by
Akf
J
=
J 0
f(X)Dk+'tf!fk(X)~ j
j
=
0
II_ ,-..,
(3.5)
,
is a dual basis of {B[k_i}fzO fk = 0,...,n).Thus we get another integral-form dual basis (2: 1a: E it?!& Icr/ G n} of B, with
nzj-=
J
f((l
-xl)**q(l
-Xm),X,,(l
-x1)xz,...,(l
--xl)**.
(1 -x,_*)xm)
Ktll”
x ~~+l~~~(~~~}
mrjD”-c:=i.“l~ai,,
‘-c;=,
cLJ
(Xi+ I 1 dX
i=l =
J XV f(l
-
Ixl,x)D”+‘w~~
(x1)
Example 3.3. If we take in ExampIe 3.2 as duai basis of {&I,k_i}fCOthe one defined by
Iff := ~(-l)‘ok-‘O”(7i)o’(/ri),
i = O,...,k;
7jE
[0, I]
(3.7)
r=O
(cf. [de Boor ‘76b, Section ,511, we obtain a di~erential-fog any point of B,.
dual basis, expanding at
Wu Dong-Bing / Dual bases of a Bernstein polynomial basis on slmplices
489
References de Boor, C. (1976a), On local linear functionais which vanish at all B-splines but one. in: A. Law and A. Sahney, eds., Theory of ,-lppro.rimafion wirh Applications, hcademic Press, New York, 120- 145. de Boor, C. ( 1976b). Splines as linear combinations of El-splines, a survey, Report MRC TSR $1667. de Boor, C. (1986). B-form basics, in: G. Farin, ed., Geometric Modeling, SIAM, Philadelphia, PA, 131-148. Hiillig, K. (1982), Multivariate splines, SIAM J. Numer. Anal. 19, 1013-1031. Schumaker, L.L. ( 1981), Spline Funcfions. Basic Theory, Wiley, New York. Zhao, K. and Sun, J. (1987). Dual bases of multi-variate Bernstein-Bezier polynomials, Computer Aided Geometric Design 5, 119-125.