mathematics of computation
volume 57,number 195
july
1991,pages 299-308
THE STRUCTUREOF MULTIVARIATESUPERSPLINE SPACES
OF HIGH DEGREE
PETER ALFELDAND MARITZASIRVENT
Abstract. We consider splines (of global smoothness r, polynomial degree d,
in a general number k of independent variables, defined on a k-dimensional
triangulation ¡T of a suitable domain ii ) which are r2 ~m~ -times differentiate across every m-face { m = 0, ■■■ , k - 1) of a simplex in !T . For the
case d > rl we identify a structure that allows the construction of a minimally
supported basis.
1. Introduction
A k-simplex K (0 < k < k) is the convex hull of k + X points in R called
the vertices of K. K is nondegenerate if its k-dimensional volume is nonzero,
and degenerate otherwise. The dimension of a nondegenerate k-simplex is k .
The convex hull of a subset of p + 1 vertices of K is a //-face of K .
Let FcR
be a given set of A^ distinct points.
A triangulation J~ of the set W is a set of nondegenerate /c-simplices satisfying the following requirements:
1. All vertices of each simplex in ¡T are elements of W.
2. The interiors of the simplices in y are pairwise disjoint.
3. The set
(1)
£1- \J KcRk
is homeomorphic to [0, 1] .
4. Each (k - l)-face of a simplex in ST is either on the boundary of Q,
or else is a common face of exactly two simplices in ÏT.
5. No simplex in ¿7" contains any points of "V other than its vertices.
Note that a /¿-face of a simplex in y is itself a /¿-dimensional simplex. On
the triangulation !T we define a spline space Srd(J~) as usual by
(2)
Srd(F) = {s £ Cr(Q) : s\r £ &Ï Vt e ^},
where ¿Pd is the (kdd)-dimensional linear space of all /c-variate polynomials
of total degree less than or equal to d.
Received February 27, 1990; revised September 21, 1990.
1980MathematicsSubjectClassification(1985 Revision).Primary 65D07, 41A63,41A15.
©1991 American Mathematical Society
0025-5718/91 $1.00+ $.25 per page
299
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PETER ALFELD AND MARITZA SIRVENT
300
In this paper, we consider subspaces of Srd(!T) obtained by increasing the
smoothness requirements across faces of the underlying simplices. More precisely, denoting by 5? the set of all /¿-faces of the simplices in ¿7" ( p =
0, ... , k - 1) and letting 5? = LLIo^u' we define the (superspline) space
§d(&~) as a subspace of Srd(&~)as follows:
§d(^) = {5 £ Sd(¿7~): s is /Mimes differentiable across a Va £ 5*},
^ '
,
~,k—dimrj—1
where p = r2
The concept of supersplines was introduced in Chui and Lai [8], [9]. The
area in between finite elements and full spline spaces was further explored by
Schumaker [16] and Ibrahim and Schumaker [11].
2. The generalized
Bézier-Bernstein
form
Crucial to analyzing the dimension of spline spaces is the Bézier-Bernstein
form of a multivariate polynomial. In the case k < 2 this form is used widely
and is well known. A review of the Bézier-Bernstein form for a general number
of variables is in de Boor [6]. In this paper, we use a notation that is particularly
suitable for our purposes. However, generalized barycentric coordinates and
global control nets have also been proposed in Alfeld [2] and de Boor [6].
We use T" as an index set and denote by N the set of nonnegative integers.
For vectors I = [iv]veT £ N
(4)
and a = [av]veT el
we define
III= £ ', >
vëv
(5)
■l=iiiin«vní.,>
vçT
'
VÇ.T
where
(6)
0°:=1.
We also use the notation
(7)
a(\) = conv{u : iv > 0},
cr(a) = conv{v : av ^ 0}
where conv X denotes the convex hull of a point set X .
We now define generalized barycentric coordinates as cardinal piecewise linear
functions bv £ Sx (Q) by the requirement
( X if v = w
(8)
W4.-{0
else
*.«'•
Clearly, in each /c-simplex K £ ¡T the functions bv , where v is a vertex of
K , reduce to the ordinary barycentric coordinates. Globally, i.e., for all x £ Q,
they satisfy
(9)
¡>>„ = 1,
bv>0
\lv£T,
and
x = ^
ver
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veîr
bv(x)v.
THE STRUCTURE OF MULTIVARÍATESUPERSPLINESPACESOF HIGH DEGREE
301
For a given polynomial degree d, we use the domain index set
;i0)
\dd = {l£HN:\l\ = dando(l)£Si'\j3r}
Letting
(11)
b = b(x) = [bv(x))ver,
it is clear that every function s £ Sd(&~) can be written as
(12)
5 = EcibIlei,
The coefficients cx are the Bézier ordinales of 5.
3. The de Casteljau
algorithm
Let 5 e Sd(^), K £ ZT, and s\K=p £&>%. Without loss of generality we
may relabel the vertices and assume that K is the k-simplex
(13)
K = {a = (ax,--- , ak+x): ax + ••• + %+, =d,
a} >0,
a.eR}.
Then, with V] denoting the vertices of K, we have
k+l
(14)
k+l
p= J2 c.b1 where x = £>,*}'
ie/cni^
;=1
T,bj = L
;=1
The Bernstein polynomials b (x) satisfy a recurrence relation, see Farin [10]
fc+i
(15)
b\x) = J2b/~eJ(x),
;=i
\l\=d,
and eJ = (0,--- , X,-- ,0).
This relation allows one to expand p in terms of Bernstein polynomials of lower
degree with (polynomial) coefficients p[(b).
Theorem 1. We have
(16)
p=
£
p,(b)b\
0<r<¿,
|I|=rf-r
where
(17)
rf(b)= c„
fc+i
(18)
prl(b)= ^2b/l^l(b),
|I| = rf-r,
0<r<d.
The intermediate coefficients p[(b) can also be written explicitly as
(19)
P,»=Eci/'
|I| = d-r.
|M|=r
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PETER ALFELD AND MARITZA SIRVENT
302
The formulas in Theorem 1 may be used to evaluate p at a given point, and
are referred to as the de Casteljau Algorithm.
4. Smoothness across an interface
Given a vector e £ R , the directional derivative of p in the direction of
a =
db.k+l
i?'---'
denoted Da , is given by (Alfeld [2])
'
de i '
(20)
Dap = d £
p\{0L)b\
HM-i
where
(21)
p\(a)=
£
cl+MaM,
\I\=d-X.
|M| = 1
In general, the rth directional derivative of p in the direction of a is given by
(Farin [10])
(22)
D>= jrhv
S *W,
1
'■ \l\=d-r
where
(23)
p[(a) = £
cl+MaM,
\I\ = d - r,
and
p°(a) = cv
|M|=r
The following theorem was proved by Farin [ 10] in the bivariate case; here we
state without proof the result in any number of variables. Let x be an m-face
of K ; without loss of generality, we will assume that
(24)
r = {*£K:
am+2 = ■• • = ak+x = 0} .
Theorem 2. Let p, q £ ¿?d be such that p\T = q\x. Then Dsap = Dsaq on t for
all directions a, 0 < s < r, if and only if Pj = qr3for all J = (/',,...
0,... ,0), |J| =d-r.
5. Subsimplices and subpolynomials
Let K £ y
(25)
be a fc-simplex and x be a face of K with dim x = n ,
K=U£RN:
{
(26)
Y,av=d,av>o\
vex
x=U£K: Y,av= A{
V€T
J
For J e K n ld , so that Y,veT jv = d- p, define
(2V)
,
J
Tj = {a = [av]v£K: jv>av,v£x}.
Clearly, Tj is a subsimplex of K similar to K .
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, jm+x,
THE STRUCTUREOF MULTIVARIATESUPERSPLINESPACESOF HIGH DEGREE
303
Let p £ ¿Pd , and (I, cjjK be the control points of p with respect to K.
The control points (I, c,)l€T define a polynomial p} of degree p. We call
Tj the subsimplex of K associated with J, and p} the subpolynomial of p
associated with J.
De Boor [6] introduces the concepts of subsimplices and subpolynomials and
proves most of the results of this section. Since our definitions are slightly
different, we restate two key facts involving subpolynomials. Theorem 2 can be
restated as follows:
Theorem 3. Let p, q £ 3?d , and x be a face of K with dim x = m such that
P\T = a\T. Then
(28)
Dsap = Dsaq onx,
for all directions a and
0 < s < p,
if and onlyif
(29)
Pi = ai Vie A"with dist(I, x) = p.
The above theorem immediately yields the following:
Theorem 4. Let K, K1 eJ,
x C K ni', p, q £&>d , and let s £ Sd(Q) with
s\K = p, s\Ki = q. Then s £ Cp(x) if and only if px = qx VI e K1 with
dist(I, x) = p.
6. Determining
sets
We now generalize the concept of a determining set known from the bivariate
case (see Alfeld and Schumaker [4]).
Definition 5. A set D c Id is a determining set of §rd(&~)if, for all s £ Sd(^),
(30)
c, = 0
VI G D => s = 0.
D is a minimal determining set if there is no determining set which has fewer
elements than D.
It is clear from elementary linear algebra that the number of elements in
a determining set of Sd(Sf) provides an upper bound on the dimension of
S^(y), and that the number of elements in a minimal determining set is unique
and equals the dimension of S^(^).
7. A minimal determining
set of Sd(^)
For each simplex ueyuJ
let p = r2 ~dim(T~1as before; we define two
sets of domain indices recursively by
(31)
2(o) = \l£\d:
{
^iv>d-p\
vea
and
(32)
2(o) = 3(o)\{j2(x),
T.<a
where x is a proper face of a .
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)
304
PETER ALFELD AND MARITZASIRVENT
Definition 6. Let it 6 yu^.
2(a) if, for all s £ STd{&~),
(33)
A set s/(a) c D(o) is a determining set of
cx = 0 V\£S¿(o)U(2(o)\2(o))^cl
= 0 VleW{o).
The set s/ is a minimal determining set of ^"(a) if there is no determining
set of 3¡(a) with fewer elements.
Lemma 7. Let d > r2 . Then, for all I £ Id there exists a unique ireyuJ
such that I £ 3(a).
Proof. To prove the lemma, we have to show first that for all a, x £ S* U!T :
(34)
o¿x^2(o)n2(x)
= 0.
To establish this, suppose there is a domain index I £ 2(a) (~)2(x), for two
simplices a ,x £S^ \J¡T, such that dim a > dim x, and x is not a face of a .
Thus,
(35)
J2iv>d-r2h-dima-1
w6cr
and £/„
> rf - r2^dimT-1,
«6r
which implies
/t^,
(36)
V~* •
E i„ +, V^
£i„ • ^.> ->
2dj - r2»Ar-dimrJ-l -rl -fc-dimr-l
vEo
vÇx
Moreover,
(37)
£'« +£', = E '«+ E '»■
Now, rearranging (37), substituting (36) into (37), and using J2veaUTiv < d,
we obtain
(38)
£
iv>d-r2k-dimr.
v€of~ir
So there exists a face f of t such that I £ 2(x), which is a contradiction.
Finally, we need to prove that ld = \Jae^>IJ^-2(a).
To see this, we only need to show that for each domain index le I¿ there
exists a simplex a G S? U !T such that I 6 2(a).
However, this is trivial,
since for each domain index I there exists at least one Ä>simplex K such that
J2veKiv = d. Thus, I G 2(K), and there must be a simplex a -< K such that
I £ 2(a).
D
Lemma 8. Let d > r2k, p = r2k~d>ma-[, and sé'(a) := 2(a) n_K, where
K £ y and a <K. Then sé (a) is a minimal determining set of 2(a).
Proof. First we establish that sf(o) is determining.
Let s G Sd(^), and let s\K = p , with p g &>d. If
c, = 0 VI G s/(a) U (2'(a) \2(a)),
then
p, = 0 Vl£s/(o)U(2(a)\2(o)),
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THE STRUCTUREOF MULTIVARIATESUPERSPLINESPACESOF HIGH DEGREE
305
where px is the subpolynomial of p of degree < p associated with I (notice
that dist (I, a) < p VI £sé(a) ). Let K1 be any other /c-simplex of 3~ such
that a <K' and K ¿ K', and let q = s\K,, q £ 9°k . Now, s £ C(a) ; then
VI G (2(a) n K') U (2(a)\2(a))pl
= ql, where ql is the subpolynomial of q
of degree < p associated with I. But,
(39)
p, = 0 Vl£(2(o)nK)u(2(o)\2(a))
(40)
=► q1= 0 VL£(2(a)nK')[J(2(a)\2(o))
(41)
=*
Cj= 0 VI G (2(a) r\K')ö (2(a) \2(a))
(
(42)
=*
c, = 0 VIG
\
(J 2(a) HK U(2(a)\2(a))
1 K£3-
\^K
(43)
=»
J
C, = 0 VlG^'((T)U(^((T)\Sr((7)).
Therefore,
(44)
cx= 0 Yl£2(o).
To see that sé (a) is indeed minimal, take I G sé (a) and consider the set
sé (a) = sé (a) \ {1}, define a polynomial on ATwhose Bézier coefficients are
equal to zero except at the domain point I, where c, = 1, and extend this
polynomial globally on the rest of the triangulation.
The coefficients cs are equal to zero on 2(a) \2(a),
since the smoothness
conditions there only involve domain indices in 2(x) for x <a .
Hence, c} = 0 V/ GsT(a) U (2(a) \2(a)),
Cj = 0 V7 G2(a) since in particular / G 2(a)
but this does not imply that
and c¡ = X.
Therefore, sé (a) cannot be determining and sé (a) is a minimal determining set of 2(a).
D
The following theorem is the central result of this paper.
Theorem 9. Let r > 0, d > r2k and let sé (a) = 2(a) n K, where K £ 9~ is
a k-simplex so that a is a face of K. Then
(45)
sé := (J sé (a)
is a minimal determining set of §d(¿7~)■
Proof. Let 5 e §¿(7) and assume that c, = 0 V/ g sé . Using induction on
dim a , we first establish that sé is a determining set.
(1) If dim a = 0 then c, = 0 V/ Gsé (a) implies c, = 0 VI G2(a), since
2(a) = 2(a).
(2) We assume now that c, = 0 V/ G 2(a)
and Va with dimrj < n .
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306
PETER ALFELD AND MARITZA SIRVENT
(3) Let dim ff = n, c, = 0 V/ G sé (a). Let 3ê(a) = sé(a)U(2(a)\2(a))
;
notice that B(a) csé(a)o (\J^a2(x)).
By the induction hypothesis, c, = 0 VI G \JT^,a2(x) implies that cx = 0
VI G ¿®(o). But se (a) is a determining set of 2(a) ; thus, c¡ = 0 V/ G S^ff),
hence
(46)
cI = 0 V/G [J 2(a),
rre^ur
and by Lemma 7 we get that
(47)
cx= 0 Vlelrf.
To show that j/ is indeed minimal, we give arbitrary Bézier ordinates cx
V\£sé , and we construct the rest of the Bézier ordinates in such a way that
the piecewise polynomial they determine is a superspline.
So, we assume Cj VI G sé are given, and let J G \d . Then by Lemma 7 there
is a unique a GS? U T such that J G 2(a).
We define Cj inductively over dim a .
(1) If ff is a vertex, then there exists K G y so that ct -<ATand sé(a) c K.
Let ffj be the subsimplex of K associated with J, a3 c sé(a). Then the
polynomial ps is defined; so, for J g 2(a) n AT'with K' g y,
A"' ?¿ A, let
tj be the subsimplex of K' associated with J. Then there is a unique way to
give Bézier ordinates to the domain points in Tj in such a way that they define
a polynomial q} which equals p3. Furthermore, in this manner, c3 can be
defined for all J G2(a)\sé(a).
(2) Suppose Cj has been defined for all J G \J¿[ma<n^(^) ■
(3) Let J g 2(a) with a an «-simplex and K g y such that se (a) c K.
And let again a-, be the subsimplex of K associated with J. Note that
(48)
oJc2(a)r)Kcsé(o)\j{\J2(x)nK.\
.
Then by the induction hypothesis and by the fact that the Bézier ordinates have
been defined on sé , we have that all of the Bézier ordinates on rjj are defined;
therefore p3 is defined. So as before, for J G 2(a) n K', K' ^ K, we can
define q} in the same way we did when a was a vertex. Thus, cx has been
defined for all I G Id .
Next, we need to show that the piecewise polynomial function defined by the
Cj's is well defined.
Suppose pL = qh for L G K and £u6i lv = d - p .
Let J G 2(a) n K' and suppose that J G £L with J 7¿L, where
(49)
{l = {l6Jf':/,>/,OE{1
íeyur,
and
(50)
rJ = {leK':iv>jv
v£o}.
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and ff ^¿}
THE STRUCTURE OF MULTIVARIATESUPERSPLINESPACESOF HIGH DEGREE
307
Then J G £L implies jv > lv , which in turn implies Tj c ¿;l . Therefore, qs is a
subpolynomial of qh . Similarly, p3 is a subpolynomial of pL . Hence, p3 = q-,
since qLUph£ C°°(Ç). Therefore, the Bézier ordinates are well defined and by
construction, the piecewise polynomial defined is a superspline. D
Corollary 10. We have dimS^(y) = E^es^r
W(<?)\■
In view of the above corollary, to compute dimS^(y) we need only to know
the cardinality of sé (a) for every a G y. Clearly, |j/(ff)| depends only on
m = dim a . The following theorem is proved in Alfeld and Sirvent [5].
r
Theorem 11. We have dimS¿(y)
k
= \^m=0(t>(m)fm, where fm is the number
of m-simplices of S? Uy, and for m = 0, ... ,k, (¡)(m) = (f>m(0),where, with
k-m-l
, the quantities <p™(p)are defined recursively by
Pm = rl
pq-p
j + m-q
j
*» = E
;=0
(51)
P-J + Q
Q
9-1
<7+ l
i+X
tíü + j)
if 0 < q < m: and
Po-P
(52)
¿o» = E
(53)
•rí(O)
j +m
j
Po - p + m
m
k-l
-E
d+k
k
m=0
8. Minimally
k+ X
m + X ¿(0).
supported
bases
Definition 12. The star of a simplex a £ S? U T, denoted star(a), is the set of
all fc-simplices A"G y such that a is a face of K.
Definition 13. A basis {I : p = X, 2, ... , dimS^(y)}
is said to be minimally
supported if for each basis function / there exists a simplex a £ S? U T such
that the support of / is contained in star (a).
The basis functions constructed in the proof of Theorem 9 are minimally
supported: using the same construction, we can define cardinal supersplines
(54)
so that, if \£Sé(a),
/jeS^y):
/I(J) = áIJ V\,i£sé
then cA= 0 VJ G ld \ star(ff) and c, = 1.
Acknowledgments
This research was supported by the National Science Foundation with grant
DMS-8701121 to the University of Utah. The authors also appreciate the careful work of the editor and the thoughtful comments on an earlier version of this
paper by an anonymous referee.
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PETER ALFELDAND MARITZASIRVENT
308
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Department
of Mathematics,
University
of Utah, Salt Lake City, Utah 84112
E-mail address: alfeld@math.utah.edu
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