Computer Aided Geometric Design 2 (19X5) 61-67

North-Holland

61

Triangular spline algorithms

Presented at Oherwolfach 12 November 19X4

Abstract. Three simple algorithms for calculatmg btvartate box splints and their linear combinattons are gtven:

(1) a Mansfield-like recursion formula. (2) the refinement of the control net and. (3) the construction of the

Bezier net. The refined net as well as the Bezier net are generated by ‘filling and averaging’ procedures. Two

examples are added: the construction of a discrete box spline and the construction of the Bezier net of a Sabin

spline. The indicated proofs are all very geometric.

Keywords. Spline. box spline. refinement. Bezier nets. algortthms, subdivision. discrete box sphnes.

The theory of multivariate splines was developed in the early 80’s. and since then has generated increasing interest. Researchers in CAD hoped to get a new, useful tool for the representation and handling of surfaces. In this paper we present algorithms for the calculation of such splines in the most interesting case other than tensor product splines.

1. Historical remarks

Triangular patches were first considered in CAGD by P. de Casteljau in 1963. He con- structed his ‘surfaces a poles’ by repeated affine mappings of a ‘domain triangle’ to the triangles of a spatial net (see e.g. [Boehm et al. ‘841). Later his net of ‘poles’ was called a ‘Bezier net’. In 1982 Dahmen and Micchelli pointed out that the ‘algorithm of de Casteljau’ can be viewed as a special case of a recursion of simplex splines, as given by Micchelli in 1980 [Dahmen, Micchelli

‘831. Triangular splines were first considered in CAGD by Sabin in 1977. He constructed

B-sp!ines by repeated convolutions with the unit pulse, starting e.g. with a rhombic constant [Sabin ‘771. Later Sabin’s B-splines were found to be ‘box splines’ [see Dahmen, Micchelli ‘841 or ‘half-box splines’ for certain cases [Prautzsch ‘83/84].

Triangular B-splines with minimal supports were considered among others by Sablonniere in 1982 [Sablonniere ‘821.

Fig. 1. The algorithm of de Casteljau.

0167-8396/85/$3.30 0 1985, Elsevier Science Publishers B.V. (North-Holland)

62

Fig. 2. The convolutions of Sabin

2. Box splines and box spline surfaces

B-splines over regular triangular grids and their properties can easily be derived from a geometric approach due to de Boor et al. and based on a general idea of Schoenberg from 1966 (see [Dahmen, Micchelli ‘831):

Let B be a box in the affine space &I,,1 (described by affine coordinates y E OX”‘), and let fl be an affine map of ~2’“’ onto the affine space d2, such that the edges of B project onto the edges of a regular triangular grid. Let X be the subspace of points c._&“’ having the same

image x E&‘. X is called the fibre of X. Let S = B n X be the intersection of the box and the fibre. and let V = vol S be the volume of S. Then a ‘box spline’ N is defined by N(X) = vol S. See Fig. 3.

Denote by i E 2’ a point on the regular triangular grid. Therefore i + [I, - 1, l] describes the same point i. Let e, = [l, 0, 01, etc., so that i + eA is a grid point neighboring i. Let p be the

triple of numbers pI of ‘origin edges’ of B which project onto the ‘directions’ eh, when the origin of B projects onto i. The box spline generated by B will be denoted by Njp( x), Note that p is unique, but i is not. Note also that translates of B correspond to translates of Nip. Thus one can normalize the Nip(x) by

EN:(x) = 1. all ie grid.

A”

affine map ll

I

Fig. 3. Schoenberg’s construction in the case of a box.

63

Fig. 4. Grid and box spline control net.

Any linear combination

.?(I) = Cd;N;( X)

of translates of box splines, where the coefficients dj are points E&‘. will form a triangular spline surface. In the natural ordering of the i, the ‘control points’ d, form the so-called ‘control net’ of the box spline surface s”, see Fig. 4.

Note that this surface can easily be computed from the detailed knowledge of only one box

spline N,P( x), because all others are translates

Nip(x)=N,p(x-i),

where 0 denotes the common origin of the grid and its corresponding coordinate system. The Njp are piecewise polynomials of total degree nz- 2, but it should be noted that the

Nip(x) span only a subspace of p”‘p2, in general.

3. Recursion of a box spline

A box spline Nip(x) can also be expressed by box splines Njp -‘I(x) of lower degree, where -e, reduces pl, by 1: Let /3r, &, & be a ‘path’ from i to x. i.e. x = i + /3,e, + ,Bzez + &ei. Then

N!(X)=& ~(BI~~-‘A(xi+(/li-Bi)N~f’*(~~ci)).

where k varies from 1 to 3 and m = p, + pr + p3. In Fig. 5. where p = [2, 2, 11, the involved box splines are represented by their supports.

Using this recursion formula, a box spline of order p can be reduced step by step to box splines of degree 0. These lowest degree box splines form rhombic discs, as shown in Fig. 2. left. Note that, if pl, = 0. the corresponding term in the recursion formula vanishes.

The recursion formula above was first found by de Boor and Hollig from analytic considera- tions [de Boor, Hollig ‘821. A more geometric proof was given by Boehm in 1984 [Boehm ‘841.

P2

P, \ P3

i

Jzz?

PI IS p3 x

NP

i

N P-= I ’

Fig. 5. Recursion. involved box splines with arguments

64 W. Boehm / Triangular spl~e crlgor~thms

4. Subdivision of the control net

Let j = ri + (~,e, + aZeZ + a3e3, LY,_ E Z, denote the points of a refined grid, such that t-i

corresponds to the point i of the initial grid. r is called the degree of refinement. A box spline Nip(x) can also be linearly expressed by box splines fij”(x) of the same order

but over a refined grid. Thus any linear combination of the box splines Njp can be written as a

linear combination of the box splines fijp, i.e.

sp = Cd,N,P = &jip. i .i

The control points dj form a refined net and can easily be computed from the dj by the following ‘filling and averaging’ procedure:

0.

1. 2. 3.

4. 5.

6. 7.

Let i and j also denote the position of the coordinates of dj and ij in triangular schemes. Let r

he the degree of refinement.

Write down the coordinutes of di into an initial scheme.

Choose a direction ek, such that p, f 0 for I + k.

In a refined scheme fill the ‘rhombuses ’ defined by the opposite points ri and ri + (r - l)e, by

duplicates of di.

Set q=[l, 1, 11-e,. For any I, for which q, < p,, and for ail j take the average of the r dutu ut j + e,, , j + rel and

write it into a neM1 scheme at j + re,.

Set q,=q,+ 1.

If q fp, got0 5.

The final scheme represents the control points of the refined net in its natural ordering. For

repeated refinements, the refined nets will converge to the spline surface. See Fig. 6. Such refinement algorithms were considered in [Boehm ‘831, [Prautzsch ‘83/84], [Dahmen,

Micchelli ‘841, and [Cohen et al. ‘841. A geometric proof of the above algorithm by repeated subdivision of the box B into r layers was given by Prautzsch in 1984 [Prautzsch ‘84, see also

Boehm ‘84/U].

5. Example

To demonstrate the construction, it is sufficient to consider the refinement of the control net

of a single box spline, defined by the identity

N;P(x)=CS~,~N{(X), k

initial scheme filling averaging

Fig, 6. The refinement algorithm scheme.

final scheme

W. Boehm / Trm~gulur spllne dgonthms 65

initial scheme

0 1 1 1

i r

1

l

1221 12321

1 2 2 1

1 1 1 0 r 0 .;

q=11,0,11 q=ll,l 11

intermediate scheme

Fig. 7. The refinement algorithm. example

\T 2 3 3 1 -, ,-3 5 6 3 lm

-) -2 3 5

7

5

2-J

6 5 3 p’

-,I 3 3 2 p’

-,l /I 1 -\ 1 \,“, ‘G

q=12,1,1 I

final scheme

where 6; k is the well-known Kronecker symbol. More general nets can be constructed from this by superposition. Fig. 7 shows an example, r = 3, p = [l. 2, 11. The averaging is marked by

e-4 r

Schemes of numbers as shown in Fig. 7 are called ‘discrete box splines’. They depend on i, p, and r. One may visualize them by erecting the ordinates Aj over the center of the support of kjp (as a generalization of Greville’s theorem for univariate splines), and by completing the net

[see Boehm ‘84/85]. It is obvious that this net can also be constructed from a pyramid in all three coordinates

[Prautzsch ‘84/85], see Fig. 8.

6. Generating the BCzier points of a box spline surface

A triangular box spline N!(x) can also be linearly expressed by ‘truncated Bernstein polynomials’ B;lp2( x) of d e g ree m - 2. Thus any linear combination of box splines Njp can be

written as a linear combination of truncated Bernstein polynomials B,!“-‘:

sp = Cd&” = xb;l-2B;‘-2,

di

0.

The toe f ficienis b;‘~’

I

form the co-called ‘Bezier net’. They can easily be computed from the by a ‘filling and averaging’ procedure as well:

Let i and j now denote the location of the coordinates of the Bkzier points bi und bJ in the usual

triungulur schemes.

\I p2 \‘Y

P3 11

c@

I

Pl \’

‘A i I<

< I

I\ ‘I ’ r=l

Fig. 8. Nets of two centered discrete box splines, p = [l. 2, l]

66

1. 2. 3. 4.

5.

6. 7.

from step r-l filling and averaging result

Fig. 9. One step of the generating algorithm.

Setbi=d;. Setq=[l, l,l].

Write down the coordinates of bl into un initial scheme.

Set r = 4, + q2 + q3 - 2. For uny k, for which q < p, in (I refined scheme fill the ‘rhombuses’ defined by the opposite

points ri und ri + (r - l)e, his the dutu of the corresponding rhombuses defined by (r - 1)i und (r - 1)i + (r - l)e, of the previous .scheme.

For all j take the aoeruge of the r datu at j, j + e,, . , j + (r - l)e, und write it into u new scheme ut j + re,

Set qn = qn + 1.

If q zp, got0 3.

The final scheme represents the desired BCzier net in its natural ordering. This net can be used to compute the box spline surface by the algorithm of the Casteljau.

-O- ‘0 - I\ / \

step r=l \ I \ / \

step r=2

o-0-0 o-0-0 0- , \

0 0 1’ 1’ ‘1 o/ 0’ ‘0 o’_ 0 “1’ 1/ 2-y ,‘Jo “0’

o-o-/l ,‘,p2$ / \

/1, 0 0 p, 0 0 \I 1

0 1,) ,I 0 0 0

3

\ / ;.

-0 o--o--o o--o--o o- o-op/o ,0,-o-/” ,0,-s / \

step

o/ 0 \

\ J 0, /1’ 1 ‘1, ,O’ 0 ‘0, , ,o,m 0 - 0 -/l- 2 - 3 -,2,- 1 - 0 - 0 -

0 0 \ 1 2 4 4 3 1 0’ ‘0, 0 0 ‘1 ‘3 4’4 > 1’0 0 _~~~_~_l_‘~‘_3_2_~,/o-~-b/_

r=3 9 o /O “1, 1 ,l 0 0 II“

1. 0 0 O/

~~o~“,“.“:~“,~,_b’-.Q /

Fig. 10. Generating the BCzier ordinates, example

The above algorithm can easily be derived by merging the constructions of Sabin [Sabin ‘771

and Prautzsch [Prautzsch ‘83/84].

7. Example

To demonstrate the construction it is sufficient to consider the control net of a single box spline, defined as above. General nets can be constructed by superposition again. Fig. 10 shows

the example for p = [l, 2, 21, the averaging is marked as above. It is obvious that the algorithm in three coordinates constructures the 3D Bezier net. But to

avoid this in the l-dimensional case above, one must again center the schemes.

References

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de Boor. C. and Hollig. K. (1982) Recurrence relations for multivariate B-splines. Proc. Amer. Math. Sot.

Cohen. A., Lyche. T. and Riesenfeld, R. (1984). Dtscrete box splines and refinement algorithms. Computer Aided

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Prautzsch. H. (1984/85). The location of the control points in the case of box splines. IMA J. Numer. Anal.. submitted.

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