Error bounded piecewise linear approximation of freeform surfaces

Computer-AidedDesign, Vol. 28. No. I, pp. 5157, 1996 Copyright Q 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 00104485198 $15.00 + 0.00

ELSEVIER

Error bounded piecewise linear approximation of freeform surfaces Gershon Elber tion 1 for emv pair of points in an approximated region, or compute a global upper bound on Equation 1. One can only assume that Equation 1 is verified for a finite set of locations on S(u, v>, probably on the boundary of the approximated region, a test that is alias-prone. In References 1-3, other subdivision criteria are suggested as well. In Reference 1, proximity and silhouette screen space considerations are employed for static, viewing direction dependent approximation, for rendering purposes. In Reference 2, planarity and chord distance criteria are employed, but for the verticed of the considered region only. In Reference 3, the loose and alias-prone chord distance criterion of,

Two methods for piecewise linear approximation of freeform surfaces are presented. One scheme exploits an intermediate bilinear approximation and the other employs global curvature bounds. Both methods attempt to adaptively create piecewise linear approximations of the surfaces, employing the maximum norm.

Keywords: adaptive sampling, polygonization, bilinear tit, freeform curves

INTRODUCTION The (piecewise) polynominal and rational freeform rep-

u,+u, s -2

resentations are frequently employed in the fields of computer graphics and computer-aided geometric design. For many applications in both fields, the freeform shape must be approximated using piecewise linear representations. Display devices support the drawing and rendering of only polygons, in general. Fundamental tasks, such as surface intersection and strength and heat transfer analysis, are frequently addressed using lower order and mostly piecewise linear approximation of the surfaces. Hence, the problem of finding an optimal, in terms of both accuracy and space, piecewise linear approximation of a freeform surface as a set of polygons is of significant importance. In References l-3, the norm of, l-

(n,, nz>

’

(S(U,, UP) + S(u,,

Q)

2

2

(2) is exploited. This criterion implicitly assumes a uniform parameterization. In References 4-6, the 2nd partial derivatives of surface S(u, v) are used to bound the distance between a C2 surface S(u, v) and its polygonal approximation Z(u, v). In Reference 4, the bound of, sup lIS(r.4, IJ) - Z(z.4,UN > $(I@, (U, U)ET

+21,l,iu,

+ 1,2A43) (3)

(1)

is used, where ni are the unit normals of surface S(u, U) at locations pi E S(u, u). Equation 1 is exploited as a curvature or divergence measure while being independent of the size of the object. Therefore, a bound on the amount of deviation of the surface from its piecewise linear approximation cannot be established. Furthermore, it is unclear how one can bound Equa-

is derived, where,

Department of Computer Science, Technion, Israel Institute of Technology, Haifa 32000, Israel Paper receiwd: 9 November 1994. Retied:

up+uq

10 Ape11995 51

Ml =

CMU, v) sup II dU2 II (U,UkT

M2 =

II (u,U)ET

M3=

sup II (U,U)ET

sup

d%(U, v>

dudv ‘1 d2S(U, u) dLJ2

II

(4)

Error bounded piecewise linear approximation: G Elber

will consider two approximation criteria, one that is based on an intermediate bilinear surface fit and one that is based on global normal curvature bounds.

BILINEAR BASED POLYGONAL APPROXIMATION Let Rh, Figure

1

A planar

surface

with arbitrary

u) be the bilinear surface,

R(u,u) = (1 - u)(l - u)Z’, + (1 - u)up,,

large 2nd derivatives

+u(1and T c 9’ is a right triangle with vertices (A, B, C) of the form B =A + (I,, 0) and C =A + (0, &I, in the parametric space of S(u, v>. Unfortunately, this bound can be arbitrarily loose as is demonstrated by the following example. Consider the regular parametric surface (Figure I>, F(u,

u> = kzu>2b2u)2, u, UE (0,1/a],

u> - R,(u, u)ll

(7)

Lemma 1 The distance (Equation 6) between T, = A(P,,P,j,P,,,) and R(u, u), u + u I 1 is less than or equal to f of the distance from P,, to the plane containing T,.

(5)

The normal of F is non-zero and is collinear with the z axis. Hence, F is a regular surface that is also planar. Nevertheless, none of the 2nd partial derivatives of F vanish in u, u E (0, 31. Moreover, the 2nd partial derivatives of F can assume arbitrarily large values by reparameterization, or modifying the value of a. Properly noted in Reference 4, Mi, 1 I i I 3 (Equation 4) are invariant under rigid motion transformation. They are, however, parameterization dependent. That is, the bound established in Equation 3 depends on the surface parameterization (i.e. a in Equation 5) and is not an intrinsic surface property. Even more elusive is the fact that while a surface can be planar, its 2nd partial derivatives can assume arbitrary large values. One should seek bounds that are not only invariant under rigid motion transformation but are also parameterization independent. Clearly, the 2nd partial derivatives express not only intrinsic surface geometry but also differential properties of the isoparametric curve considered. These are manifested as the geodesic curvature the isoparametric curves in the tangent plane as well as changes in the speed of the parameterization’. Let S(u, v> be a C2 continuous regular parametric surface. Throughout this paper, without loss of generality and unless otherwise stated, we will assume a parametric domain of u, u E [O, 11. Let R,(u, u) be an approximation of S(u, u). Then, E = r-r+‘“,

u, UE [O, 11

Approximate R(u, u> using two triangles T, = A(PO,,P,,,Plo> and T2 = A(P,,P,,P,,). Tl and T2 cover, in the parametric domain of R(u, v), the closed triangular domains of u + u I 1 and u + u 2 1, respectively.

bzuY, 01,

a > 0

u)P,, +uvp,,,

Proof Rotate and-translate T, and R(u, v) into f, and &u, u) so that T, is contained in the Z = 0 plane. Because R(u, VI is rigid motion invariant, this transformation does not affect the distance from the R(u, u) to the triangles T, and T2 approximating ii. Then, the 2 component of R(u, v) is equal to R,(u, u) = u&f, because p(;o = pi, = P”;, = 0, where pz denotes the Z component of the transfocmed point. R,(u, v) is exactly the distance from R(u, u> to the 2 = 0 plant. Hence, (uv> provides the necessary bound. SUP Ir+c’ll, u, L’E[O, I] Clearly, uu is an increasing monotone function in the prescribed domain. The function uu achieves its upper bound for Tl along the boundary u + u = 1. Then, the single extremum (a maximum) of uu = ~(1 - u) is at u=+anduu=$. The same holds for T2. A bilinear, R(u, u), can therefore be approximated using two triangles, q, i = 1, 2, and the maximal deviation from the bilinear can be easily estimated. Let S(u, u) be a piecewise polynomial parametric surface represented as a B-spline surface. Let R&u, u) be a bilinear surface approximation to S(u, u), defined over the four comer points of S(u, u>. R&u, u) can be degree raised’ and refined9~“’ to the same function space of S(u, u). Call the new refined and degree raised bilinear Rf(u, u>. Then (see Equation 6 and also References 11 and 12),

(6)

is an upper bound on the maximal deviation between the region of the surface and its approximation. In this work, we will attempt to globally estimate E when R, is a piecewise linear approximation of S. We will restrict our discussion to an approximation that is adaptively derived using a recursive surface subdivision based approach. The divide and conquer approach is greedy and cannot ensure a globally optimal approximation. Yet, it is very simple and convenient to use. Different flatness criteria can be employed and comparisons can be easily performed. We

maxllS(u, u) - R,Ju, u>ll I‘. L u) - RI(u, u)ll

= maxllS(u,

11,L’

=

mad II,

52

I’

ci

i-1

j=O

PijBT7 (u)Bz, (U)

Error bounded piecewise linear approximation: G Elber

In Figure 3, several examples of polygonal approximations of freeform NURBs surfaces computed using the bilinear intermediate representation are shown. This approach is also summarized in Algorithm 1. A greedy approach in the subdivision process selects the subdivision direction that minimizes the approximation error from the two subdivisiom direction possibilities, u or u.

= maxll E k (P;j - Q,j> B,,., (u)BJ p (~111 1,)u i=o j-0 (8)

s ma( llpij - QijII, vi, i)

employing the partition of unity property of the B-spline basis functions, BiyT (u> and B/ lr (v>. Hence, one can put a bound on the distance between two triangles, q, i = 1, 2, approximating Sk, v) by fitting a bilinear R&u, v> to S(u, u>, bounding the maximal distance between S(u, u) and R&u, u), using Equation 8 and bounding the distance between I;., i = 1,2 and R&u, v), using Lemma 1. The bound of the distance from S(u, u) to q is the accumulative result of the bounds established from S(u, v> to R&u, v> and from R&u, v) to lJ. The established bound is sharp in the following case (see Figure 2). Let,

Algorithm

BEGIN

R&u, LJ) = a bilinear surface from the four comers of Sh.4, v); Rf(u, u) = R&u, u> degree raised and refined to same function space of S(u, u>; T,, T2 (= two triangles approximating R&u, u>; e1 e= maxllS(u, v)-RRf(u, UN; l2 = Maximal distance from RJu, u> to T,, Tz;

t PijB,? (u)Bj! (~1, i=o j=O

S(u, u> = 4:

(i, j, 1) i =j = 1

Pij =

(i, j, 0) othenvise

i

1

Input: S( u, v), freeform C’ continuous surface; E, maximal deviation from S(u, v) to its polygonal approximation; output: 9, set of polygons approximating S(u, v) to within E. Algorithm: SubdivSrfUsingBilinears (S(u,v), E);

where B,” (t) are the linear B-spline basis functions. The bilinear surface, R, (u, ZI>,fitted to the four comers of S(u, v) is clearly planar,

IF (et

+

l2 < E) THEN

return IT,, TJ; ELSE BEGIN

R,(u,

VI = i

h QijBf (u)Bf

i=() j=O

(v),

(1) S,(u, v), S,(u, v> e S(u, V) subdivided into two; return SubdivSrfUsingBilinears (Si(U,U),E) u SubdivSrfUsingBilinears (S2(u,u),~);

Qij = (2i, 2j, 0)

and the refined bilinear, elevated to the same space of S(u, v), R:(u, u> = i i QirjBi”(u)B; i=o j=l)

END;

The teapot in Figure 3 is shown with two triangles per approximated patch. The rest of the examples in Figure 3 show a polygonization of a sufficiently flat patch into four triangles by sampling the centre of the patch as a fifth interior path, PC, at u = u = i. Let R(u, LJ)be a bilinear surface as in Equation 7.

(v),

Q:j = (i, i, 0) Because, Ti, i = 1, 2 can exactly represent the planar bilinear, the magnitude of lIPI - Q;,ll is indeed equal to the extreme distance between S(u, v) and q, i = 1, 2. For higher degree surfaces, this bound can be further refined by establishing the maximal value a basis function can assume in the interior of the parametric domain.

Lemma 2 The distance (Equation 6) between Tl = A( P,,P,,P,) and R(u, u), u + us 1, u > IJ is less than or equal to i of the maximal distance from P,, or P,, to the plane containing T, . Proof Following the proof of Lemma 1, recognizing that the newly introduced constrain of Pt = 0 necessitates p;, = -&, and maximizing the resulting bilinear function of(l-2u)uatu=v=a.

Figure 2 The bilinear bound established in Lemma 1 is sharp. The bound of IlO:, - P,,II is indeed the maximal deviation of the approximation from the original surface, S(u, u), in solid lines. The (planar) bilinear, R:(u, u), is shown in dashed line

Table 1 compares the number of polygons and the computation time of each approximation, using both 53

Error bounded

piecewise

linear approximation:

G Elber

The projection of the second partial derivatives of S(u, u) onto n, the unit normal of S(u, ~1, yields,

a2s I,, = (,u,,n)

d2S

I,, = (34)

(9)

where ljj are known as the coefficients of the matrix of the second fundamental form, L7. Computing Iii for the surface in Figure I reveals that indeed all ljj have vanished. Nonetheless, the coefficients of the second fundamental form are parameterization dependent, in general. Consider, for example, the unit sphere, S’(t), $>= (cos(a~)cos(~>,

cos(a0)sin(+),

sin(a0)

Clearly, the unit normal of S2 equal S2. The second partials of S2 with respect to 13 equals -a2S2 and therefore I,, = -a2 depending on a which is a reparametrization of 8. Let p E S(u, u), S(u, u> is sufficiently differentiable and regular. Let I E TP, where TP is the tangent plane of S(u, u) at p, be represented as the linear combination of the two first partial derivatives of S(u, v) that span TP. Then,

Figure 3 Some examples of a polygonal approximation of NURBs freeform surfaces using bilinear fit. In (a) uniform subdivision is shown while in (b) the bilinear fit method is used. In all examples, the total number of polygons in (a) is larger then in (b). See also Table 1

as

(

r=((y,,y,J,

1>=r,,$+Yc.Z

d&s

(10)

-$&

y = ( y,, 7,) is a vector in the parametric space of S(u, u). The normal curmfure, K,, of SurfaCe Sk, v>at p in direction I is equal to7, where

bilinear fit and uniform subdivision, for the examples of Figure 3. All approximations were tuned so that the number of polygons in the uniform sampling method is larger than the bilinear fitting method. Computation time was measured on an SGI R4400 150 Mhz Indy.

K

II

=-

YLYT

(11)

YGY’

where G and L are the matrices of the first and second fundamental forms,

CURVATURE BASED POLYGONAL APPROXIMATION

(12)

The 2nd-order derivatives have been shown to be insufficient as curvature bounds due to their dependency on the parametrization as well as the introduction of Geodesic curvature into the expressions of the 2ndorder derivative. In this section, we examine the use of intrinsic and parametrization independent normal curvature, K,, to bound the error of the polygonal approximation.

Table 1 Time and number of polygons of the approximations

dS

and g,j= (z>

I

as

-&-

under

rigid

motion and reparameterization. In Reference 13, an upper bound on the normal curvature of a surface S(u, v) is computed as the sum of the squares of the principal normal curvatures, K:(u,

in

Figure 3

Model

Uniform subdivision, Figure 6a

(Figure 3)

Time, s

Number of polygons

Time, s

Polygons

Table Swept surface Fuselage Teapot

Z.4

2544 768 1216 896

29 16 48 21

1992 732 1084 838

7.7 1.3

%I is invariant

J

Bilinear fit, Figure 66

54

Error bounded piecewise linear approximation: G Elber

U) and K,~(u, scalar field,

v)

and represented

(b.4, U) = (K,1(U,

as a piecewise rational

U)j2 + (K,f(U,u)j2

1 =--‘%A,

(13)

< --

K mm

One can symbolically compute t(u, v)13 and employ the convex hull property of the BCzier and NURBS upper bound normal curvature from t(u, v>. Denote by 2 an upper bound on dm 2 = fiI K: 1, 2 if K;=K ,’ and ,!j=max
2

2

= (C(Si+ Si))2K,,, 1

-

(14)

8

because

o2

O( 04) = case - 1 + 2

is

non-negative

throughout. Lemma 3 can be easily extended to the symmetric case for which K(S) < 0. If K(S) can assume both positive and negative values in the domain in question, two bounding circles may be used to bound both the convex and concave regions of C(s). Lemma 3 can be employed in the generation of a more optimal piecewise linear approximation of freeform parametric curves. Given C(t), I is an arbitrary regular parametrization, one can estimate 2

Lemma 3

(ctsi+l -si))2

Kmax

8

e2

K max

*

max UC(s) -_5$II 5 S,SS
+o(e4)

=-

“Let _Y = C(s.)C(s,+i> be a linear approximation of where s is the arclength curve CL), si’SsSsi+t, parameter of C (see 4). Assume C(S) is curvature continuous and let K(S) > 0, si 5 s s si+ 1 be the curvature field of C(S). Denote by K,,,~~ = max SiSSSS,,, K(S), the maximal curvature of C(s), si 4 s 5 si+ 1. As2 sume IIcCsi+1)- C(Si)ll< -

E=

1-T

‘LX

2

K;=o.

K mm

1

1

arc length at the neighbourhood

of C(t,> using ;.

Figure 5 shows an example of using Lemma

3, compared to the uniform in parametric space sampling case. We are now ready to establish a practical bound on the maximal possible deviation from S(u, v> of every line segments connecting two points on S(u, v). Let 9 be the largest diagonal of the boundin box of S(u, v>. B Assume S(u, v) is fairly flat, that is > > Sifl -si.

Proof Let %Tibe the circle of radius -

1 through C(s,) and

C(s,+ t) (see Figure 4). C(s) sy?L s sj+ 1 is bounded between gi and z simply because K,,,~~ 2 K(S) 2 0, where K,,,=~ and 0 are constant curvatures of gi and _%$respectively. Hence, the maximal deviation between C(s) and g is bounded from above by the maximal deviation between q and Pi, E=

<

K

max IIC(s) -_CSJI Si
be employed to provide an estimate on the ilowed length of the piecewise linear approximation, E, in the local neighbourhood of the surface. Finally, it is interesting to compare the result of Lemma 3 to the known result from approximation

ll,q --gI

=--

1 K max

max

Then, and because si+ 1 - si is difficult to compute, 9 can be employed to approximate si+ 1 - si. Further, 9% since t bounds K,,,~~ and following Lemma 3, can

1

-CDS8 K mm

I!3 (4

+d

Figure 5 An fl shaped cubic and a B shaped quadratic B-spline curves approximated using uniform sampling (a) and optimal sampling based on Lemma 3 (b). Both approximations carry the same number of samples. Original curve is shown in gray, piecewise linear approximation in black.

Figure 4 A bound on c is established as a function of the maximal curvature K,~~ of C(s) and the arc length si+ 1- si

55

Error bounded piecewise linear approximation: G Elber

of, (b -a>’

Ilf(t> astib

8

sup

Ilf”(t)ll

(15)

asrb

where 10) is a linear approximation of f(t) and recalling that f”(t) = a for arc length parameterization. Assume surface S(u, u) is a cylinder and hence parabolic. Let the lines of curvature7 be the isoparametric directions such that K,: = 0, K,” = K. Clearly, there is a need to subdivide S(u, v) only in u. [(u, v) = (K,:I(U, u))’ = K*, and is constant throughout. On the other hand, the diagonal, ~3, is reducing in size by subdividing either in u or in u, providing little clues as to the obviously preferred subdivision direction, u. It is unfortunate that there is no simple way to derive a preferred subdivision direction, either u or u, using only the information that is provided by .$(u, u>. One can consider symbolically computing and exploiting the normal curvatures in the isoparametric directions, (K: (u, u))* and (K,$u, u)>*, as well as the normal curvature along the diagonal u = u. Not only that all this symbolic manipulation is computationally intensive, but even with the aid of (K,U(U, u>)* and * it is insufficient. Consider the case where the surface is planar, but with a non-rectilinear boundary. In Reference 1, it is properly noted that the boundary of the surface should also be considered by applying flatness or linearity tests to all four boundaries. A given surface S(u, u) can be flat and hence K: = 0, i = 1, 2, yet S(u, u) must be subdivided due to its freeform shaped boundary. Thus, an additional test should examine the curves of the boundary of the approximated surface.

Figure 6 A biquadratic B-spline surface S(u, ~1) with it scalar curvature fields (K~(u, cl))* and (K& LJ))~. (K~(u, u)j2 is scaled by a factor of 0.001

One can attempt to subdivide a surface at the parameter values of its interior knot instead of at the centre of the parametric domain. The existence of knots suggests the existence of high resolution information in the surface and hence provides clues on the complexity of the shape. This simple heuristic yields surprisingly good results without curvature estimation computation. In Figure 6, the maximal normal curvatures in the isoparametric direction is found to be close to the interior knots’ parameter values, manifested as the discontinuities in the figure. In Figure 3, the heuristic of subdividing at interior knots was employed yielding good results throughout. CONCLUSION

EXTENSIONS We presented two approaches to bound the maximal deviation of piecewise linear approximations of freeform surfaces. We found that the use of curvature based polygonal approximation is difficult due to its significant order of ((u, u), lack of ability to derive the preferred subdivision direction and difficulty in determining termination conditions of subdivisions. The use of a bilinear fit is not only sufficient, but is also more efficient computationally, and with an ability to determine a preferred subdivision direction. Moreover, the bilinear fit can exploit (K:(u, u>)* and (K~(u, u))* to derive a more optimal subdivision direction, either u or u. The bilinear fit method can be combined with other methods presented in the literature. For example, the use of proximity and silhouette screen space consideration’ can be embedded into the bilinear tolerance directly or, alternatively, incorporated as a parallel test. One can attempt to globally approach the problem of piecewise linear approximation of freedom surfaces. Given a freeform surface S(u, u), one would like to compute the ‘best’ approximation of S(u, u) using n triangles. The greedy subdivision methods discussed herein consider the distribution of the triangles within a uniform isoparametric grid. As a result, a different surface parameterization will result in a different polygonal approximation. This artificial constraint should be eliminated. Moreover, current schemes, including the ones exploited herein, are local and greedy in their nature. By computing the entire global solution at a

In Algorithm 1 line cl), surface S(u, LJ)is subdivided at the middle of the parametric domain. If S(u, u) is not C’ continuous, a preprocessing subdivision stage must be applied to ensure C’ continuity. The motivation for such a preprocessing stage is obvious. The discontinuities of S(u, u) must be reflected in the polygonal approximation. This streamline also motivates an attempt to subdivide S(u, u> at locations of extreme curvature. Clearly, a subdivision of S(u, u) in u at the u location with the highest normal curvature in the tangent plane direction perpendicular to iso-u parametric direction can yield a more optimal polygonal approximation. The perpendicular to an iso-u parametric direction can be approximated using the iso-u parametric direction. Herein, the symbolic computation of ( K,U(U, u))’ and (K,U(U, u)>’ can be employed to convey the locations of the extreme values of the normal curvatures in the isoparametric directions and hence provide better subdivision locations. Figure 6 shows a surface S(u, u) with its (K~(u, u))* and (K,$, u))* SCdar CUIVatUre fields. As a biquadratic surface, S(u, u) is curvature discontinuous at interior knots as can be seen from the shape of (K,U(U, VI)*. Nonetheless, the three highly curved regions of S(u, u) along the triangular cross-section clearly show up in (K,U(U, u>>*. The use of these scalar curvature fields can improve the subdivision process, creating a more optimal polygonal approximation. 56

Error bounded piecewise linear approximation: G Elber

6

single stage, more optimal solutions can be expected. This open question can also be formulated as follows: given n vertices, position them in the parametric space of surface S(u, v> so that a triangulation over these vertices will be globally optimal, under some norm. This global approach has not been investigated, to the best knowledge of the author, and yet is of great interest.

7 8

9 10

ACKNOWLEDGEMENTS 11

The author is grateful to Craig Gotsman and the reviewers of this paper for their detailed valuable remarks.

12 13

REFERENCES 14 Herzen, B V and Barr, A H ‘Accurate triangulation of deformed. intersecting surfaces’ Cornour. Gmoh. Vol 21 No 4 . Siggraph (1987) Schmidt, M F W ‘Cutting cubes - visualizing implicit surfaces by adaptive polygonization’ I&. Comput. Vol 10 No 2 (1993) pp 101-115 Vlassopoulos, V ‘Adaptive polygonization of parametric surfaces’ Vis. Comout Vol 6 No 5 (1990) DD 291-298 Filip, D Magedson, R and Markot, R”‘Surface algorithms using bounds on derivatives’ Comput. Aided Geom. Des Vol 3 (1986) pp 295-311 Nadler, E ‘Piecewise linear best Lz approximation on triangulation’ in Chui, C K Schumarker, L L and Ward, J D (Eds.) Int. Symposium on Approximation Theory V Academic Press Inc. (1986) pp 499-502

Sheng, X and Hirsh, B E ‘Triangulation of trimmed surfaces in parametric space’ Comput.-Aided Des. Vol 24 No 8 (1992) pp 437-444 Carmo, M D Differential Geometry of Curtes and Surfaces Prentice-Hall (1976) Cohen, E Lyche, T and Schumaker, L ‘Algorithms for degree raising for splines’ ACM Trans. Graph. Vol 4 No 3 (1986) pp 171-181 Boehm, W ‘Inserting new knots into B-spline curves’ Comput.Aided Des. Vol 12 No 4 (19801 pp 199-201 Cohen, E Lyche, T and Riesenfeld, R ‘Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics’ Comput. Graph. Image Proc. Vol 14 (1980) pp 87-111 Elber, G ‘Free form surface analysis using a hybrid of symbolic and numeric computation’ PhD Thesis Computer Science Department, University of Utah Schaback, R ‘Error estimates for approximations from control nets’ Cornput. Aided Geom. Des. Voi 10 (1993) pp 57-66 Elber. G and Cohen, E ‘Second order surface analvsis using hybrid symbolic and numeric operators’ Trans. Graph. $01 12 No 2 (1993) pp 160-178 de Boor, C A Practical Guide to Splines Applied Mathematical Sciences 27, Springer, Berlin (1978)

Gershon Elber is a lecturer in the Computer Science Department, Technion, Ismel. His research interests span computer-aided geometric designs and computer graphics. Dr Elber receited a BS in computer engineering and an MS in computer science from the Technion, Ismel in 1986 and 1987, respectiwly, and a PhD in computer science from the Uniwrsity of Utah, USA in 1992. He is a member of the ACM and IEEE.

57