Some insight into the properties of beta-splines is gained by applying the techniques of urn models. Urn models are used to construct beta-spline basis functions and to derive the basic properties of these blending functions and the corresponding beta-spline curves. Only the simple notion of linear geometric continuity and with the most elementary beta parameter are outlined. Non-linear geometric continuity leads to additional beta parameters and to more complicated basis functions. Whether urn models can give us any insight into these higher order concepts still remains to be investigated

Goldman, R. N.

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URN MODELS AND BETA-SPLINES 
R. N. Goldman 
Control Data Corporation 
Minneapolis, Minnesota 
1. Introduction 
A well-established connection exists between discrete urn models and the 
standard curves and surfaces used in computer-aided geometric design (CAGD) [1],[2], 
[3], [4], [5]. The ~&ier and B-spline blending functions both model elementary 
stochastic processes, and many of the geometric properties of ~6zier and B-spline 
curves and surfaces can be derived by studying these probabilistic models [4],[5]. 
Recently Barsky has introduced a new type of spline into CAGD called the 
beta-spline [6],[7],[8]. The purpose of this paper is to try to gain some insight 
into the properties of beta-splines by applying the techniques of urn models. 
2. Beta-Splines 
Beta-splines are generalizations of B-splines. They were developed in order 
to replace the somewhat artificial concept of parametric continuity by the more 
natural notion of geometric continuity. Brief1 the idea is this: Two curves L(t) 
to<t<tl, - - R(u) ug<u<ul - - are said to meet with ntk-order parametric continuity (Cn) 
if and only if 
Unfortunately this definition depends on more than just the geometry of the curves 
L(t),R(u); it also depends on the specific choice of their parametric 
representations. A linear change of parameter u=tv l>O will not change the shape 
of the curve R(u), but by the chain rule 
Thus R(pv) and L(t) do not meet with nth-order parametric continuity even 
though the curves R($v),R(u) are geometrically identical. To rectify this anomaly, 
the concept of geomekric continuity is introduced. 
Two curves L(t) ,R(u) are said to meet with linear nth-order geometric 
continuity (LGn) if and only if there exists a constantP>O such that 
https://ntrs.nasa.gov/search.jsp?R=19850020268 2020-03-22T19:36:35+00:00Z
It is easy to check that the notion of linear nth-order geometric continuity is 
invariant under linear changes of parameter. Of course, this concept is not 
invariant under non-linear changes of parameter. A more general notion of 
geometric continuity (Gn) and more general constraint equations invariant under 
non-linear changes of parameter are given in [8]. 
Splines have typically been defined in terms of parametric continuity, and the 
B-splines form a convenient basis for these parametric splines. The more general 
notion of geometric continuity requires us to search for a new set of basis 
functions suitable for these new types of sglines. These basis functions are 
called beta-splines. We shall now use urn models to construct beta-splines and 
study their properties. 
3 .  An Urn Model for Beta-Splines 
Consider an urn initially containing w white balls and b black balls. One 
ball at a time is drawn at random from the urn, its color inspected, and then 
returned to the urn. If the ball was the jth white ball to be chosen, then 
$j(+b) additional black balls are added to the urn; if the ball was the jth 
black ball to be chosen, then !-j(w+b) additional white balls are added to the 
urn. 
We now introduce the following notation: 
W t = --- = 
wtb probability of selecting a white ball on the first trial 
N N 
sj(t) = sj(p,t) = probability of selecting a white ball after selecting 
exactly j white balls in the first N trials 
N N fj(t) = fj((l,t) = probability of selecting a black ball after selecting 
exactly j white balls in the first N trials 
N N Bj(t) = Bj(p,t) = probability of selecting exactly j white balls in the 
first N trials 
For each fixed it can be shown that the functions ~!(t), ..., ~i(t) 
are linearly independent polynomials of degree N and they satisfy the constraint 
equations(*) 
k N d B.. k k N  P d B .  
The functions ~d(t), . . . ,~&t) are the beta-spline basis functions. If 
p=1, these functions are the uniform B-spline basis functions and the urn model is 
the standard urn model for B-splines [ I ] ,  [ 4 ] .  
Given a sequence of control points P=(PO, ..., PM), we can use these 
beta-spline basis functions as blending functions to construct LG~'~ continuous 
beta-spline curves in much the same way that we use the uniform B-s line basis 
functions to define cN-l continuous B-spline curves. Define the itR curve 
seguient by setting 
and define the beta-spline curve by setting 
,PI (t) = Bi[P ,PI (t) i - < t - < i+l 0 - < i - < M-N 
From the constraint equations (*) it follows immediately that 
Thus B[P ,P] (t ) is an L G ~ - ~  continuous beta-spline curve. 
Without moving the control points, we can alter the shape of the beta-spline 
curve B[l,P](t) simply by changing the scalar parameter g. The effect of 
increasing (3 is to move the curve closer to its control polygon and to.bias the 
curve towards its initial control points. Thus our Q corresponds to Barsky's bias 
parameter [7]. 
In table 1 we summarize those properties of beta-spline basis functions and 
beta-spline curves which are directly derivable from the beta-spline urn model. 
Many of these properties are new and are presented here for the first time. 
4. Conclusion 
Urn models can be used to construct beta-spline basis functions and to derive 
the basic properties of these blending functions and the corresponding beta-spline 
curves. This is only the beginning; much work remains to be done. Here we have 
dealt only with the simple notion of linear geometric continuity and with the most 
elementary beta parameter. Non-linear geometric continuity leads to additional 
beta parameters and to more complicated basis functions [a]. Whether urn models 
can give us any insight into these higher order concepts still remains to be 
investigated. 
References 
1. Goldman, R. N., An Urnful of Blending Functions, IEEE Computer Graphics and 
Applications, vol. 3, no. 7, 1983, pp. 49-54. 
2. Goldman, R. N., An Intuitive Approach to Bgzier and Other Random Curves and 
Surfaces, SIGGRAPH Tutorial on Freeform Curves and Surfaces, 1983. 
3. Goldman, R. N., Geometry and Probability, SIGGRAPH Tutorial on Freeform 
Curves and Surfaces, 1984. 
4. Goldman, R. N., Polya's Urn Model and Computer Aided Geometric Design, SIAM 
Journ. on Algebraic and Discrete'Methods, vol. 6, no. 1, 1985, pp. 1-28. 
5. Goldman, R. N., Urn Models, Approximations, and Splines, Submitted to the 
Journal of Approximation Theory. 
6. Barsky, B. A., The Beta-Spline: A Local Representation Based on Shape 
Parameters and Fundamental Geometric Measures, Ph.D. Thesis, University of 
Utah, Salt Lake City, Utah, 1981. 
7. Barsky, B. A. and Beatty, J. C., Local Control of Bias and Tension in 
Beta-Splines, ACM Trans. on Graphics, vol. 2, no. 2, 1982, pp. 109-134. 
8. Barsky, B. A. and DeRose, T. D., Geometric Continuity of Parametric Curves, 
Tech. Rept. No. UCB/CSD 841205, Computer Science Division University of 
California, Berkeley, October 1984. 
TABLE 1 - PROPERTIES OF BETA-SPLINE BASIS FUNCTIONS AND CURVES 
Urn 
-
Basis Functions Curve 
1. Probability ==> Z ~fl(t) 3 = 1 ==> Coordinate Free 
Distribution !! 
" 
==> BN(t) , 0 O<t(l j ==> Local Convex Hull Property 
2. Symmetry between ==> Symmetry Formula ==> Curve Symmetry 
white, p and 
black, p -' f )  = B ~ , -  B[P  PI(^) = B [ ~ - ~ , P ~ ~ ~ ~ ~ ~ ~ ]  (~- +l-t) 
3. Counting ==> Explicit Formulas ==> Recursion Formula 
N- j ==> Locality of End Points 
==> Explicit Recursion Formula 
N- j 
t +G (see 4) 
=> Pi does not affect 
j=l dkBi[(I,p] ---- - -- - k=O, 1,. . . , N-1 
tN 
==> Pi+N does not affect 
j=l --- dkBi[p,p] - -- -- k=0.1,. . . .N-1 
4. Relationship ==> Recursion Formula ==> Geometric Construction 
between first BN+l(t) = f;(t)B;(t) Algorithm 
N and first 3 Let i<tli+l 
N+l picks 
Pq(t)*~~+~ J O<j<N 
Then B[P ,PI (t)=P;(t) 
5. Recursion Formula ==> Polynomial Functions ==> Polynomial Spline 
==> Differentiability ==) LGN-l 
Constraints 
6. Linear Inde~endence ==> Linear Independence ==> Locallv Non-Deeenerate 
of first N+i - 
rmoments ==> Polynomial Basis ==> Local Subdivision Algorithm 
7. Limitine Conditions Limits Tension/Bias 
1f p = -  at most 1 =-> Lim Bfl(t) - ( ~ - t ) ~  j=O ==> Lim B[(),P](~) is the 
white ball can be I)->- J 
selected. = 1-(1-~)~ j=l $:::On determined by 
= o jH,l po.....pM-wl. 
~f a-0 at most 1 ==> Lim BN(t) = tN j=N =-> Lim BlP,PI(t) is the 
black ball can be p ->o P->O = l-tN jq-1 
selected polygon determined by 
= 0 j*N-1, N pNL1,.-* ,pM- 
8 .  Adding only balls ==> Total Positivity ==> Variation Diminishing 
of the opposite Property 
color 
9. Two urns each with =-> Independent =-> Rectangular Tensor 
two colors Distributions Product Surfaces 
~y(s,t) = B:(s)B;(~) 


Urn Models and Beta-splines

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