Recursive Triangles

Abstract

In approximation theory and computer aided geometric design many polynomial and piecewise polynomial curve schemes can be generated from simple triangular recursive evaluation algorithms. Most prominent are de Casteljau’s algorithm for Bezier curves [24], de Boor’s algorithm for B-splines [15], and Neville’s algorithm for Lagrange polynomials [19], but similar recursive evaluation algorithms are known for Catmull-Rom splines [4], for convolution curves [1], and for curves defined in terms of the normalized power basis [10], the Newton basis [9], or even the standard monomial basis [10].

These algorithms all have the same basic triangular structure. Points (or vectors or scalars) are placed at the base of a triangular array and combinations of adjacent points (or vectors or scalars), two points at a time, are calculated to generate the next level in the array. These algorithms continue in this fashion until a single value emerges at the apex of the triangle. This value depends linearly on the input points, which are called control points; the coefficients of the control points are called blending functions. The combinations generally depend on some scalar parameter t, and the point that emerges from the apex of the triangle is the point on the parametric curve at the parameter value t. The functions which define the combinations are linear in t, and therefore for triangles of depth N the curve which is generated is a polynomial or piecewise polynomial of degree N in t. Usually die combinations are affine, so generally these curves are affine invariant and their blending functions are partitions of unity.

We call such algorithms recursive triangles. We are going to study these recursive schemes in some detail. Many examples will be provided, and various interrelationships between these different triangular schemes will be highlighted. Connections between recursive curve schemes, discrete probability distributions, and simple stochastic models will be underscored. Change of basis algorithms based on recursive triangles will also be explored.

In Section 1 we introduce recursive triangles with two fundamental examples: Pascal’s triangle for the binomial coefficients and de Casteljau’s algorithm for Bezier curves. Pascal’s triangle for the binomial coefficients is the easiest and best known example of a recursive triangle, and de Casteljau’s algorithm for Bezier curves is the simplest and most basic recursive curve scheme in computer aided geometric design. General triangular recursive curve schemes are then presented and their common properties are recorded. The relationship between these triangular recursive schemes and simple stochastic models is reviewed with special attention given to random walks and urn models. A multiprocessor hardware for fast computation of points along these recursive curves is also briefly discussed.

The two most important general classes of recursive curve schemes are Polya polynomials and B-spline curves. In Section 2 we investigate these two general schemes and tabulate their properties. By broadening the definition of these schemes, we include the Lagrange basis, the Newton basis, the power basis, and the monomial basis. The Polya and B-spline schemes have many dual properties, and we examine this duality and it ramifications in some detail in Section 3. Dual functionals, blossoming, change of basis methods, and knot insertion techniques are our main focus here.

In Section 4 we introduce a few lesser known recursive triangular curve schemes of possible interest in computer aided geometric design by concatenating some simpler or better known schemes. In particular, this approach is used to study Catmull-Rom splines and convolution curves. In Section 5 we very briefly examine recursive surface schemes. We close in Section 6 with a summary of our main results, a few conclusions, and some open questions for possible future investigations.