an:04157750
Zbl 0705.65008
Ramshaw, Lyle
Blossoms are polar forms
EN
Comput. Aided Geom. Des. 6, No. 4, 323-358 (1989).
00172559
1989
j
65D07 41A15
affine interpolation; B-spline; B??zier point; blossoming; de Boor algorithm; de Casteljau algorithm; dual functional; homogeneity; multiaffine function; polar form; quasi-interpolant; spline reproductivity; tensor
Summary: Consider the functions \(H(t):=t^ 2\) and \(h(u,v):=uv\). The identity \(H(t)=h(t,t)\) shows that H is the restriction of h to the diagonal \(u=v\) in the uv-plane. Yet, in many ways, a bilinear function like h is simpler than a homogeneous quadratic function like H. More generally, if F(t) is some n-ic polynomial function, it is often helpful to study the polar form of F, which is the unique symmetric, multiaffine function \(f(u_ 1,...,u_ n)\) satisfying the identity \(F(t)=f(t,...,t)\). The mathematical theory underlying splines is one area where polar forms can be particularly helpful, because two pieces F and G of an n-ic spline meet at a point r with \(C^ k\) parametric continuity if and only if their polar forms f and g agree on all sequences of n arguments that contain at least n-k copies of r.
This polar approach to the theory of splines emerged in rather different guises in three independent research efforts: Paul de Faget de Casteljau called it `shapes through poles'; Carl de Boor called it `B-splines without divided differences'; and the author called it `blossoming'. This paper reviews the work of de Casteljau, de Boor, and the author in an attempt to clarify the basic principles that underly the polar approach. It also proposes a consistent system of nomenclature as a possible standard.