Blossoms are polar forms

Abstract

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. The polar approach to the theory of splines emerged in rather different guises in three independent research efforts: Paul de Faget Casteljau called it ‘shapes through poles’; Carl de Boor called it ‘B-splines without divided differences’; and Lyle Ramshaw called it ‘blossoming’. This paper reviews the work of de Casteljau, de Boor, and Ramshaw 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.