Blossoming off the diagonal

Generalizations of the Bezier and the B-spline schemes are investigated using the technique of blossoming.; The Bernstein basis functions {dollar}{lcub}Bsbsp{lcub}k{rcub}{lcub}n{rcub}(t){rcub}{dollar} can be generated by taking the discrete convolution of the sequence {dollar}langle(1 - t),trangle{dollar} with itself {dollar}n - 1{dollar} times. The Convolution blending functions {dollar}{lcub}Csbsp{lcub}k{rcub}{lcub}n{rcub}(t){rcub}{dollar} are generated by replacing the parameter t with an arbitrary linear function {dollar}Xsb{lcub}j{rcub}(t)=asb{lcub}j{rcub}t+bsb{lcub}j{rcub}{dollar} in each factor of the discrete convolution. Equivalently, {dollar}Csbsp{lcub}k{rcub}{lcub}n{rcub}(t){dollar} is the blossom of {dollar}Bsbsp{lcub}k{rcub}{lcub}n{rcub}(t){dollar} evaluated off the diagonal at the linear functions {dollar}Xsb1(t),...,Xsb{lcub}n{rcub}(t).{dollar} The polynomials {dollar}Csbsp{lcub}k{rcub}{lcub}n{rcub}(t){dollar} form a basis for the space of polynomials of degree n or less if and only if the elementary symmetric functions {dollar}ssb{lcub}j{rcub}(asb1,...,asb{lcub}n{rcub})ne 0{dollar} for {dollar}j=1,...,n.{dollar} We investigate how this Convolution basis extends many of the properties of the Bernstein basis, and we show that as generalizations of Bezier curves, the Convolution curves {dollar}sumsb{lcub}k{rcub}Psb{lcub}k{rcub}Csbsp{lcub}k{rcub}{lcub}n{rcub}(t){dollar} have many interesting and desirable properties for computer aided geometric design (CAGD).; To obtain even more general schemes, we apply the blossoming-and-evaluating-off-the-diagonal paradigm to B-splines. These Blossomed B-splines (BB-splines) subsume the continuous B-splines, the discrete B-splines, the Convolution basis functions and the Bernstein basis functions. Locally (per segment) the BB-splines of degree n form a basis for the space of polynomials of degree n or less under the same conditions as the Convolution basis functions. We explore how this BB-spline scheme extends many of the properties of the B-spline scheme. BB-spline curves also exhibit some interesting new continuity phenomena because each knot is now associated with several values called snarls.; As an immediate ramification of these generalizations, we obtain various extensions of Marsden's Identity. From these generalized identities, we derive transformations between different well-known representations for curves in CAGD. These basis transformations are fundamental for CAGD since many problems in CAGD can be reduced to finding transformations between different polynomial or spline bases.; We also investigate two bivariate schemes where the parameter domains are triangles. Under some mild restrictions, the bivariate Convolution polynomials of degree n, which can be obtaining by blossoming the bivariate Bernstein basis functions of degree n and evaluating off the diagonal, form a basis for the space of bivariate polynomials of total degree n or less. Analogous results hold for the BB-weights which are the blossoms of the B-weights evaluated off the diagonal.