Applications of the polynomial s-power basis in geometry processing

Abstract

We propose a unified methodology to tackle geometry processing operations admitting explicit algebraic expressions. This new approach is based on representing and manipulating polynomials algebraically in a recently basis, the symmetric analogue of the power form (s-power basis for brevity), so called because it is associated with a “Hermite two-point expansion” instead of a Taylor expansion. Given the expression of a polynomial in this basis over the unit interval u ε[0, 1], degree reduction is trivally obtained by truncation, which yields the He many terms as desired of the corresponding Hermite interpolant and build “s-power series,” akin to Taylor series. Applications include computing integral approximations of rational polynomials, or approximations of offset curves.