Applications of the polynomial s-power basis in geometry processing

We propose a unified methodology to tackle geometry processing operations a dmitting explicit algebraic expressions. This new approach is based on repr esenting and manipulating polynomials algebraically in a recently presented 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 is an element of [0, 1], degree reduction i s trivially obtained by truncation, which yields the Hermite interpolant th at matches the original derivatives at u = {0, 1}. Operations such as divis ion or square root become meaningful and amenable in this basis, since we c an compute as many terms as desired of the corresponding Hermite interpolan t and build "s-power series," akin to Taylor series. Applications include c omputing integral approximations of rational polynomials, or approximations of offset curves.