Optimal interpolation with spatial rational Pythagorean hodograph curves

Partial fraction decomposition for rational Pythagorean hodograph curves

“…In what follows, we are going to discuss and compare these three methods with each other and also with prior attempts to construct rational PH curves with rational arc length. Since all three methods can be viewed as generalizations of computation algorithms for rational PH curves [7,8,[14][15][16], the scope of this discussion extends to rational PH curves without the rational arc length constraint. Let's first talk about specifics of the rational arc length case.…”

“…In the past, is was solved by various methods, e.g. in [8,9,[14][15][16]. The purpose of the present paper is to extend these methods so that the both integrals (4) and ( 5) are rational.…”

“…The paper [16] presents a method to compute rational PH curves by directly studying the rationality of the integral (4) which can be insured by imposing "zero residue" conditions on the integrand. These are formulated as linear equations in terms of the partial fraction coefficients of the rational function λ(t) of Equations ( 4) and (5).…”

“…The first method adapts a recent approach of [14] for computing rational PH curves and requires solving a modestly sized and well structured system of linear equations. The second constructs the curves by imposing linear zero-residue conditions on the hodograph, thus extending ideas of the previous papers [16,21]. The third method generalizes the dual approach of [7] from planar to spatial curves.…”

“…A different approach of a more algebraic flavor was used in [13] to construct a particular kind of planar rational PH curves. All spatial rational PH curves (containing the planar ones) were algebraically constructed in [14,15] via solving a system of linear equations and in [16] by imposing zero residue conditions on hodograph. Whereas all polynomial PH curves have polynomial arc length functions, only a proper subset of the rational PH curves admits rational arc lengths.…”