Range functions are a fundamental tool for certified computations in geometric modelling, computer graphics, and robotics, but traditional range functions have only quadratic convergence order ( ). For “superior” convergence order (i.e., ), we exploit the Cornelius–Lohner framework in order to introduce new bivariate range functions based on Taylor, Lagrange, and Hermite interpolation. In particular, we focus on practical range functions with cubic and quartic convergence order. We implemented them in Julia and provide experimental validation of their performance in terms of efficiency and efficacy. • Classical bivariate range functions have only quadratic convergence order. • We derive bivariate range functions with cubic and quartic convergence order. • The theoretically proven convergence orders are validated by numerical examples.

We propose a new algorithm to compute the topology of a real algebraic space curve . The novelties of this algorithm are a new technique to achieve the lifting step which recovers points of the space curve in each plane fiber from several projections and a weaker notion of generic position. As distinct to previous work, our sweep generic position does not require that x -critical points have different x -coordinates. The complexity of achieving this sweep generic position property is thus no longer a bottleneck in term of complexity. The bit complexity of our algorithm is O ˜ ( d 18 + d 17 τ ) where d and τ bound the degree and the bitsize of the integer coefficients, respectively, of the defining polynomials of the curve and polylogarithmic factors are ignored. To the best of our knowledge, this improves upon the best currently known results at least by a factor of d 2 .

Computing the intersection between two parametric surfaces (SSI) is one of the most fundamental problems in geometric and solid modeling. Maintaining the SSI topology is critical to its computation robustness. We propose a topology-driven hybrid symbolic-numeric framework to approximate rational parametric surface-surface intersection (SSI) based on a concept of interval algebraic topology analysis (IATA) , which configures within a 4D interval box the SSI topology. We map the SSI topology to an algebraic system's solutions within the framework, classify and enumerate all topological cases as a mixture of four fundamental cases (or their specific sub-cases). Various complicated topological situations are covered, such as cusp points or curves, tangent points (isolated or not) or curves, tiny loops, self-intersections, or their mixtures. The theoretical formulation is also implemented numerically using advanced real solution isolation techniques, and computed within a topology-driven framework which maximally utilizes the advantages of the topology maintenance of algebraic analysis, the robustness of iterative subdivision, and the efficiency of forward marching. The approach demonstrates improved robustness under benchmark topological cases when compared with available open-source and commercial solutions, including IRIT, SISL, and Parasolid.