Bivariate range functions with superior convergence order
This work provides a practical improvement for certified computations in geometric modeling, computer graphics, and robotics, but is incremental as it extends existing frameworks.
The authors introduce new bivariate range functions with cubic and quartic convergence order (m>2), improving upon traditional quadratic convergence (m=2). Experimental validation in Julia demonstrates their efficiency and efficacy.
Range functions are a fundamental tool for certified computations in geometric modeling, computer graphics, and robotics, but traditional range functions have only quadratic convergence order ($m=2$). For ``superior'' convergence order (i.e., $m>2$), 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.